MT5821 Advaced Combiatorics 1 Coutig subsets I this sectio, we cout the subsets of a -elemet set. The coutig umbers are the biomial coefficiets, familiar objects but there are some ew thigs to say about them. These will lead o to topics later i the otes. We begi with a simple observatio. Propositio 1.1 The umber of subsets of a -elemet set is 2. Proof A subset is specified by sayig which elemets of the set it cotais. For each elemet, we have the choice of icludig it or leavig it out. The result of biary choices is that there are 2 possible subsets. Alteratively, we ca match up the subsets of {1,...,} with the -tuples of zeros ad oes (of which there are 2 ): the subset A is matched to the -tuple e, where { 1 if i A, e i 0 if i / A. We will sometimes abbreviate -elemet set to -set. 1.1 Biomial coefficiets Defiitio The biomial coefficiet ( ) is the umber of -subsets of a -set. Propositio 1.2 ( ) ( 1)( 2) ( + 1), ( 1)( 2) 1 the product of descedig itegers startig at divided by a similar product startig at. 1
Proof We choose a -subset of the -set by picig its elemets oe at a time: there are choices for the first, 1 for the secod,..., ad + 1 for the th. However, we have over-couted, sice the same elemets i a differet order mae up the same subset. The umber of orders i which elemets ca be chose is ( 1) 1, ad we have to divide by this to get the aswer. Immediately from the defiitio ad the result of the precedig sectio we have ( ) Propositio 1.3 2. 0 Defiitio The factorial fuctio is defied by! ( 1)( 2) 1, the product of the itegers from 1 to. Usig this, we ca write the biomial coefficiet more succictly as ( )!!( )!, sice the ( )! i the deomiator cacels the factors i! from dow to 1, leavig oly those i the umerator i the propositio. However, this ( ice) compact formula is ot always the best i practice. Cosider evaluatig. The formula i the Propositio gives 200 2 ( ) 200 2 200 199 2 1 19900, while the oe usig factorials gives ( ) 200 200! 2 2!198! 78865786736479050355236321393218506229513597768717326329474253324435 2 19815524305648002601818171204326257846611456725808373449616646563 The what do you do? ( ) You probably thi of the expressio as maig sese whe ad are positive itegers ad <. However, the formula i the Propositio wors fie i other cases too: 2
( ) 1 sice a -set has a uique subset with o elemets (the empty 0 ( ) set). Similarly 1. If >, the the formula gives 0, which is correct sice there are o - subsets i this case. 1.2 The Biomial theorem The reaso for the ame biomial coefficiets is that these umbers occur as coefficiets i the biomial theorem: Theorem 1.4 (Biomial Theorem) For ay o-egative iteger, ( ) (x + y) x y. 0 How is such a theorem proved? We have two ways of thiig about the coefficiets (as coutig somethig, or as a formula), so there are potetially two ways to prove the theorm. It is istructive to compare them. First proof The formula suggests a proof by iductio. It is clearly true whe 0. So let us assume it for a value ad prove it for + 1. We have (x + y) +1 (x + y)(x + y) (x + y) ( 0x y ). O the right, expadig the bracets will give us a sum of terms, each of which will have the sum of the expoets of x ad y equal to + 1. Cosider the term x +1 y with 1. This will be the sum of two cotributios, x ( ) x y ad y ( 1 ) x +1 y 1. So we eed: Lemma 1.5 ( ) ( ) + 1 ( + 1 ). 3
This ca be proved usig the formula for the biomial coefficiet. The terms for 0 ad + 1 each oly come from oe place i the expressio, ad the coefficiet of each of them is 1, as required. So the iductio step is complete. Secod proof Cosider (x + y) (x + y) (x + y) ( factors). Expadig all the bracets, we get a sum of terms, each a product of a power of x ad a power of y. The term x y comes( by) selectig y from of the bracets ad x from the remaiig oes. There are ways to choose the bracets from which to select y. The result follows. I this case the coutig proof is much simpler! Very ofte we will have two proofs of a result, oe by coutig ad oe aalytic; judge for yourself which you prefer. 1.3 Biomial coefficiet idetities There is a huge idustry ivolvig provig idetities coectig biomial coefficiets. Here are a few. I have give hits for coutig proofs i some cases. You should write out detailed proofs of some of these, ad compare them to proofs usig the formulae. ( ) ( ) Propositio 1.6. This says that you ca match up the -subsets of a -set with the ( )- subsets; match each set with its complemet. Propositio 1.7 ( ) ( ) + 1 ( + 1 We saw this earlier, as a lemma i the proof of the Biomial Theorem. Suppose you ( have ) to pic a team of from a class of + 1 pupils. You ca do this i + 1 ways. Alteratively, focus attetio o oe of the childre, say A. There ). 4
( ) are teams icludig A, sice the remaider of the team must be piced 1 ( ) from the remaiig pupils. Similarly there are teams ot icludig A. The result follows. This is the basic property which is used to costruct ( ) Pascal s Triagle, a triagular array i which the th etry i the th row is. (Both row umbers, ad etry umbers i each row, start from 0.) The rule is: each etry of the table is the sum of the two etries i the row above, to the left ad the right of the positio we are looig at. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 Pascal s triagle was ot iveted by Pascal; it was ow i Chia some ceturies earlier, ad possibly brought to Europe by missioaries. Figure 1 shows Chu Shi-Chieh s versio, from his boo Ssu Yua Yü Chie, dated 1303. ( ) ( ) 1 Propositio 1.8. 1 Cosider choosig a team of from a class of pupils, ad the choosig a captai for the team. We could alteratively choose the captai first, ad the pic the remaiig 1 members of the team. ( )( ) ( ) m m + Propositio 1.9, where the sum is over all values of i i i for which the biomial coefficiets mae sese (that is, 0 i m ad 0 i. Pic a team of players from a class of m girls ad boys. This result is ow as the Vadermode covolutio. 5
1.4 Geeratig fuctios Figure 1: Chu Shi-Chieh s Triagle Our formulatio of the Biomial Theorem is slightly wasteful: sice the expoets of x ad y sum to, oe ca be deduced from the other. It will be more useful to state the theorem i this form: Theorem 1.10 (Biomial Theorem) For a o-egative iteger, ( ) (1 + x) x. 0 The expressio o the right is what is ow as the geeratig fuctio for the biomial coefficiets. I geeral, if we have a sequece of umbers a 0,a 1,a 2,..., the geeratig fuctio for the sequece is a x, so that the coefficiet of the th power of the idetermiate is the th umber i the sequece. The sum is over all relevat values. I the case of the biomial theorem, it does t matter if we do t stop i time, sice the biomial coefficiets vaish for > ; so eve if we thi of the sum as over all atural umbers, there are oly fiitely may op-zero terms. 6 0
This form is already useful. Here is a example. A special case of the Vadermode covolutio (taig m ad usig Propositio 1.6) is 0 ( ) 2 ( ) 2. If we put alteratig sigs i the sum, somethig differet happes: Propositio 1.11 0( 1) ( ) 2 Proof We start with the idetity { 0 ) if is odd, if is eve.. ( 1) /2( /2 (1 x) (1 + x) (1 x 2 ). Now we calculate the coefficiet of x o both sides. O the left, taig the coefficiet of x i the first factor ad x i the secod, we obtai 0( 1) ( )( ) 0( 1) ( ) 2. O the right, we have oly eve powers of x, so the coefficiet is zero if is odd. If is eve, the required coefficiet is obtaied from the biomial expasio of (1 x 2 ). ( ) The biomial coefficiet has two parameters. So oe could as about the geeratig fuctio for the parameter, amely ( ) y 0 for fixed, or eve the bivariate geeratig fuctio, where we have oe idetermiate for each parameter. Ca we calculate these? We have 0 0 ( ) x y (1 + x) y 0 7 1 1 (1 + x)y.
Puttig x 1 says that we do t care about the value of, so we get the geeratig fuctio for the total umber of subsets of a -set; what we fid is i agreemet with Propositio 1.1. Cotiuig the calculatio above, ( ) x y 0 0 1 1 2y 2 y, 0 1 1 y 1 1 (y/(1 y)x) y 0 (1 y) +1 x. Now we ca equate the coefficiets of x o the two sides to deduce 0 ( ) y y (1 y) +1. This is the required geeratig fuctio. Note that the power series for the right-had side begis with the term y ; this is because the biomial coefficiets vaish for <. We ca deduce from this a form of the Biomial Theorem for egative expoet. Divide by y ad put m + 1, x y, ad l m + 1 : ( ) (1 + x) m ( x) m+1 m + 1 ( 1) m ( m + 1) 1 ( 1) m+1 x m+1 ( m)( m 1) ( m l + 1) x l ( ) l! m x l. l ( ) m Here we have defied the biomial coefficiet by the same rule as for l the usual biomial coefficiets. Ideed, we ca go further: 8
Defiitio Let be a o-egative iteger ad a ay real umber. The let ( ) a a(a 1) (a l + 1). l! ( ) a Theorem 1.12 (Geeral biomial theorem) (1 + x) a x. 0 This is, at preset, a theorem of aalysis, sice we eed aalysis to defie the fuctio (1 + x) a for real umbers a ad to calculate its Taylor series. However, we will come bac to this result later. 1.5 Uimodality ad estimates The biomial coefficiets form a uimodal sequece: ( ) Propositio 1.13 For fixed, the biomial coefficets icrease with for < /2, ad decrease ( as) icreases for > /2. If is eve, the the cetral biomial coefficiet is the largest; if is odd, the the two o either side /2 ( ) ( ) of the cetre (that is, ad ) are equal ad are larger ( 1)/2 ( + 1)/2 tha all the others. Proof From the formula, we see that ( ) ( ). + 1 + 1 ( ) ( ) So is greater tha, equal to, or less tha accordig as is greater + 1 tha, equal to, or less tha + 1; that is, accordig as is less tha, equal to, or greater tha ( 1)/2. The result follows from this. We will meet other sequeces of combiatorial umbers which also have the uimodal property. I cases where we do t have a formula for the umbers i questio, we caot use such a simple argumet. Istead, we will develop a test for uimodality, based o properties of the geeratig fuctio. As we have see, the + 1 biomial coefficiets sum to 2. So we have: 9
Propositio 1.14 If is eve, the 2 + 1 ( ) 2. /2 A similar result holds for odd. I fact, the precise asymptotics of the cetral biomial coefficiet ca be calculated usig Stirlig s formula, which we will meet later. Its value is close to c 2 / for a suitable costat c. 1.6 Matrices By left-aligig the etries of Pascal s Triagle, ( ) we ca covert it ito a ifiite i lower-triagular matrix B, with (i, j) etry. (Note that this correctly gives j B i j 0 for i < j.) Multiplicatio of lower-triagular matrices is possible; o ifiite sums are ivolved. For if A ad B are lower-triagular, the the (i, j) etry of AB is A i B j, ad the terms i the sum are o-zero oly if j i.) This also shows that the product AB is lower-traigular. This meas that, i practice, if we multiply the top left-had corers of lower-triagular matrices, we get the top left-had corer of the product. For example, 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 2 1 0 1 3 1 1 1 2 1 0 1 3 3 1 This is a special case of somethig much more geeral: 0 0 1 0 0 0 0 1 Theorem 1.15 Let B be the matrix of biomial ( ) coefficiets defied above, ad let i B be the matrix with (i, j) etry ( 1) i j. The B is the iverse of B: that j is, BB B B I. Proof The set of all real polyomials is a real vector space, with basis {1,x,x 2,x 3,...}. Aother basis is {1,x + 1,(x + 1) 2,(x + 1) 3,...}. Now the Biomial Theorem shows that B is the trasitio matrix expressig the vectors of the secod basis i terms of the first. So its iverse is the trasitio matrix i the reverse directio. If y x+1, the x y 1, ad so the coefficiets of the iverse are foud from the expasios of powers of y 1 i terms of y, givig the result. 10.
I fact this ca ( ) be exteded. For ay real umber c, let B(c) be the matrix with i (i, j) etry c i j. The B(1) B ad B( 1) B. More geerally, B(c) is the j trasitio matrix from the basis {1,x,x 2,x 3,...} to the basis {1,x + c,(x + c) 2,(x + c) 3,...}. So arguig as above we fid that B(c 1 )B(c 2 ) B(c 1 + c 2 ). I other words, the map c B(c) is a homomorphism from the additive group of real umbers to the multiplicative group of upper triagular matrices with diagoal etries 1. Exercises 1.1. (a) I Vacouver i 1984, I saw a Dutch pacae house advertised a thousad ad oe combiatios of toppigs. What do you deduce? (b) More recetly McDoald s offered a meal deal with a choice from eight compoets of your meal, ad advertised 40,312 combiatios. What do you deduce? 1.2. Prove the Vadermode covolutio usig geeratig fuctios. 1.3. Show that the umber of subsets of {1,...,} of eve cardiality is 2 1. Calculate the umber of subsets of cardiality divisible by 4. Your aswer should deped o the cogruece of mod 8. (Hit: calculate (1 + i).) 1.4. Write dow Pascal s triagle mod 2; that is, record oly whether each etry is odd or eve. You should fid that the triagle has a fractal structure; ca you explai why? 1.5. Lucas Theorem gives a expressio for the cogruece of biomial coefficiets modulo a prime. Let p be prime, ad let a a r p r + a r 1 p r 1 + + a 1 p + a 0 ad b b r p r +b r 1 p r 1 + +b 1 p+b 0 be the expressios for a ad b i base p, so that 0 a i,b i p 1 for all i Prove that ( ) ( )( ) ( )( ) a ar ar 1 a1 a0 (mod p). b b r b r 1 b 1 b 0 11
Hit: Show first that, if a ps + c ad b pt + d with 0 c,d p 1, the ( ) ( )( ) a s c (mod p). b t d Remar: This exercise may help you with the precedig oe. ( ) 1.6. The multiomial coefficiet, where 1,..., r are o-egative 1,..., r itegers with sum, is defied to be! 1! r!. ( ) ( ) (So, i this otatio,.), Prove the Multiomial Theorem: (x 1 + x 2 + + x r ) 1 + 2 + + r ( 1, 2,..., r ) x 1 1 x 2 2 x r r. 1.7. Costruct a bijectio betwee the set of all -elemet subsets of {1, 2,..., } cotaiig o two cosecutive elemets, ad the set of all -elemet subsets of {1,2,..., + 1}. Hece show that the umber of such subsets is ( +1). I the UK Natioal Lottery, six umbers are chose radomly from the set {1,...,49}. What is the probability that the selectio cotais o two cosecutive umbers? 1.8. Prove that the geeratig fuctio for the cetral biomial coefficiets is ( ) 2 x (1 4x) 1/2, ad deduce that 0 1 ( 2 )( 2( ) ) 4. [Note: Fidig a coutig proof of this idetity is quite challegig!] 12