WEBVTT 00:00:00.000 --> 00:00:02.475 align:middle line:90% [SQUEAKING] 00:00:02.475 --> 00:00:03.960 align:middle line:90% [RUSTLING] 00:00:03.960 --> 00:00:14.180 align:middle line:90% [CLICKING] 00:00:14.180 --> 00:00:15.760 align:middle line:90% PETER SHOR: OK. 00:00:15.760 --> 00:00:16.343 align:middle line:90% recall. 00:00:16.343 --> 00:00:17.760 align:middle line:84% So of course, I'm going to go over 00:00:17.760 --> 00:00:20.200 align:middle line:90% a few things from last time. 00:00:20.200 --> 00:00:33.900 align:middle line:84% Suppose we have a sub i things of size i. 00:00:33.900 --> 00:00:39.840 align:middle line:90% 00:00:39.840 --> 00:00:42.960 align:middle line:90% So these could be-- 00:00:42.960 --> 00:00:47.600 align:middle line:84% well, I mean, they could be Dyck walks, in which case, a sub i 00:00:47.600 --> 00:00:49.640 align:middle line:90% is the Catalan number. 00:00:49.640 --> 00:01:00.400 align:middle line:84% And the generating function is summation a sub i x. 00:01:00.400 --> 00:01:01.560 align:middle line:90% OK. 00:01:01.560 --> 00:01:04.180 align:middle line:84% The notes generally use n for this, 00:01:04.180 --> 00:01:07.640 align:middle line:84% so I'm going to try to do that, but I'm probably 00:01:07.640 --> 00:01:11.920 align:middle line:90% going to not be consistent. 00:01:11.920 --> 00:01:13.720 align:middle line:84% The generating function is a of x. 00:01:13.720 --> 00:01:17.560 align:middle line:90% Summation, a sub n x to the n. 00:01:17.560 --> 00:01:20.580 align:middle line:84% n equals-- well, let's say 0 to infinity. 00:01:20.580 --> 00:01:26.360 align:middle line:84% But, I mean, maybe we want to start 00:01:26.360 --> 00:01:31.040 align:middle line:90% counting at 1 in certain cases. 00:01:31.040 --> 00:01:32.680 align:middle line:90% OK. 00:01:32.680 --> 00:01:52.640 align:middle line:84% And suppose we have two classes of things, a sub i and b sub i. 00:01:52.640 --> 00:01:55.200 align:middle line:84% Make that a sub n and b sub n, and c 00:01:55.200 --> 00:02:04.410 align:middle line:84% sub n equals a sub n plus b sub n, where c is the union of A 00:02:04.410 --> 00:02:09.990 align:middle line:90% and B. Or the disjoint union. 00:02:09.990 --> 00:02:17.930 align:middle line:84% Then c of x is equal to A of x plus B of x. 00:02:17.930 --> 00:02:27.150 align:middle line:90% 00:02:27.150 --> 00:02:38.630 align:middle line:84% And suppose c equals the direct product of A and B, 00:02:38.630 --> 00:02:40.250 align:middle line:90% where the size of c-- 00:02:40.250 --> 00:02:43.630 align:middle line:90% 00:02:43.630 --> 00:02:51.230 align:middle line:84% so c is a, comma, b, something from a and something from b. 00:02:51.230 --> 00:02:58.410 align:middle line:84% So the size of c is equal to the size of a plus size of b. 00:02:58.410 --> 00:03:03.070 align:middle line:90% 00:03:03.070 --> 00:03:11.150 align:middle line:84% Then c sub n equals summation i equals 0 to n, a sub i, 00:03:11.150 --> 00:03:17.750 align:middle line:90% b sub n minus i. 00:03:17.750 --> 00:03:23.150 align:middle line:84% Because if you want two things to sum to n, one should be i 00:03:23.150 --> 00:03:25.350 align:middle line:84% and the other should be n minus i. 00:03:25.350 --> 00:03:34.770 align:middle line:84% And c of x is equal to A of x times B of x. 00:03:34.770 --> 00:03:39.630 align:middle line:90% 00:03:39.630 --> 00:03:44.550 align:middle line:84% And there's one last way of manipulating 00:03:44.550 --> 00:03:50.275 align:middle line:84% generating functions that I did not discuss last time. 00:03:50.275 --> 00:03:51.650 align:middle line:84% Actually, there's more than that. 00:03:51.650 --> 00:03:54.430 align:middle line:84% I mean, you can differentiate them and integrate them. 00:03:54.430 --> 00:03:57.070 align:middle line:84% But I gave you an example of that last time. 00:03:57.070 --> 00:04:24.340 align:middle line:84% But let's let sequence of A be the set of finite sequences 00:04:24.340 --> 00:04:30.670 align:middle line:90% of things in A. 00:04:30.670 --> 00:04:36.270 align:middle line:84% So let's do an example that will illustrate this. 00:04:36.270 --> 00:04:47.010 align:middle line:90% Suppose A is equal to 0 and 1. 00:04:47.010 --> 00:04:53.540 align:middle line:90% 00:04:53.540 --> 00:05:00.470 align:middle line:84% Sequence of A is equal to binary strings. 00:05:00.470 --> 00:05:06.560 align:middle line:90% 00:05:06.560 --> 00:05:07.060 align:middle line:90% Oops. 00:05:07.060 --> 00:05:11.100 align:middle line:90% 00:05:11.100 --> 00:05:17.970 align:middle line:90% 0, 0 is in A cubed, for example. 00:05:17.970 --> 00:05:20.740 align:middle line:90% 00:05:20.740 --> 00:05:23.440 align:middle line:84% So what is the generating function for sequence of A? 00:05:23.440 --> 00:05:57.260 align:middle line:90% 00:05:57.260 --> 00:06:19.820 align:middle line:84% I want to claim S of x is equal to sum from i equals 0, 00:06:19.820 --> 00:06:24.060 align:middle line:90% n equals 0 to infinity. 00:06:24.060 --> 00:06:33.880 align:middle line:90% A. OK. 00:06:33.880 --> 00:06:36.420 align:middle line:90% Let's write it out first. 00:06:36.420 --> 00:06:48.180 align:middle line:84% Is equal to 1 plus A of x plus A of x squared plus A of x cubed. 00:06:48.180 --> 00:06:52.780 align:middle line:90% So this is the empty sequence. 00:06:52.780 --> 00:06:55.526 align:middle line:90% This is 0, 1. 00:06:55.526 --> 00:06:58.820 align:middle line:90% 00:06:58.820 --> 00:07:05.440 align:middle line:84% This would be 0, 0, 0, 1, 1, 0, 1, 1 for binary sequences, 00:07:05.440 --> 00:07:06.330 align:middle line:90% et cetera. 00:07:06.330 --> 00:07:11.700 align:middle line:90% 00:07:11.700 --> 00:07:22.380 align:middle line:84% And A of x equals 2x for binary sequences, because here, 00:07:22.380 --> 00:07:32.020 align:middle line:84% there are two binary sequences of length 1 and x means length 1 00:07:32.020 --> 00:07:37.020 align:middle line:84% here, and 2 means there are two of them. 00:07:37.020 --> 00:07:48.530 align:middle line:84% So S of x is equal to 1 plus 2x plus 4x squared plus 8x cubed 00:07:48.530 --> 00:07:54.490 align:middle line:84% plus dot, dot, dot, which you probably all recognize as 1 over 00:07:54.490 --> 00:07:55.676 align:middle line:90% 1 minus 2x. 00:07:55.676 --> 00:07:59.770 align:middle line:90% 00:07:59.770 --> 00:08:06.110 align:middle line:84% We can go up here and say this equals 1 over 1 minus A of x. 00:08:06.110 --> 00:08:16.170 align:middle line:90% 00:08:16.170 --> 00:08:18.130 align:middle line:84% I guess this calculation is not very 00:08:18.130 --> 00:08:22.330 align:middle line:84% impressive for binary strings because everybody 00:08:22.330 --> 00:08:27.510 align:middle line:84% knows how to come up with this in much easier ways. 00:08:27.510 --> 00:08:29.010 align:middle line:84% Next what we're going to do is we're 00:08:29.010 --> 00:08:31.370 align:middle line:90% going to do Fibonacci numbers. 00:08:31.370 --> 00:08:38.230 align:middle line:84% And Fibonacci numbers are a little bit harder, 00:08:38.230 --> 00:08:41.169 align:middle line:84% and we will see that the generating function actually 00:08:41.169 --> 00:08:44.240 align:middle line:84% gives you interesting things for Fibonacci numbers. 00:08:44.240 --> 00:08:52.650 align:middle line:90% 00:08:52.650 --> 00:08:54.830 align:middle line:90% OK. 00:08:54.830 --> 00:09:03.530 align:middle line:84% So Let's start with another question. 00:09:03.530 --> 00:09:12.370 align:middle line:84% Suppose we want to send information over a-- 00:09:12.370 --> 00:09:32.010 align:middle line:84% we want to send information over a telegraph type channel 00:09:32.010 --> 00:09:39.650 align:middle line:84% where we can send a dot and a dash. 00:09:39.650 --> 00:09:43.450 align:middle line:84% And the question is, how much information or how many messages 00:09:43.450 --> 00:09:44.140 align:middle line:90% can we send? 00:09:44.140 --> 00:09:53.610 align:middle line:90% 00:09:53.610 --> 00:10:01.450 align:middle line:90% Can we send n time n. 00:10:01.450 --> 00:10:07.570 align:middle line:84% And I forgot to tell you, a dot has length 1 00:10:07.570 --> 00:10:09.510 align:middle line:90% and a dash has length 2. 00:10:09.510 --> 00:10:19.250 align:middle line:90% 00:10:19.250 --> 00:10:21.050 align:middle line:90% OK. 00:10:21.050 --> 00:10:27.210 align:middle line:90% Let's start calculating it. 00:10:27.210 --> 00:10:30.855 align:middle line:90% 00:10:30.855 --> 00:10:35.270 align:middle line:84% With time 1, All we can do is send a dot. 00:10:35.270 --> 00:10:38.490 align:middle line:90% So that's not really a message. 00:10:38.490 --> 00:10:40.770 align:middle line:84% I mean, if you can only send one message, 00:10:40.770 --> 00:10:43.290 align:middle line:90% it contains no information. 00:10:43.290 --> 00:10:46.290 align:middle line:84% At time 2, we could either send a dash or two dots. 00:10:46.290 --> 00:10:49.760 align:middle line:90% 00:10:49.760 --> 00:10:52.800 align:middle line:84% At time three, so that's two possibilities. 00:10:52.800 --> 00:10:53.800 align:middle line:90% This is one possibility. 00:10:53.800 --> 00:10:57.560 align:middle line:84% In time 3, we can either send a dash and a dot, 00:10:57.560 --> 00:11:00.240 align:middle line:84% or a dot and a dash, or three dots. 00:11:00.240 --> 00:11:02.040 align:middle line:90% That's three. 00:11:02.040 --> 00:11:10.060 align:middle line:84% In time 4, we send dot, dot, dot, or dash, dash, dash, dot, 00:11:10.060 --> 00:11:16.000 align:middle line:84% dot, dot, dot, dash, and dot, dot, dot, dot. 00:11:16.000 --> 00:11:17.300 align:middle line:90% So that's five messages. 00:11:17.300 --> 00:11:18.970 align:middle line:84% Does anybody recognize this sequence? 00:11:18.970 --> 00:11:23.460 align:middle line:90% 00:11:23.460 --> 00:11:23.960 align:middle line:90% OK. 00:11:23.960 --> 00:11:27.160 align:middle line:90% Let's do the next one. 00:11:27.160 --> 00:11:32.840 align:middle line:84% Well, there are two dashes and a dot, or dot and two dashes, 00:11:32.840 --> 00:11:35.520 align:middle line:90% or dash, dot, dash. 00:11:35.520 --> 00:11:43.300 align:middle line:84% And we could dash and three dots, dot, dash, 00:11:43.300 --> 00:11:47.800 align:middle line:84% dot, dot dot, dot, dot, dash, dot, dot, dot, dot, dash, 00:11:47.800 --> 00:11:52.600 align:middle line:84% and dot, dot, dot, dot, dot, dot. 00:11:52.600 --> 00:11:53.990 align:middle line:84% So there's eight of these things. 00:11:53.990 --> 00:11:56.560 align:middle line:90% 00:11:56.560 --> 00:11:57.240 align:middle line:90% OK. 00:11:57.240 --> 00:12:01.000 align:middle line:84% Do people recognize the sequence now? 00:12:01.000 --> 00:12:02.960 align:middle line:90% What is it? 00:12:02.960 --> 00:12:04.060 align:middle line:90% Fibonacci numbers. 00:12:04.060 --> 00:12:04.720 align:middle line:90% Yeah. 00:12:04.720 --> 00:12:06.840 align:middle line:90% So we get the Fibonacci numbers. 00:12:06.840 --> 00:12:08.798 align:middle line:84% And actually, there's a very easy way 00:12:08.798 --> 00:12:10.590 align:middle line:84% to prove that we get the Fibonacci numbers. 00:12:10.590 --> 00:12:16.520 align:middle line:90% 00:12:16.520 --> 00:12:21.640 align:middle line:84% I want to claim f sub n equals f sub n minus 1 00:12:21.640 --> 00:12:25.120 align:middle line:90% plus f sub n minus 2. 00:12:25.120 --> 00:12:34.960 align:middle line:84% So this is if something ends with a dot. 00:12:34.960 --> 00:12:38.640 align:middle line:84% So sequence of length n is either 00:12:38.640 --> 00:12:40.200 align:middle line:84% going to end with a dot or a dash. 00:12:40.200 --> 00:12:42.240 align:middle line:84% If it ends with a dot, then there 00:12:42.240 --> 00:12:47.920 align:middle line:84% are fn minus 1 ways of filling the first n minus 1 times units. 00:12:47.920 --> 00:12:50.240 align:middle line:84% If it ends with a dash, there are 00:12:50.240 --> 00:12:56.760 align:middle line:84% fn minus 2 ways of filling the first n minus 2 times. 00:12:56.760 --> 00:13:00.900 align:middle line:84% So you get this recurrence for the Fibonacci numbers, 00:13:00.900 --> 00:13:05.120 align:middle line:84% which I'm sure everybody already knows about. 00:13:05.120 --> 00:13:07.600 align:middle line:84% So what we're going to do is we're going to ask, 00:13:07.600 --> 00:13:09.560 align:middle line:84% how can we use generating functions 00:13:09.560 --> 00:13:12.490 align:middle line:84% to find a formula for the i-th Fibonacci number? 00:13:12.490 --> 00:13:57.190 align:middle line:90% 00:13:57.190 --> 00:14:01.910 align:middle line:90% And I wanted to say something. 00:14:01.910 --> 00:14:05.270 align:middle line:84% Back when our lecture notes were first written, 00:14:05.270 --> 00:14:12.106 align:middle line:90% I used f1 equals 1, f0. 00:14:12.106 --> 00:14:14.950 align:middle line:90% 00:14:14.950 --> 00:14:21.030 align:middle line:84% There's one way of sending message of like 0, 00:14:21.030 --> 00:14:23.190 align:middle line:90% which is the empty message. 00:14:23.190 --> 00:14:26.750 align:middle line:90% So f0 equals 1, f1 equals 1. 00:14:26.750 --> 00:14:31.670 align:middle line:84% f2 equals 2, f3 equals 3, f4 equals 5, et cetera. 00:14:31.670 --> 00:14:50.430 align:middle line:84% So since then, Wikipedia has installed itself 00:14:50.430 --> 00:15:00.350 align:middle line:84% as the default arbiter of definitions, et cetera. 00:15:00.350 --> 00:15:06.170 align:middle line:84% And today, Wikipedia says f0 equals 0, f1 equals 1, 00:15:06.170 --> 00:15:12.270 align:middle line:84% f2 equals 1, f3 equals 2, and everything is shifted by one. 00:15:12.270 --> 00:15:15.070 align:middle line:84% And I have not had time to rewrite the notes. 00:15:15.070 --> 00:15:19.430 align:middle line:84% So the notes disagree with Wikipedia's definition 00:15:19.430 --> 00:15:26.710 align:middle line:84% of Fibonacci numbers, and today, Wikipedia basically 00:15:26.710 --> 00:15:33.230 align:middle line:84% is the default definition for everything, 00:15:33.230 --> 00:15:37.382 align:middle line:84% even of some references disagree. 00:15:37.382 --> 00:15:37.882 align:middle line:90% OK. 00:15:37.882 --> 00:15:43.070 align:middle line:90% 00:15:43.070 --> 00:15:47.610 align:middle line:84% And it's much easier to do this calculation. 00:15:47.610 --> 00:15:49.610 align:middle line:84% I think it's much easier to do this calculation. 00:15:49.610 --> 00:15:51.750 align:middle line:84% It's a little easier to do this calculation. 00:15:51.750 --> 00:15:55.270 align:middle line:84% And we use the alternate definition rather than 00:15:55.270 --> 00:15:56.750 align:middle line:90% Wikipedia's definition. 00:15:56.750 --> 00:16:03.670 align:middle line:84% But let's write down the generating function for f of x 00:16:03.670 --> 00:16:12.950 align:middle line:84% of x equals summation f of x summation the i-th the induction 00:16:12.950 --> 00:16:15.270 align:middle line:90% number times x to the-- 00:16:15.270 --> 00:16:17.630 align:middle line:84% n-th Fibonacci number times x to the n, 00:16:17.630 --> 00:16:27.990 align:middle line:84% which is equal to 1 plus x plus x squared plus 3x cubed plus 5x 00:16:27.990 --> 00:16:33.150 align:middle line:84% to the fourth, plus 8x to the fifth, plus 13x to the sixth, 00:16:33.150 --> 00:16:35.140 align:middle line:90% dot, dot, dot. 00:16:35.140 --> 00:16:36.700 align:middle line:90% OK. 00:16:36.700 --> 00:16:40.480 align:middle line:90% So how do we compute the value? 00:16:40.480 --> 00:16:43.480 align:middle line:84% How do we find a closed form for f of x? 00:16:43.480 --> 00:16:46.700 align:middle line:90% 00:16:46.700 --> 00:16:58.140 align:middle line:84% Well, let's take f of x equals x plus x squared plus 2x cubed 00:16:58.140 --> 00:17:03.740 align:middle line:84% plus 3x cubed plus 5x to the third, 00:17:03.740 --> 00:17:14.660 align:middle line:84% and x squared f of x equals x squared plus x cubed plus 2x 00:17:14.660 --> 00:17:19.359 align:middle line:84% fourth plus 3x to the fourth plus dot, dot, dot. 00:17:19.359 --> 00:17:22.300 align:middle line:90% 00:17:22.300 --> 00:17:24.525 align:middle line:90% OK. 00:17:24.525 --> 00:17:27.160 align:middle line:84% So now we have these three formulas. 00:17:27.160 --> 00:17:31.060 align:middle line:84% Can anybody tell me what I should do with them? 00:17:31.060 --> 00:17:32.060 align:middle line:90% AUDIENCE: Subtract them. 00:17:32.060 --> 00:17:33.143 align:middle line:90% PETER SHOR: Subtract them. 00:17:33.143 --> 00:17:38.400 align:middle line:90% Very good f of x minus x. 00:17:38.400 --> 00:17:42.500 align:middle line:90% f of x minus x squared. 00:17:42.500 --> 00:17:46.540 align:middle line:90% f of x equals. 00:17:46.540 --> 00:17:48.240 align:middle line:84% Can anybody tell me what this is? 00:17:48.240 --> 00:17:56.180 align:middle line:90% 00:17:56.180 --> 00:17:56.820 align:middle line:90% One. 00:17:56.820 --> 00:17:58.300 align:middle line:90% Exactly. 00:17:58.300 --> 00:18:03.620 align:middle line:84% So it's one, because anything after the first term, we have 00:18:03.620 --> 00:18:07.240 align:middle line:84% f sub i, f sub i minus 1, and f sub i minus 2, 00:18:07.240 --> 00:18:10.140 align:middle line:84% and we take this and subtract that and subtract that. 00:18:10.140 --> 00:18:13.180 align:middle line:90% We get zero. 00:18:13.180 --> 00:18:16.200 align:middle line:84% The first term doesn't quite work like that. 00:18:16.200 --> 00:18:18.770 align:middle line:84% We take 1 minus 0 minus 0 and we get 1. 00:18:18.770 --> 00:18:22.140 align:middle line:90% 00:18:22.140 --> 00:18:27.780 align:middle line:84% And of course, this gives you the equation f of x 1 00:18:27.780 --> 00:18:36.020 align:middle line:84% minus x minus x squared equals 1 or f of x equals 1 over 1 00:18:36.020 --> 00:18:38.110 align:middle line:90% minus x minus x squared. 00:18:38.110 --> 00:18:40.940 align:middle line:90% 00:18:40.940 --> 00:18:42.644 align:middle line:90% OK. 00:18:42.644 --> 00:18:48.300 align:middle line:84% So how do we find a formula for f of x from this? 00:18:48.300 --> 00:18:53.380 align:middle line:84% How many of you remember from calculus the method 00:18:53.380 --> 00:18:54.720 align:middle line:90% of partial fractions? 00:18:54.720 --> 00:18:57.700 align:middle line:90% 00:18:57.700 --> 00:18:58.960 align:middle line:90% So some of you do. 00:18:58.960 --> 00:19:03.780 align:middle line:84% How many have never seen the method of partial fractions? 00:19:03.780 --> 00:19:05.680 align:middle line:90% OK, so most of you remember it. 00:19:05.680 --> 00:19:09.580 align:middle line:90% 00:19:09.580 --> 00:19:21.060 align:middle line:84% I mean, I guess there's some question as to when 00:19:21.060 --> 00:19:24.660 align:middle line:84% we have Mathematica, does anybody 00:19:24.660 --> 00:19:28.900 align:middle line:84% need to remember how to integrate expressions? 00:19:28.900 --> 00:19:37.010 align:middle line:84% But I guess they haven't taken away all the integration 00:19:37.010 --> 00:19:41.050 align:middle line:90% methods from calculus yet. 00:19:41.050 --> 00:19:41.710 align:middle line:90% OK. 00:19:41.710 --> 00:19:45.490 align:middle line:90% So what do we do? 00:19:45.490 --> 00:19:47.590 align:middle line:84% We use the method of partial fractions. 00:19:47.590 --> 00:19:58.050 align:middle line:90% 00:19:58.050 --> 00:20:04.730 align:middle line:84% We say 1 minus x minus x squared is equal to 1 minus phi 00:20:04.730 --> 00:20:08.460 align:middle line:90% plus 1 minus phi minus. 00:20:08.460 --> 00:20:15.730 align:middle line:90% 00:20:15.730 --> 00:20:27.228 align:middle line:84% And phi plus, phi minus are the roots of-- 00:20:27.228 --> 00:20:28.770 align:middle line:84% well, what we're going to do is we're 00:20:28.770 --> 00:20:32.050 align:middle line:84% going to reverse the order of these coefficients 00:20:32.050 --> 00:20:34.790 align:middle line:84% and say y squared minus y minus 1. 00:20:34.790 --> 00:20:40.090 align:middle line:90% 00:20:40.090 --> 00:20:44.930 align:middle line:90% And ah. 00:20:44.930 --> 00:20:46.840 align:middle line:84% I forgot to put the x's in this equation. 00:20:46.840 --> 00:20:53.630 align:middle line:90% 00:20:53.630 --> 00:20:54.130 align:middle line:90% Yeah. 00:20:54.130 --> 00:20:58.530 align:middle line:84% Maybe I should explain why phi plus and phi minus are roots 00:20:58.530 --> 00:21:01.850 align:middle line:90% of y squared minus 1 minus 1. 00:21:01.850 --> 00:21:11.730 align:middle line:84% So y squared minus y minus 1 equals y minus y plus y minus y 00:21:11.730 --> 00:21:14.250 align:middle line:90% minus. 00:21:14.250 --> 00:21:20.170 align:middle line:84% And let's let x equal 1 over y, 1 over x2 squared minus 1 00:21:20.170 --> 00:21:22.140 align:middle line:90% over x minus 1. 00:21:22.140 --> 00:21:26.770 align:middle line:90% 00:21:26.770 --> 00:21:27.270 align:middle line:90% Wait. 00:21:27.270 --> 00:21:31.530 align:middle line:90% 00:21:31.530 --> 00:21:34.770 align:middle line:84% This should be y squared plus y minus 1. 00:21:34.770 --> 00:21:37.090 align:middle line:90% y squared plus y minus 1. 00:21:37.090 --> 00:21:47.050 align:middle line:84% And now 1 over x squared plus 1 over x minus 1 00:21:47.050 --> 00:21:59.680 align:middle line:84% equals 1 over x minus phi plus 1 over x minus phi minus. 00:21:59.680 --> 00:22:14.370 align:middle line:90% 00:22:14.370 --> 00:22:15.800 align:middle line:84% I think I've done something wrong. 00:22:15.800 --> 00:22:33.720 align:middle line:90% 00:22:33.720 --> 00:22:35.560 align:middle line:90% Let me think about this. 00:22:35.560 --> 00:22:41.120 align:middle line:84% So we have 1 minus x minus x squared equals k. 00:22:41.120 --> 00:22:45.780 align:middle line:84% And now we want y squared plus y. 00:22:45.780 --> 00:22:49.240 align:middle line:90% 00:22:49.240 --> 00:23:03.160 align:middle line:84% This should be if y is 1 over x, then this corresponds to the 1. 00:23:03.160 --> 00:23:04.500 align:middle line:90% This corresponds to y. 00:23:04.500 --> 00:23:05.860 align:middle line:90% This corresponds to y squared. 00:23:05.860 --> 00:23:08.560 align:middle line:84% But this really should be minus y, minus y 00:23:08.560 --> 00:23:10.680 align:middle line:90% squared plus 1 equals that. 00:23:10.680 --> 00:23:11.710 align:middle line:90% And now we have-- 00:23:11.710 --> 00:23:20.280 align:middle line:90% 00:23:20.280 --> 00:23:21.220 align:middle line:90% that doesn't work. 00:23:21.220 --> 00:23:31.380 align:middle line:90% 00:23:31.380 --> 00:23:31.880 align:middle line:90% OK. 00:23:31.880 --> 00:23:48.560 align:middle line:90% 00:23:48.560 --> 00:23:49.300 align:middle line:90% Oh, yeah. 00:23:49.300 --> 00:24:10.320 align:middle line:90% 00:24:10.320 --> 00:24:11.380 align:middle line:90% Yeah, this is right. 00:24:11.380 --> 00:24:14.960 align:middle line:90% 00:24:14.960 --> 00:24:15.560 align:middle line:90% OK. 00:24:15.560 --> 00:24:16.060 align:middle line:90% Good. 00:24:16.060 --> 00:24:20.520 align:middle line:90% 00:24:20.520 --> 00:24:25.680 align:middle line:84% And multiplying both sides of this by x squared, 00:24:25.680 --> 00:24:31.360 align:middle line:84% we get 1 minus phi plus x, 1 minus phi 00:24:31.360 --> 00:24:33.080 align:middle line:90% minus x on the right side. 00:24:33.080 --> 00:24:42.120 align:middle line:84% And we should get 1 minus x minus x squared. 00:24:42.120 --> 00:24:46.365 align:middle line:84% So this should be a minus, and this should be a minus, 00:24:46.365 --> 00:24:47.490 align:middle line:90% and this should be a minus. 00:24:47.490 --> 00:24:53.500 align:middle line:90% 00:24:53.500 --> 00:24:54.000 align:middle line:90% Yeah. 00:24:54.000 --> 00:24:54.440 align:middle line:90% OK. 00:24:54.440 --> 00:24:54.940 align:middle line:90% Good. 00:24:54.940 --> 00:24:58.560 align:middle line:90% 00:24:58.560 --> 00:25:01.800 align:middle line:90% So the roots of y squared. 00:25:01.800 --> 00:25:10.160 align:middle line:84% Roots are negative 1, 1 plus or minus 00:25:10.160 --> 00:25:16.280 align:middle line:84% the square root of 5 over 2 by the quadratic formula. 00:25:16.280 --> 00:25:21.270 align:middle line:84% And these are phi plus and phi minus. 00:25:21.270 --> 00:25:23.710 align:middle line:84% And phi plus will have the plus sign and phi 00:25:23.710 --> 00:25:25.250 align:middle line:90% minus will have a minus sign. 00:25:25.250 --> 00:25:34.290 align:middle line:90% 00:25:34.290 --> 00:25:34.790 align:middle line:90% Wait, no. 00:25:34.790 --> 00:25:36.810 align:middle line:90% I wanted-- I'm sorry. 00:25:36.810 --> 00:25:40.992 align:middle line:90% 00:25:40.992 --> 00:25:42.200 align:middle line:90% I want to keep this equation. 00:25:42.200 --> 00:25:48.570 align:middle line:90% 00:25:48.570 --> 00:25:49.070 align:middle line:90% OK. 00:25:49.070 --> 00:25:54.830 align:middle line:90% 00:25:54.830 --> 00:25:55.910 align:middle line:90% AUDIENCE: Sorry. 00:25:55.910 --> 00:26:00.790 align:middle line:84% How did it get there, root of y squared minus 1? 00:26:00.790 --> 00:26:01.696 align:middle line:90% PETER SHOR: Oh. 00:26:01.696 --> 00:26:06.710 align:middle line:84% So what I did was I replaced y by 1 00:26:06.710 --> 00:26:10.040 align:middle line:84% over x and then multiplied by y squared. 00:26:10.040 --> 00:26:18.990 align:middle line:90% 00:26:18.990 --> 00:26:23.210 align:middle line:84% And I clearly did this too fast because I confused myself, 00:26:23.210 --> 00:26:25.030 align:middle line:90% so let's do it. 00:26:25.030 --> 00:26:27.570 align:middle line:84% And I confused you too by going too fast, 00:26:27.570 --> 00:26:30.350 align:middle line:90% so let's go over more slowly. 00:26:30.350 --> 00:26:30.850 align:middle line:90% OK. 00:26:30.850 --> 00:26:34.470 align:middle line:90% Let's let y equal 1 over x. 00:26:34.470 --> 00:26:35.790 align:middle line:90% x. 00:26:35.790 --> 00:26:41.190 align:middle line:84% So this is 1 minus 1 over phi minus 1 over phi squared 00:26:41.190 --> 00:26:49.910 align:middle line:84% is equal to 1 minus phi plus over y. 00:26:49.910 --> 00:26:54.830 align:middle line:90% 1 minus phi minus over y. 00:26:54.830 --> 00:26:57.812 align:middle line:90% OK. 00:26:57.812 --> 00:27:01.150 align:middle line:84% And now multiply both sides by y squared. 00:27:01.150 --> 00:27:06.950 align:middle line:84% So this is y squared minus y minus 1 equals y minus phi 00:27:06.950 --> 00:27:10.980 align:middle line:90% plus y minus phi minus. 00:27:10.980 --> 00:27:13.590 align:middle line:90% 00:27:13.590 --> 00:27:20.070 align:middle line:84% So phi plus is the largest root of y squared minus y minus 1, 00:27:20.070 --> 00:27:25.590 align:middle line:84% and y minus is the smaller root of y squared minus y minus 1. 00:27:25.590 --> 00:27:35.470 align:middle line:84% So we have phi plus is equal to 1 plus root phi over 2, 00:27:35.470 --> 00:27:40.930 align:middle line:84% and phi minus equals 1 minus phi over 2. 00:27:40.930 --> 00:27:45.470 align:middle line:90% 00:27:45.470 --> 00:27:45.970 align:middle line:90% OK. 00:27:45.970 --> 00:28:09.710 align:middle line:90% 00:28:09.710 --> 00:28:10.880 align:middle line:90% OK. 00:28:10.880 --> 00:28:13.310 align:middle line:90% So what do we have? 00:28:13.310 --> 00:28:21.700 align:middle line:84% We have f of x equals 1 over 1 minus 1, 00:28:21.700 --> 00:28:32.140 align:middle line:84% x minus x squared equals 1 over 1 minus phi plus x, 1 minus phi 00:28:32.140 --> 00:28:34.140 align:middle line:90% minus x. 00:28:34.140 --> 00:28:36.100 align:middle line:84% And by partial fractions, we know 00:28:36.100 --> 00:28:42.620 align:middle line:84% this is equal to a over 1 minus phi 00:28:42.620 --> 00:28:50.200 align:middle line:84% plus x plus b over 1 minus phi minus x. 00:28:50.200 --> 00:29:08.793 align:middle line:90% 00:29:08.793 --> 00:29:09.293 align:middle line:90% OK. 00:29:09.293 --> 00:29:14.500 align:middle line:90% So what do we do with this? 00:29:14.500 --> 00:29:23.500 align:middle line:84% Well, let's try to solve this equation. 00:29:23.500 --> 00:29:29.020 align:middle line:84% So this is equal to a times 1 minus phi 00:29:29.020 --> 00:29:40.220 align:middle line:84% minus x plus b times 1 minus phi plus x over 1 00:29:40.220 --> 00:29:42.920 align:middle line:90% minus x minus x squared. 00:29:42.920 --> 00:29:50.300 align:middle line:84% Because this times this is 1 minus x minus x squared. 00:29:50.300 --> 00:29:53.460 align:middle line:84% So now we have this is equal to this, 00:29:53.460 --> 00:29:56.060 align:middle line:90% so this gives us two equations. 00:29:56.060 --> 00:30:12.260 align:middle line:84% We have a plus b equals 1, and a phi minus x plus b phi a-- 00:30:12.260 --> 00:30:19.420 align:middle line:84% yeah. a phi minus x plus b phi plus x equals 0x. 00:30:19.420 --> 00:30:23.620 align:middle line:90% 00:30:23.620 --> 00:30:26.500 align:middle line:90% OK, so we have two equations. 00:30:26.500 --> 00:30:32.220 align:middle line:84% We have two unknowns, a and b, and you know how to solve them. 00:30:32.220 --> 00:30:36.520 align:middle line:84% And I am not going to do it because it's a lot of algebra 00:30:36.520 --> 00:30:41.020 align:middle line:84% and I quite possibly would screw up somewhere along the way. 00:30:41.020 --> 00:30:45.380 align:middle line:84% And you know how to do it anyway. 00:30:45.380 --> 00:30:53.340 align:middle line:84% So the answer comes out to a equals 1 over root phi times 00:30:53.340 --> 00:31:01.380 align:middle line:84% 1 plus root phi over 2, and b equals 1 over root phi 00:31:01.380 --> 00:31:05.120 align:middle line:90% times 1 minus root phi 2. 00:31:05.120 --> 00:31:07.660 align:middle line:90% 00:31:07.660 --> 00:31:10.490 align:middle line:84% So this is phi plus and this is phi minus. 00:31:10.490 --> 00:31:35.890 align:middle line:90% 00:31:35.890 --> 00:31:40.330 align:middle line:90% So the formula for f. 00:31:40.330 --> 00:31:52.490 align:middle line:84% Well, f of x equals a over 1 minus y plus x plus b over 1 00:31:52.490 --> 00:32:01.050 align:middle line:84% minus y minus x is equal to a times 1 plus phi 00:32:01.050 --> 00:32:06.690 align:middle line:84% plus x plus phi plus squared x squared 00:32:06.690 --> 00:32:10.580 align:middle line:90% plus phi plus cubed x cubed. 00:32:10.580 --> 00:32:13.970 align:middle line:90% 00:32:13.970 --> 00:32:26.930 align:middle line:84% And plus b times 1 plus phi minus x plus phi minus x 00:32:26.930 --> 00:32:31.130 align:middle line:84% squared x squared plus phi minus cubed 00:32:31.130 --> 00:32:33.390 align:middle line:90% x cubed plus dot, dot, dot. 00:32:33.390 --> 00:32:37.610 align:middle line:90% 00:32:37.610 --> 00:32:47.090 align:middle line:84% And of course, f of x was equal to summation x sub i, 00:32:47.090 --> 00:32:50.770 align:middle line:90% x sub n, x to the n. 00:32:50.770 --> 00:32:54.650 align:middle line:84% So to find the n-th Fibonacci number, 00:32:54.650 --> 00:32:58.160 align:middle line:84% we need to find the n-th term in the sequence. 00:32:58.160 --> 00:33:02.810 align:middle line:90% 00:33:02.810 --> 00:33:14.930 align:middle line:84% So we get f sub n is equal to a phi plus to the n plus b phi 00:33:14.930 --> 00:33:17.674 align:middle line:90% minus to the n. 00:33:17.674 --> 00:33:20.390 align:middle line:84% And let's go back and see what a and b were. 00:33:20.390 --> 00:33:24.810 align:middle line:84% a was 1 over root phi times phi plus 00:33:24.810 --> 00:33:28.250 align:middle line:84% and b was 1 over root phi times phi minus. 00:33:28.250 --> 00:33:39.050 align:middle line:84% So this is equal to 1 over root phi phi plus to the n plus 1 00:33:39.050 --> 00:33:43.550 align:middle line:90% plus phi minus n plus 1. 00:33:43.550 --> 00:33:48.050 align:middle line:90% 00:33:48.050 --> 00:33:57.990 align:middle line:84% Where phi plus is 1 over root phi plus 1 plus root phi over 2. 00:33:57.990 --> 00:34:02.690 align:middle line:84% And phi minus is 1 minus root phi over 2. 00:34:02.690 --> 00:34:07.500 align:middle line:84% So here, we have a term for the n-th Fibonacci, formula 00:34:07.500 --> 00:34:08.750 align:middle line:90% for the n-th Fibonacci number. 00:34:08.750 --> 00:34:16.679 align:middle line:90% 00:34:16.679 --> 00:34:17.550 align:middle line:90% Oh, wow. 00:34:17.550 --> 00:34:24.080 align:middle line:90% 00:34:24.080 --> 00:34:24.580 align:middle line:90% OK. 00:34:24.580 --> 00:35:03.870 align:middle line:90% 00:35:03.870 --> 00:35:10.700 align:middle line:84% So how fast does the n-th Fibonacci number grow? 00:35:10.700 --> 00:35:28.370 align:middle line:84% How many messages can we send in time n with dots and dashes? 00:35:28.370 --> 00:35:31.240 align:middle line:90% 00:35:31.240 --> 00:35:34.150 align:middle line:90% Well, it's just this formula. 00:35:34.150 --> 00:35:45.200 align:middle line:90% 00:35:45.200 --> 00:35:53.480 align:middle line:84% It's 1 over root phi, phi plus to the n plus 1 phi 00:35:53.480 --> 00:35:55.140 align:middle line:90% minus to the n plus 1. 00:35:55.140 --> 00:35:58.520 align:middle line:90% 00:35:58.520 --> 00:36:01.160 align:middle line:90% OK. 00:36:01.160 --> 00:36:11.600 align:middle line:84% Now I want to say that phi plus is bigger than 1, 00:36:11.600 --> 00:36:14.640 align:middle line:90% and phi minus is less than 1. 00:36:14.640 --> 00:36:19.000 align:middle line:84% So this is just very approximate and very close to 1 00:36:19.000 --> 00:36:23.260 align:middle line:84% over root phi, phi plus to the n plus 1. 00:36:23.260 --> 00:36:27.040 align:middle line:90% 00:36:27.040 --> 00:36:36.080 align:middle line:84% So the information transmission rate 00:36:36.080 --> 00:36:43.960 align:middle line:84% is log base 2 phi plus to the n plus 1 00:36:43.960 --> 00:36:50.720 align:middle line:84% over root phi, which is approximately-- 00:36:50.720 --> 00:36:55.790 align:middle line:84% well, we want to divide this by the time per time unit. 00:36:55.790 --> 00:37:07.470 align:middle line:90% 00:37:07.470 --> 00:37:15.710 align:middle line:84% Let's say information rate per time unit is log 2 phi plus 2n 00:37:15.710 --> 00:37:20.270 align:middle line:84% plus 1 root phi over n, which is approximately-- 00:37:20.270 --> 00:37:25.190 align:middle line:84% so this n plus 1 comes close to canceling this n. 00:37:25.190 --> 00:37:28.670 align:middle line:84% Root phi is just a constant factor, a constant addition 00:37:28.670 --> 00:37:30.710 align:middle line:90% factor which is negligible. 00:37:30.710 --> 00:37:35.930 align:middle line:90% So it's log 2 of phi plus. 00:37:35.930 --> 00:37:39.910 align:middle line:90% 00:37:39.910 --> 00:37:43.590 align:middle line:84% So if you have a channel that uses dots and dashes to transmit 00:37:43.590 --> 00:37:47.550 align:middle line:84% information, this is how much you can transmit in time n. 00:37:47.550 --> 00:37:50.430 align:middle line:84% And we're actually going to be going back to this later 00:37:50.430 --> 00:37:54.070 align:middle line:84% in the course and talking about information theory 00:37:54.070 --> 00:37:58.310 align:middle line:84% and how much information you can transmit in general. 00:37:58.310 --> 00:37:59.738 align:middle line:90% OK. 00:37:59.738 --> 00:38:15.470 align:middle line:90% So let's see. 00:38:15.470 --> 00:38:21.270 align:middle line:84% I'm trying to think if I have time to do this next bit. 00:38:21.270 --> 00:38:22.210 align:middle line:90% I think I do. 00:38:22.210 --> 00:38:26.630 align:middle line:90% 00:38:26.630 --> 00:38:28.180 align:middle line:84% So let's answer another question. 00:38:28.180 --> 00:38:44.630 align:middle line:90% 00:38:44.630 --> 00:38:45.630 align:middle line:90% Oh, wait. 00:38:45.630 --> 00:38:48.870 align:middle line:84% Before we answer another question, I want to go back 00:38:48.870 --> 00:38:54.840 align:middle line:84% and say, we actually could have gotten this formula. 00:38:54.840 --> 00:38:58.510 align:middle line:90% 00:38:58.510 --> 00:38:59.450 align:middle line:90% Where is it? 00:38:59.450 --> 00:39:06.950 align:middle line:90% 00:39:06.950 --> 00:39:15.030 align:middle line:84% f of x equals 1 over 1 minus x minus x squared, much more 00:39:15.030 --> 00:39:17.550 align:middle line:90% easily. 00:39:17.550 --> 00:39:19.470 align:middle line:84% So how could we have gotten the generating 00:39:19.470 --> 00:39:25.270 align:middle line:84% function for this dots and dashes problem much more easily? 00:39:25.270 --> 00:39:25.870 align:middle line:90% Anybody? 00:39:25.870 --> 00:39:31.970 align:middle line:90% 00:39:31.970 --> 00:39:32.470 align:middle line:90% OK. 00:39:32.470 --> 00:39:38.190 align:middle line:84% So yeah, it's not completely obvious. 00:39:38.190 --> 00:39:50.380 align:middle line:84% So f of x is number of sequences of dot and dash. 00:39:50.380 --> 00:39:54.590 align:middle line:90% 00:39:54.590 --> 00:40:02.550 align:middle line:84% So the generating function at x equals 1, 00:40:02.550 --> 00:40:12.430 align:middle line:84% I'm sorry x plus x squared for dot, dash, 00:40:12.430 --> 00:40:16.790 align:middle line:84% because dots have time 1, which is x, 00:40:16.790 --> 00:40:20.110 align:middle line:84% dashes have time 2, which corresponds to x squared. 00:40:20.110 --> 00:40:23.550 align:middle line:84% And we just add them because the generating 00:40:23.550 --> 00:40:26.310 align:middle line:84% function when you have two disjoint 00:40:26.310 --> 00:40:30.350 align:middle line:90% things, you take the sum. 00:40:30.350 --> 00:40:38.150 align:middle line:84% Does the sequence of dots and dashes is equal to. 00:40:38.150 --> 00:40:42.470 align:middle line:84% Well, we use this formula up here, 1 over 1 minus A of x 00:40:42.470 --> 00:40:46.680 align:middle line:84% is 1 over 1 minus x minus x squared. 00:40:46.680 --> 00:40:49.510 align:middle line:90% 00:40:49.510 --> 00:40:52.590 align:middle line:84% And we just saved ourselves a lot of effort 00:40:52.590 --> 00:40:57.020 align:middle line:84% by using that formula rather than deriving it 00:40:57.020 --> 00:41:01.460 align:middle line:84% through writing out the recurrence 00:41:01.460 --> 00:41:08.010 align:middle line:84% and adding the equations up and seeing what the result was. 00:41:08.010 --> 00:41:14.140 align:middle line:90% 00:41:14.140 --> 00:41:19.930 align:middle line:84% So maybe we should try applying this to a different question. 00:41:19.930 --> 00:41:26.100 align:middle line:90% 00:41:26.100 --> 00:41:27.780 align:middle line:84% It's a tiling question, which you're not 00:41:27.780 --> 00:41:30.325 align:middle line:90% getting on your homework. 00:41:30.325 --> 00:41:32.450 align:middle line:84% I think there's a tiling question on your homework. 00:41:32.450 --> 00:41:44.980 align:middle line:90% 00:41:44.980 --> 00:41:58.060 align:middle line:84% The generating function or number of tilings of 2 00:41:58.060 --> 00:42:13.268 align:middle line:84% by n strip with two kinds of tiles. 00:42:13.268 --> 00:42:19.850 align:middle line:84% A triangular tile and a zigzag tile. 00:42:19.850 --> 00:42:22.540 align:middle line:90% 00:42:22.540 --> 00:42:24.180 align:middle line:90% AUDIENCE: Can you-- 00:42:24.180 --> 00:42:25.965 align:middle line:90% PETER SHOR: Say something? 00:42:25.965 --> 00:42:27.340 align:middle line:84% AUDIENCE: Can you rotate or flip? 00:42:27.340 --> 00:42:29.465 align:middle line:84% PETER SHOR: Yeah, you're allowed to rotate or flip. 00:42:29.465 --> 00:42:31.920 align:middle line:90% 00:42:31.920 --> 00:42:36.380 align:middle line:84% Otherwise you couldn't tile the strip without flipping this one. 00:42:36.380 --> 00:42:41.240 align:middle line:90% So for three, you get-- 00:42:41.240 --> 00:42:44.380 align:middle line:90% 00:42:44.380 --> 00:42:58.570 align:middle line:90% and it's for six, you get-- 00:42:58.570 --> 00:43:05.860 align:middle line:90% 00:43:05.860 --> 00:43:07.410 align:middle line:90% no, let's do five first. 00:43:07.410 --> 00:43:10.020 align:middle line:90% 00:43:10.020 --> 00:43:12.560 align:middle line:90% For five, you get. 00:43:12.560 --> 00:43:22.780 align:middle line:90% 00:43:22.780 --> 00:43:24.040 align:middle line:90% Yeah, that's five. 00:43:24.040 --> 00:43:26.060 align:middle line:90% 1 plus 2 plus 2. 00:43:26.060 --> 00:43:27.130 align:middle line:90% And it's flipped. 00:43:27.130 --> 00:43:32.260 align:middle line:90% 00:43:32.260 --> 00:43:33.410 align:middle line:90% I won't draw it. 00:43:33.410 --> 00:43:38.140 align:middle line:90% 00:43:38.140 --> 00:43:41.080 align:middle line:84% And for six what you can get is you can get. 00:43:41.080 --> 00:43:48.660 align:middle line:90% 00:43:48.660 --> 00:43:51.010 align:middle line:90% I'm doing this wrong. 00:43:51.010 --> 00:43:57.400 align:middle line:84% Six, you can get, and now you can join another unit of six. 00:43:57.400 --> 00:44:01.290 align:middle line:90% 00:44:01.290 --> 00:44:01.790 align:middle line:90% OK. 00:44:01.790 --> 00:44:06.290 align:middle line:90% 00:44:06.290 --> 00:44:09.390 align:middle line:84% So how would you find the generating function for this? 00:44:09.390 --> 00:44:33.650 align:middle line:90% 00:44:33.650 --> 00:44:46.170 align:middle line:84% Well, let's use this sequence formula, Ax is equal to. 00:44:46.170 --> 00:44:52.510 align:middle line:84% Well, there's two ways of tiling a 3 by 2 strip. 00:44:52.510 --> 00:44:58.690 align:middle line:90% So that's x cubed times 2 plus-- 00:44:58.690 --> 00:45:03.150 align:middle line:84% well, there are two ways of tiling of a five strip. 00:45:03.150 --> 00:45:06.210 align:middle line:90% So that's 2x to the fifth. 00:45:06.210 --> 00:45:08.740 align:middle line:84% There are two ways of tiling a seven strip. 00:45:08.740 --> 00:45:12.210 align:middle line:90% 00:45:12.210 --> 00:45:17.170 align:middle line:84% I mean you just stick two z tiles instead of one z tile. 00:45:17.170 --> 00:45:21.170 align:middle line:84% So that's plus 2x cubed plus dot, dot, dot. 00:45:21.170 --> 00:45:26.720 align:middle line:84% And that is equal to 2x cubed over 1 minus x squared. 00:45:26.720 --> 00:45:40.370 align:middle line:90% 00:45:40.370 --> 00:45:49.810 align:middle line:84% So g of x is equal to 1 over 1 minus 2x cubed over 1 minus x 00:45:49.810 --> 00:45:57.500 align:middle line:90% squared is equal to 1 over-- 00:45:57.500 --> 00:46:00.450 align:middle line:90% 00:46:00.450 --> 00:46:09.170 align:middle line:84% I won't say 1 minus x squared minus 2x cubed. 00:46:09.170 --> 00:46:11.920 align:middle line:84% And I want to put in 1 minus x squared on top. 00:46:11.920 --> 00:46:17.750 align:middle line:90% 00:46:17.750 --> 00:46:18.250 align:middle line:90% OK. 00:46:18.250 --> 00:46:28.850 align:middle line:90% 00:46:28.850 --> 00:46:29.550 align:middle line:90% OK. 00:46:29.550 --> 00:46:34.890 align:middle line:84% And you can figure out what the recurrence for this 00:46:34.890 --> 00:46:38.890 align:middle line:90% is, which is really weird. 00:46:38.890 --> 00:46:43.370 align:middle line:90% But maybe I shouldn't say weird. 00:46:43.370 --> 00:46:47.320 align:middle line:84% Which is really quite non-intuitive. 00:46:47.320 --> 00:46:59.880 align:middle line:84% But we had the recurrence fn equals fn minus 1 00:46:59.880 --> 00:47:01.980 align:middle line:90% plus fn minus 2. 00:47:01.980 --> 00:47:05.960 align:middle line:84% That came up to 1 minus x minus x squared. 00:47:05.960 --> 00:47:15.920 align:middle line:84% So this is going to be fn is equal to fn 00:47:15.920 --> 00:47:19.730 align:middle line:90% minus 2 minus 2 fn minus 3. 00:47:19.730 --> 00:47:23.760 align:middle line:90% 00:47:23.760 --> 00:47:28.820 align:middle line:84% And the top says what the initial conditions are. 00:47:28.820 --> 00:47:32.280 align:middle line:90% 00:47:32.280 --> 00:47:34.200 align:middle line:90% f0. 00:47:34.200 --> 00:47:36.720 align:middle line:90% Wait, this should be a plus. 00:47:36.720 --> 00:47:42.580 align:middle line:84% f0 equals 1 and f2 equals minus 1. 00:47:42.580 --> 00:47:48.400 align:middle line:90% 00:47:48.400 --> 00:47:57.400 align:middle line:84% And I'm not going to have time to explain 00:47:57.400 --> 00:48:02.260 align:middle line:84% why going from here to here makes this, 00:48:02.260 --> 00:48:07.380 align:middle line:84% but let's see what sequence we get. 00:48:07.380 --> 00:48:21.720 align:middle line:90% 00:48:21.720 --> 00:48:24.590 align:middle line:84% And I should not say this is f2 equals 93. 00:48:24.590 --> 00:48:33.000 align:middle line:90% 00:48:33.000 --> 00:48:43.240 align:middle line:90% Because, I mean, 1 minus x-- 00:48:43.240 --> 00:48:43.900 align:middle line:90% what was that? 00:48:43.900 --> 00:48:49.800 align:middle line:84% 1 over 1 minus x squared minus 2x cubed. 00:48:49.800 --> 00:49:00.280 align:middle line:84% The initial condition is 2 equals 1, f of 0 equals 1, 00:49:00.280 --> 00:49:08.120 align:middle line:84% and f sub i equals f of i minus 2 plus 2 and minus 3. 00:49:08.120 --> 00:49:17.840 align:middle line:84% And x squared over 1 minus x squared minus 2x cubed is 00:49:17.840 --> 00:49:23.240 align:middle line:84% the same thing, except the initial condition is f of 2 00:49:23.240 --> 00:49:28.760 align:middle line:84% equals minus 1 and the same recurrence. 00:49:28.760 --> 00:49:33.440 align:middle line:84% So we can look at the recurrence and see if we actually 00:49:33.440 --> 00:49:36.730 align:middle line:90% get what we wanted. 00:49:36.730 --> 00:49:40.850 align:middle line:90% 00:49:40.850 --> 00:49:43.430 align:middle line:90% So 1. 00:49:43.430 --> 00:49:45.298 align:middle line:90% So this is 0, 1. 00:49:45.298 --> 00:49:47.270 align:middle line:90% 2. 00:49:47.270 --> 00:49:48.870 align:middle line:90% We get 1 here. 00:49:48.870 --> 00:49:50.460 align:middle line:90% 3, we get 1. 00:49:50.460 --> 00:49:53.030 align:middle line:90% 00:49:53.030 --> 00:49:55.590 align:middle line:90% We get 2 here. 00:49:55.590 --> 00:49:56.870 align:middle line:90% 4. 00:49:56.870 --> 00:50:03.480 align:middle line:84% We look back 2 and we get another 1. 00:50:03.480 --> 00:50:09.550 align:middle line:90% 00:50:09.550 --> 00:50:11.030 align:middle line:90% Is that right? 00:50:11.030 --> 00:50:12.110 align:middle line:90% Yeah, that's right. 00:50:12.110 --> 00:50:17.790 align:middle line:84% And now we subtract minus 1, 0, 1. 00:50:17.790 --> 00:50:23.970 align:middle line:84% For 5, we look back 2 and we get 2 and we add 2 times this. 00:50:23.970 --> 00:50:25.190 align:middle line:90% So this should be 4. 00:50:25.190 --> 00:50:33.830 align:middle line:84% And now we subtract 1, 0, 1, 0, because subtracting-- 00:50:33.830 --> 00:50:36.830 align:middle line:84% shifting over 2 and subtracting is the same as multiplying 00:50:36.830 --> 00:50:40.230 align:middle line:90% by minus x squared. 00:50:40.230 --> 00:50:41.770 align:middle line:90% And we get this. 00:50:41.770 --> 00:50:50.510 align:middle line:90% 00:50:50.510 --> 00:50:53.700 align:middle line:90% Oh, this should be 2. 00:50:53.700 --> 00:50:56.430 align:middle line:90% 00:50:56.430 --> 00:50:58.010 align:middle line:90% 1, 0, 1, 2. 00:50:58.010 --> 00:51:02.630 align:middle line:90% 00:51:02.630 --> 00:51:07.352 align:middle line:90% And we get 1, 0, 0, 2, 0, 2. 00:51:07.352 --> 00:51:12.030 align:middle line:90% 00:51:12.030 --> 00:51:15.030 align:middle line:90% And indeed there were 2 for 5. 00:51:15.030 --> 00:51:17.310 align:middle line:90% There were 2 for-- 00:51:17.310 --> 00:51:23.790 align:middle line:84% there was 2 for 3, and there was 1 for length 0, which 00:51:23.790 --> 00:51:26.910 align:middle line:90% is just the empty tiling. 00:51:26.910 --> 00:51:29.950 align:middle line:84% And the next one should give us-- 00:51:29.950 --> 00:51:32.090 align:middle line:84% the next element should really give us 4 00:51:32.090 --> 00:51:33.830 align:middle line:90% if I've calculated it right. 00:51:33.830 --> 00:51:35.390 align:middle line:90% And it probably does. 00:51:35.390 --> 00:51:38.870 align:middle line:90% 00:51:38.870 --> 00:51:42.470 align:middle line:90% I'm not going to check it. 00:51:42.470 --> 00:51:45.460 align:middle line:84% So that is how you could do another tiling. 00:51:45.460 --> 00:51:52.510 align:middle line:84% OK, so now let's switch to a different question. 00:51:52.510 --> 00:51:53.980 align:middle line:90% Let's look at Pascal's triangle. 00:51:53.980 --> 00:51:57.590 align:middle line:90% 00:51:57.590 --> 00:51:58.090 align:middle line:90% OK. 00:51:58.090 --> 00:52:30.030 align:middle line:90% 00:52:30.030 --> 00:52:30.690 align:middle line:90% OK. 00:52:30.690 --> 00:52:37.860 align:middle line:84% Pascal's triangle is 0, 0, 1 choose 0, 1 choose one, 00:52:37.860 --> 00:52:45.220 align:middle line:84% 2 choose 0, 2 choose 1, 2 choose 2, et cetera. 00:52:45.220 --> 00:52:47.180 align:middle line:90% And let's look at n numbers. 00:52:47.180 --> 00:52:58.140 align:middle line:84% 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 00:52:58.140 --> 00:53:01.940 align:middle line:90% 5, 1, et cetera. 00:53:01.940 --> 00:53:05.740 align:middle line:90% So let's do this. 00:53:05.740 --> 00:53:10.940 align:middle line:84% Now suppose you're just looking at this and you start to sum 00:53:10.940 --> 00:53:16.360 align:middle line:84% and you ask, what happens if I sum, I guess, diagonals of it? 00:53:16.360 --> 00:53:21.540 align:middle line:84% I mean, let's take 1 plus 1 plus 2. 00:53:21.540 --> 00:53:25.060 align:middle line:90% 1 plus 3 plus 1. 00:53:25.060 --> 00:53:25.750 align:middle line:90% That's 5. 00:53:25.750 --> 00:53:31.820 align:middle line:90% 00:53:31.820 --> 00:53:37.740 align:middle line:90% 1 plus 6 plus 5 plus 1. 00:53:37.740 --> 00:53:40.460 align:middle line:90% That's 13. 00:53:40.460 --> 00:53:46.100 align:middle line:84% And you happen to notice that these are Fibonacci numbers. 00:53:46.100 --> 00:53:49.760 align:middle line:84% I mean, 2 is the third Fibonacci number, 00:53:49.760 --> 00:53:51.380 align:middle line:84% 5 is the fifth of Fibonacci number, 00:53:51.380 --> 00:53:56.780 align:middle line:84% 13 is the seventh Fibonacci number, et cetera. 00:53:56.780 --> 00:53:58.300 align:middle line:90% OK. 00:53:58.300 --> 00:54:03.103 align:middle line:84% So once you've noticed this, you can ask, how can I prove it? 00:54:03.103 --> 00:54:05.520 align:middle line:84% And actually, there are a whole bunch of ways to prove it. 00:54:05.520 --> 00:54:08.820 align:middle line:84% You can prove it by counting something two different ways. 00:54:08.820 --> 00:54:11.980 align:middle line:84% You can prove it by straightforward induction, 00:54:11.980 --> 00:54:14.700 align:middle line:84% and you can prove it using generating functions. 00:54:14.700 --> 00:54:17.200 align:middle line:84% And I'm going to prove it using generating functions. 00:54:17.200 --> 00:54:22.940 align:middle line:90% 00:54:22.940 --> 00:54:28.180 align:middle line:84% And I have to say, I think this proof is written up fairly badly 00:54:28.180 --> 00:54:29.700 align:middle line:90% in the notes. 00:54:29.700 --> 00:54:32.200 align:middle line:90% But it's in the notes. 00:54:32.200 --> 00:54:32.700 align:middle line:90% OK. 00:54:32.700 --> 00:54:48.246 align:middle line:90% 00:54:48.246 --> 00:54:56.760 align:middle line:84% Proof using generating functions. 00:54:56.760 --> 00:54:57.260 align:middle line:90% OK. 00:54:57.260 --> 00:55:02.660 align:middle line:90% 00:55:02.660 --> 00:55:09.620 align:middle line:84% Well, first, let's compute the generating 00:55:09.620 --> 00:55:14.210 align:middle line:90% function for Pascal's triangle. 00:55:14.210 --> 00:55:19.460 align:middle line:90% 00:55:19.460 --> 00:55:26.080 align:middle line:84% Now, this is a two dimensional array of numbers 00:55:26.080 --> 00:55:29.020 align:middle line:84% so we're going to need two variables to do this. 00:55:29.020 --> 00:55:32.690 align:middle line:84% And we'll call this x to the 0, y to the 0. 00:55:32.690 --> 00:55:34.450 align:middle line:90% x to the y and y to the 0. 00:55:34.450 --> 00:55:38.410 align:middle line:90% 00:55:38.410 --> 00:55:42.550 align:middle line:84% x to the 0, y to the 1, et cetera. 00:55:42.550 --> 00:55:46.170 align:middle line:84% So what we get is we get x plus y 00:55:46.170 --> 00:55:51.290 align:middle line:84% choose x, x equals 1, 0 to infinity. 00:55:51.290 --> 00:55:54.490 align:middle line:90% Some y equals 0 to infinity. 00:55:54.490 --> 00:55:58.430 align:middle line:84% x plus y, which is x, times x to the-- 00:55:58.430 --> 00:56:03.290 align:middle line:90% 00:56:03.290 --> 00:56:04.850 align:middle line:90% OK. 00:56:04.850 --> 00:56:09.930 align:middle line:84% I am not looking at my notes because in my notes 00:56:09.930 --> 00:56:17.250 align:middle line:84% I'm sure I did not use x for two different things. 00:56:17.250 --> 00:56:18.270 align:middle line:90% So let's see. 00:56:18.270 --> 00:56:24.450 align:middle line:90% 00:56:24.450 --> 00:56:27.090 align:middle line:90% Oh. 00:56:27.090 --> 00:56:27.690 align:middle line:90% Yeah. 00:56:27.690 --> 00:56:33.935 align:middle line:84% Some a equals 0 to infinity, b equals 0 to infinity. 00:56:33.935 --> 00:56:36.938 align:middle line:90% a plus b, choose a. 00:56:36.938 --> 00:56:40.842 align:middle line:90% 00:56:40.842 --> 00:56:43.190 align:middle line:90% x to the a, y to the b. 00:56:43.190 --> 00:56:48.130 align:middle line:90% 00:56:48.130 --> 00:56:52.130 align:middle line:84% So this is the generating function for Pascal's triangle. 00:56:52.130 --> 00:56:54.050 align:middle line:84% Two dimensional generating functions, so you 00:56:54.050 --> 00:56:55.910 align:middle line:90% get two different variables. 00:56:55.910 --> 00:57:01.530 align:middle line:90% 00:57:01.530 --> 00:57:03.570 align:middle line:90% So what is it? 00:57:03.570 --> 00:57:06.174 align:middle line:90% 00:57:06.174 --> 00:57:12.230 align:middle line:84% a plus b, which is x to the a, y to the b. 00:57:12.230 --> 00:57:21.570 align:middle line:90% 00:57:21.570 --> 00:57:22.070 align:middle line:90% OK. 00:57:22.070 --> 00:57:27.860 align:middle line:90% 00:57:27.860 --> 00:57:36.570 align:middle line:84% So what I want to do is I want to break this up into a equals 0 00:57:36.570 --> 00:57:49.930 align:middle line:84% to n and b equals n minus a, and then-- so n 00:57:49.930 --> 00:57:51.793 align:middle line:90% equals 0 to infinity. 00:57:51.793 --> 00:58:00.490 align:middle line:90% 00:58:00.490 --> 00:58:02.730 align:middle line:84% So I'm going to break this up and just 00:58:02.730 --> 00:58:05.650 align:middle line:84% sum it along the rows first and then 00:58:05.650 --> 00:58:09.210 align:middle line:90% sum all the rows after that. 00:58:09.210 --> 00:58:26.850 align:middle line:84% So single row is sum n choose a, a equals 0 to n, x to the a, y 00:58:26.850 --> 00:58:30.320 align:middle line:90% to the n minus a. 00:58:30.320 --> 00:58:31.900 align:middle line:84% Can someone tell me what this is? 00:58:31.900 --> 00:58:36.400 align:middle line:90% 00:58:36.400 --> 00:58:38.260 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:58:38.260 --> 00:58:38.892 align:middle line:90% 00:58:38.892 --> 00:58:39.600 align:middle line:90% PETER SHOR: Yeah. 00:58:39.600 --> 00:58:40.720 align:middle line:90% AUDIENCE: x plus y. 00:58:40.720 --> 00:58:41.520 align:middle line:90% PETER SHOR: Right. 00:58:41.520 --> 00:58:48.000 align:middle line:90% This is x plus y to the n. 00:58:48.000 --> 00:58:50.800 align:middle line:90% Good. 00:58:50.800 --> 00:58:59.640 align:middle line:84% So sum n equals 0 to infinity, x plus y to the n 00:58:59.640 --> 00:59:03.940 align:middle line:84% is equal to 1 over 1 minus x minus y. 00:59:03.940 --> 00:59:20.220 align:middle line:90% 00:59:20.220 --> 00:59:20.720 align:middle line:90% OK. 00:59:20.720 --> 00:59:27.720 align:middle line:90% 00:59:27.720 --> 00:59:31.120 align:middle line:90% Well, what do we do with that? 00:59:31.120 --> 00:59:32.640 align:middle line:84% Well, what we're going to have to do 00:59:32.640 --> 00:59:36.400 align:middle line:84% is we're going to have to go back and look 00:59:36.400 --> 00:59:38.500 align:middle line:84% at the sum a little more carefully. 00:59:38.500 --> 00:59:55.960 align:middle line:90% 00:59:55.960 --> 01:00:08.660 align:middle line:84% OK, so there are 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1. 01:00:08.660 --> 01:00:16.760 align:middle line:90% 01:00:16.760 --> 01:00:27.200 align:middle line:90% Let's take 1, 5, 10, 10, 5, 1. 01:00:27.200 --> 01:00:29.400 align:middle line:90% And there's a 1 over here. 01:00:29.400 --> 01:00:32.170 align:middle line:90% So let's look at this diagonal. 01:00:32.170 --> 01:00:45.720 align:middle line:90% 01:00:45.720 --> 01:00:51.320 align:middle line:90% 1, 6, 5. 01:00:51.320 --> 01:00:54.360 align:middle line:90% That's 13. 01:00:54.360 --> 01:00:55.656 align:middle line:90% OK. 01:00:55.656 --> 01:01:07.200 align:middle line:84% So this turns out to be 3 choose 0 plus 3 plus 0 choose 01:01:07.200 --> 01:01:15.550 align:middle line:90% 0 times x cubed y to the 0. 01:01:15.550 --> 01:01:21.280 align:middle line:90% 01:01:21.280 --> 01:01:27.910 align:middle line:90% Plus 2 plus 2 choose 2. 01:01:27.910 --> 01:01:34.370 align:middle line:84% x squared y squared plus 1 plus 4 choose 4. 01:01:34.370 --> 01:01:36.950 align:middle line:90% 01:01:36.950 --> 01:01:38.130 align:middle line:90% That's 5 choose 4. 01:01:38.130 --> 01:01:39.910 align:middle line:90% That's this. 01:01:39.910 --> 01:01:57.130 align:middle line:84% xy cubed plus 0 plus 6 choose 6, x to the 0, y to the. 01:01:57.130 --> 01:02:04.430 align:middle line:90% 01:02:04.430 --> 01:02:06.510 align:middle line:90% This is y4. 01:02:06.510 --> 01:02:09.590 align:middle line:90% This is y 6. 01:02:09.590 --> 01:02:12.070 align:middle line:84% So what we're doing is we're going along 01:02:12.070 --> 01:02:21.500 align:middle line:84% and we're subtracting 1 from m and adding 2 to n each time. 01:02:21.500 --> 01:02:57.190 align:middle line:90% 01:02:57.190 --> 01:03:15.310 align:middle line:84% So I want to say this is summation i equals 0 to k. 01:03:15.310 --> 01:03:29.750 align:middle line:90% 01:03:29.750 --> 01:04:06.550 align:middle line:84% Let's call this n minus i plus 2i choose 2i x to the 2i, 01:04:06.550 --> 01:04:08.670 align:middle line:90% x to the n minus i, y to the 2i. 01:04:08.670 --> 01:04:18.400 align:middle line:90% 01:04:18.400 --> 01:04:18.900 align:middle line:90% OK. 01:04:18.900 --> 01:04:21.780 align:middle line:90% 01:04:21.780 --> 01:04:28.300 align:middle line:84% So what we want is we want to somehow group 01:04:28.300 --> 01:04:31.940 align:middle line:84% all these terms together in the generating function. 01:04:31.940 --> 01:04:33.840 align:middle line:84% Can anybody tell me how to do that? 01:04:33.840 --> 01:04:48.680 align:middle line:90% 01:04:48.680 --> 01:04:49.180 align:middle line:90% OK. 01:04:49.180 --> 01:04:57.300 align:middle line:90% 01:04:57.300 --> 01:05:00.580 align:middle line:90% x equals z squared. 01:05:00.580 --> 01:05:03.840 align:middle line:90% And let's let y equal z. 01:05:03.840 --> 01:05:07.080 align:middle line:90% 01:05:07.080 --> 01:05:07.580 align:middle line:90% OK. 01:05:07.580 --> 01:05:15.940 align:middle line:84% So now what we're going to do is this becomes summation i equals 01:05:15.940 --> 01:05:24.980 align:middle line:84% 0 2n, n plus i choose 2i, x to the-- 01:05:24.980 --> 01:05:31.620 align:middle line:84% or z to the 2n minus 2i times z to the 2i. 01:05:31.620 --> 01:05:36.380 align:middle line:90% And this is z to the 2n. 01:05:36.380 --> 01:05:40.380 align:middle line:90% 01:05:40.380 --> 01:05:44.300 align:middle line:84% So if we plug the generating function, 01:05:44.300 --> 01:05:47.780 align:middle line:84% if we plug x equals z squared and y equals 01:05:47.780 --> 01:05:52.700 align:middle line:84% z into the generating function for Pascal's triangle, 01:05:52.700 --> 01:05:58.460 align:middle line:84% the generating function we get will give us these, 01:05:58.460 --> 01:06:01.820 align:middle line:90% I guess diagonal sums. 01:06:01.820 --> 01:06:04.860 align:middle line:90% Is this clear? 01:06:04.860 --> 01:06:05.520 align:middle line:90% OK. 01:06:05.520 --> 01:06:06.700 align:middle line:90% Great. 01:06:06.700 --> 01:06:12.860 align:middle line:84% So what was the generating function for Pascal's triangle? 01:06:12.860 --> 01:06:15.280 align:middle line:90% It was 1 minus x minus y. 01:06:15.280 --> 01:06:51.460 align:middle line:90% 01:06:51.460 --> 01:06:52.732 align:middle line:90% OK. 01:06:52.732 --> 01:07:00.620 align:middle line:90% So substitute x equals-- 01:07:00.620 --> 01:07:02.800 align:middle line:84% what was that? x equals z squared. 01:07:02.800 --> 01:07:11.620 align:middle line:84% xz squared, y equals z, and 1 over 1 minus x minus y. 01:07:11.620 --> 01:07:22.010 align:middle line:84% You get 1 over 1 minus z minus 1 over z squared minus z. 01:07:22.010 --> 01:07:25.810 align:middle line:84% And this is a generating function 01:07:25.810 --> 01:07:26.920 align:middle line:90% for the Fibonacci numbers. 01:07:26.920 --> 01:07:41.170 align:middle line:90% 01:07:41.170 --> 01:07:44.690 align:middle line:84% And you can go back and you can look at this. 01:07:44.690 --> 01:07:50.010 align:middle line:84% And you can realize that our proof not only gives you 01:07:50.010 --> 01:07:52.530 align:middle line:84% the Fibonacci numbers for these diagonals, 01:07:52.530 --> 01:08:07.970 align:middle line:84% it also, if you look at the other diagonals, 2 plus 1 is 3. 01:08:07.970 --> 01:08:18.790 align:middle line:84% And I want to say 3 plus 4 plus 1 is 8. 01:08:18.790 --> 01:08:25.410 align:middle line:90% 01:08:25.410 --> 01:08:32.050 align:middle line:84% I mean, these diagonals starting on the left side of the triangle 01:08:32.050 --> 01:08:34.689 align:middle line:84% give you the odd numbered Fibonacci 01:08:34.689 --> 01:08:39.090 align:middle line:84% numbers, or the every other Fibonacci numbers. 01:08:39.090 --> 01:08:44.410 align:middle line:84% These diagonals starting on the right side 01:08:44.410 --> 01:08:48.630 align:middle line:84% give you the Fibonacci numbers between them. 01:08:48.630 --> 01:08:54.810 align:middle line:90% 01:08:54.810 --> 01:08:56.542 align:middle line:90% OK. 01:08:56.542 --> 01:09:00.729 align:middle line:84% And that's actually all I wanted to say about generating 01:09:00.729 --> 01:09:02.930 align:middle line:90% functions today. 01:09:02.930 --> 01:09:04.850 align:middle line:84% We're going to be doing generating functions 01:09:04.850 --> 01:09:08.930 align:middle line:90% for Catalan numbers on Tuesday. 01:09:08.930 --> 01:09:14.569 align:middle line:84% But I want to hold that off because I want to do everything 01:09:14.569 --> 01:09:15.470 align:middle line:90% on the same day. 01:09:15.470 --> 01:09:18.050 align:middle line:84% So I thought maybe I should show you 01:09:18.050 --> 01:09:24.154 align:middle line:84% the proof of the same recurrence with counting something in two 01:09:24.154 --> 01:09:24.779 align:middle line:90% different ways. 01:09:24.779 --> 01:09:41.890 align:middle line:90% 01:09:41.890 --> 01:09:49.529 align:middle line:90% So Fibonacci numbers. 01:09:49.529 --> 01:09:54.650 align:middle line:90% Number of length n. 01:09:54.650 --> 01:10:02.370 align:middle line:84% Sequences of dots and dashes where 01:10:02.370 --> 01:10:07.550 align:middle line:84% the dashes are length 2 and the dots are length 1. 01:10:07.550 --> 01:10:08.050 align:middle line:90% OK. 01:10:08.050 --> 01:10:15.590 align:middle line:90% 01:10:15.590 --> 01:10:16.090 align:middle line:90% OK. 01:10:16.090 --> 01:10:35.000 align:middle line:90% 01:10:35.000 --> 01:10:50.600 align:middle line:84% And if n equals 5, the number of sequences is 8. 01:10:50.600 --> 01:10:53.240 align:middle line:90% So we will be doing that one. 01:10:53.240 --> 01:11:05.710 align:middle line:84% So let's group sequences by number of dots. 01:11:05.710 --> 01:11:17.588 align:middle line:90% 01:11:17.588 --> 01:11:18.546 align:middle line:90% OK. 01:11:18.546 --> 01:11:22.640 align:middle line:90% So we can have eight dots. 01:11:22.640 --> 01:11:24.000 align:middle line:90% Well, we can have zero. 01:11:24.000 --> 01:11:25.030 align:middle line:90% We can have one dot. 01:11:25.030 --> 01:11:27.600 align:middle line:90% 01:11:27.600 --> 01:11:30.520 align:middle line:90% Two dashes. 01:11:30.520 --> 01:11:33.010 align:middle line:90% We could have two dots. 01:11:33.010 --> 01:11:39.640 align:middle line:90% 01:11:39.640 --> 01:11:41.020 align:middle line:90% Well, we can't have two dots. 01:11:41.020 --> 01:11:44.000 align:middle line:84% We can have three dots and one dash, 01:11:44.000 --> 01:11:48.130 align:middle line:84% or we could have five dots and zero dashes. 01:11:48.130 --> 01:11:50.880 align:middle line:90% 01:11:50.880 --> 01:11:53.960 align:middle line:84% So if you have one dot and two dashes, 01:11:53.960 --> 01:11:56.070 align:middle line:84% how many ways are there of arranging these things? 01:11:56.070 --> 01:12:04.640 align:middle line:90% 01:12:04.640 --> 01:12:05.520 align:middle line:90% AUDIENCE: Three. 01:12:05.520 --> 01:12:06.720 align:middle line:90% PETER SHOR: Yeah. 01:12:06.720 --> 01:12:12.160 align:middle line:90% It's 1 plus 2 choose 2. 01:12:12.160 --> 01:12:19.760 align:middle line:84% If you have three dots and one dash, you get 3 plus 1 choose 1. 01:12:19.760 --> 01:12:24.420 align:middle line:84% And if you have five dots and zero dashes, you get 5 plus 0 01:12:24.420 --> 01:12:27.160 align:middle line:90% choose 0. 01:12:27.160 --> 01:12:34.640 align:middle line:84% And I want to claim that this is essentially this sum. 01:12:34.640 --> 01:12:35.820 align:middle line:90% So summation. 01:12:35.820 --> 01:12:45.640 align:middle line:90% 01:12:45.640 --> 01:12:54.800 align:middle line:84% I'm going to say an plus i, i equals 0 to something choose. 01:12:54.800 --> 01:13:04.190 align:middle line:90% 01:13:04.190 --> 01:13:06.910 align:middle line:90% This is 3. 01:13:06.910 --> 01:13:12.550 align:middle line:90% This is 3 plus 1, plus 1 is 4. 01:13:12.550 --> 01:13:14.790 align:middle line:90% And this is 1. 01:13:14.790 --> 01:13:20.050 align:middle line:84% And that corresponds to this diagonal 3 plus 4 plus 1. 01:13:20.050 --> 01:13:24.870 align:middle line:90% 01:13:24.870 --> 01:13:32.750 align:middle line:84% So this is a way of proving this formula by counting 01:13:32.750 --> 01:13:35.030 align:middle line:90% things in two different ways. 01:13:35.030 --> 01:13:37.670 align:middle line:84% You count the number of sequences 01:13:37.670 --> 01:13:40.070 align:middle line:90% of dots and dashes of length n. 01:13:40.070 --> 01:13:42.950 align:middle line:84% And first you do it by just saying that the Fibonacci 01:13:42.950 --> 01:13:44.630 align:middle line:90% number. 01:13:44.630 --> 01:13:49.350 align:middle line:84% The second way you do it is by grouping them 01:13:49.350 --> 01:13:55.070 align:middle line:84% by the number of dots and dashes in each one, and twice 01:13:55.070 --> 01:13:57.110 align:middle line:84% the number of dots plus the number of dashes 01:13:57.110 --> 01:14:02.310 align:middle line:84% is equal to the length of the sequence, and summing them up 01:14:02.310 --> 01:14:05.330 align:middle line:90% gets you these diagonals. 01:14:05.330 --> 01:14:11.190 align:middle line:90% 01:14:11.190 --> 01:14:12.930 align:middle line:90% OK, any questions about this? 01:14:12.930 --> 01:14:15.310 align:middle line:90% 01:14:15.310 --> 01:14:17.310 align:middle line:84% AUDIENCE: Can you explain, what's the last thing 01:14:17.310 --> 01:14:18.710 align:middle line:90% that you connected it to? 01:14:18.710 --> 01:14:20.310 align:middle line:90% The last thing you said. 01:14:20.310 --> 01:14:22.322 align:middle line:84% You're connecting it to Pascal's triangle? 01:14:22.322 --> 01:14:24.780 align:middle line:84% PETER SHOR: Yeah, I'm connecting it with Pascal's triangle. 01:14:24.780 --> 01:14:27.910 align:middle line:90% 01:14:27.910 --> 01:14:29.690 align:middle line:84% Actually, I probably should have. 01:14:29.690 --> 01:14:48.930 align:middle line:90% 01:14:48.930 --> 01:14:49.430 align:middle line:90% OK. 01:14:49.430 --> 01:14:51.160 align:middle line:84% I have a formula in here, I believe. 01:14:51.160 --> 01:15:08.030 align:middle line:90% 01:15:08.030 --> 01:15:08.530 align:middle line:90% OK. 01:15:08.530 --> 01:15:32.150 align:middle line:90% 01:15:32.150 --> 01:15:39.710 align:middle line:84% Summation m plus j choose 2j, j equals 0 2m. 01:15:39.710 --> 01:15:51.750 align:middle line:90% 01:15:51.750 --> 01:15:52.250 align:middle line:90% Claim. 01:15:52.250 --> 01:15:54.830 align:middle line:90% 01:15:54.830 --> 01:16:09.540 align:middle line:84% This is the number of ways of arranging 01:16:09.540 --> 01:16:15.910 align:middle line:90% dots and dashes to have length. 01:16:15.910 --> 01:16:19.940 align:middle line:90% 01:16:19.940 --> 01:16:20.880 align:middle line:90% Let's see. 01:16:20.880 --> 01:16:24.540 align:middle line:90% 01:16:24.540 --> 01:16:41.740 align:middle line:84% So I'm going to call this m minus j plus 2j choose 2j, 01:16:41.740 --> 01:16:45.210 align:middle line:90% and to have length m. 01:16:45.210 --> 01:16:59.540 align:middle line:90% 01:16:59.540 --> 01:17:12.000 align:middle line:84% If we have m minus j dashes, 2j dots, you get length m. 01:17:12.000 --> 01:17:18.800 align:middle line:90% 01:17:18.800 --> 01:17:19.300 align:middle line:90% Yeah. 01:17:19.300 --> 01:17:23.820 align:middle line:84% I mean, the dots are length 1, the dashes are length 2, 01:17:23.820 --> 01:17:24.900 align:middle line:90% so this is length 2m. 01:17:24.900 --> 01:17:30.420 align:middle line:90% 01:17:30.420 --> 01:17:50.620 align:middle line:84% And number of ways to arrange m minus j functions, 01:17:50.620 --> 01:18:02.410 align:middle line:84% 2j dots is m minus j plus 2j choose 2j. 01:18:02.410 --> 01:18:05.540 align:middle line:90% 01:18:05.540 --> 01:18:09.550 align:middle line:90% Which is m plus j choose 2j. 01:18:09.550 --> 01:18:13.260 align:middle line:90% 01:18:13.260 --> 01:18:26.580 align:middle line:84% So sum m plus j choose 2j is a number of ways of breaking 2m. 01:18:26.580 --> 01:18:27.080 align:middle line:90% 2m. 01:18:27.080 --> 01:18:29.762 align:middle line:90% 01:18:29.762 --> 01:18:30.660 align:middle line:90% No. 01:18:30.660 --> 01:18:33.820 align:middle line:84% Of arranging dots and dashes to have length 2m total. 01:18:33.820 --> 01:18:40.460 align:middle line:84% So this is equal to Fibonacci number of length 2m, 01:18:40.460 --> 01:18:41.700 align:middle line:90% maybe plus or minus. 01:18:41.700 --> 01:18:45.480 align:middle line:84% I mean, or maybe this is 2m plus 1 or negative 1 or something. 01:18:45.480 --> 01:18:48.690 align:middle line:84% But anyway, this gives you the formula. 01:18:48.690 --> 01:19:17.290 align:middle line:90% 01:19:17.290 --> 01:19:18.370 align:middle line:90% OK. 01:19:18.370 --> 01:19:21.330 align:middle line:84% So next time we're going to derive the formula 01:19:21.330 --> 01:19:27.130 align:middle line:84% for the Catalan number, which, I can't remember if I told you 01:19:27.130 --> 01:19:28.350 align:middle line:90% what it was already or not. 01:19:28.350 --> 01:19:29.170 align:middle line:90% I think I did. 01:19:29.170 --> 01:19:32.390 align:middle line:90% 01:19:32.390 --> 01:19:35.090 align:middle line:90% We derived it using Dyck walks. 01:19:35.090 --> 01:19:36.730 align:middle line:84% Next time, we will derive it using 01:19:36.730 --> 01:19:39.730 align:middle line:84% generating functions and a recurrence for the Catalan 01:19:39.730 --> 01:19:41.050 align:middle line:90% number. 01:19:41.050 --> 01:19:45.530 align:middle line:84% And that will, I think, finish our section 01:19:45.530 --> 01:19:46.750 align:middle line:90% on generating functions. 01:19:46.750 --> 01:19:48.560 align:middle line:90% And I think-- 01:19:48.560 --> 01:19:54.000 align:middle line:90%