4 Generating functions in two variables
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1 4 Generating functions in two variables (Wilf, sections.5.6 and ) Definition. Let a(n, m) (n, m 0) be a function of two integer variables. The 2-variable generating function of a(n, m) is F (x, y) a(n, m)x n y m. There are also two sequences of -variable ordinary power series generating functions, G n (y) a(n, m)y m for each n, n 0, and H m (x) a(n, m)x n for each m, m 0. Clearly, F (x, y) G n (y)x n Examples. (Wilf, section.5) Let a(n, m) H m (x)y m. ( ) n, then it satisfies m a(n, m) a(n, m ) + a(n, m) (4.) unless n m 0. (If n < 0 or m < 0, we set a(n, m) 0.) We shall use this recurrence relation to calculate the 2-variable generating function of a(n, m). Let s fix n and calculate G n (y). By multiplying (4.) by y m and summing over m we obtain a(n, m)y m a(n, m )y m + a(n, m)y m The left-hand side is G n (y) and the second term on the right-hand side is G n (y). The first term on the right-hand side is a(n, m )y m y y m a(n, m )y m a(n, m)y m y a(n, m)y m yg n (y), therefore G n (y) yg n (y) + G n (y) (y + )G n (y). 2
2 Now G 0 (y) hence and F (x, y) a(0, m)y m, since a(0, 0) and a(0, m) 0 if m > 0, G n (y)x n G n (y) (y + ) n (y + ) n x n (xy + x) n x m x xy. Alternatively, we can prove that H m (x) (see question 3 on ( x) m+ problem sheet 8) and use this to calculate F (x, y). In this case it is also possible to give a direct proof that F (x, y) x xy without using G n or H m, but this is an exception. 2. (Wilf, section.6) Let b(n, m) the number of ways of partitioning {, 2,..., n} into exactly m subsets (b(0, 0) 0, b(n, m) 0 if m < 0 or n < 0). These numbers are called the Stirling numbers of the 2nd kind. Given a partition of {, 2,..., n} into m subsets, if n is in a subset on its own, by omitting it we obtain a a partitioning of {, 2,..., n } into m subsets, if n is in a subset of at least 2 elements, by omitting it we obtain a a partitioning of {, 2,..., n } into m subsets and n can be added to any of the m subsets, therefore b(n, m) b(n, m ) + mb(n, m) for n. Let s fix n and let s try to calculate G n (y). By multiplying the recurrence relation by y m and summing over m we obtain b(n, m)y m b(n, m )y m + mb(n, m)y m The left-hand side is G n (y), the first term on the right-hand side is yg n (y) just like in the previous example and mb(n, m)ym yg n (y) by Theorem. (iii), hence G n (y) yg n (y) + yg n (y). G 0 (y) b(0, m)y m, since b(0, 0) and b(0, m) 0 if m > 0, ( ( therefore G n (y) y + d )) n (). This can be used to calculate G n (y) for dy 22
3 any particular n, for example, G (y) y, G 2 (y) y 2 +y, G 3 (y) y 3 +3y 2 +y, G 4 (y) y 4 +6y 3 +7y 2 +y, it does but this does not give an explicit expression for G n (y), F (x, y) or b(n, m). There is a connection with the exponential generating function of powers of n, {n k } egf G k (x)e x, because the polynomials occurring in the exponential generating function of n k satisfy exactly the same recurrence relation. 4. Cards, decks and hands: the unrestricted problem (Wilf, ) Definition. A family F is a set of objects called cards. The weight is a function w : F Z + associating a positive integer to each card. (In applications the weight is usually a measure of size or value.) A deck is a set of cards of the same weight, D r {x w(x) r} is the set of cards of weight r for each positive integer r, and d r D r is the number of cards of weight r. We shall assume that d r is finite for each r, but F may be infinite. A hand is a finite collection of cards with possible repetitions, so the same card may be used several times. The weight of a hand is the sum of the weights of the cards in it. h(n, k) is the number of hands of weight n consisting of k cards. h(n) h(n, k) is the total number of hands of weight n. The 2-variable hand k0 enumerator of F is H(x, y) k0 h(n, k)x n y k, and the -variable hand enumerator is H(x) h(n)x n H(x, ). Note: Wilf uses prefab instead of family in this context and reserves the term family for the labelled cards used in the rest of Chapter 4. Lemma 4. (Wilf, p. 93, Fundamental lemma of unlabelled counting) Let F and F 2 be disjoint families of cards with 2-variable hand enumerators H (x, y) and H 2 (x, y), respectively. Let F F F 2 and let H(x, y) be its 2-variable hand enumerator. Then H(x, y) H (x, y)h 2 (x, y). 23
4 Proof. Key idea: Given a hand of weight n consisting of k cards from F and a hand of weight n 2 consisting of k 2 cards from F 2, their union is a hand of weight n + n 2 consisting of k + k 2 cards from F. Conversely, any hand of weight n consisting of k cards from F can be split uniquely into the union of a hand containing cards from F and a hand containing cards from F 2, and if these hands have parameters n, k and n 2, k 2, respectively, then n n + n 2 and k k + k 2. Therefore h(n, k) h(n, k )h(n 2, k 2 ). Hence H(x, y) k0 n +n 2 n k +k 2 k n 0,n 2 0 k 0,k 2 0 h(n, k)x n y k k0 n +n 2 n k +k 2 k n 0,n 2 0 k 0,k 2 0 k0 n +n 2 n k +k 2 k n 0,n 2 0 k 0,k 2 0 n 0 k 0 h(n, k )x n y k H (x, y)h 2 (x, y). h(n, k )h(n 2, k 2 )x n y k h(n, k )x n y k h(n 2, k 2 )x n 2 y k 2 n 2 0 k 2 0 h(n 2, k 2 )x n 2 y k 2 Theorem 4.2 (Wilf, Theorem 3.4.) Let F be a family of cards and let d r be the number of cards of weight r in F, as usual. Then the hand enumerators of F are H(x, y) r ( x r y) dr and H(x) r ( x r ) dr. Proof. It is sufficient to prove the theorem for the 2-variable hand enumerator H(x, y), by substituting y into it we obtain the formula for the -variable hand enumerator H(x). The proof consists of three steps, proving the theorem gradually for more and more general cases. Step. Assume that F consists of a single card of weight s. Then d s 24
5 and d r 0 for r s. The only possibly hand of k cards consists of k copies of the only card and therefore has weight ks, so { if n ks, h(n, k) 0 otherwise. Hence H(x, y) x ks y k k0 (x s y) k k0 x s y. Step 2. Assume that F is finite. We shall use induction on q F. If q 0, then F, h(0, 0) and h(n, k) 0 if (n, k) (0, 0), so H(x, y). d r 0 for all r, therefore, too, so the ( x r dr y) r theorem holds in this case. Let now q and let s assume that the theorem holds for families consisting of q cards. Let F be a family consisting of q cards. Let s choose an arbitrary card from F, let F be the family consisting of this single card and let F 2 be the family consisting of all the other cards in F. Let s be the weight of the chosen card, and let d s d s and d r d r if r s. d r is exactly the number of cards of weight r in F 2. By Step, the 2-variable hand enumerator of F is x s y. F 2 contains q cards, so by the induction hypothesis, its hand enumerator is. ( x r y) d r r Hence by Lemma 4., the 2 variable hand enumerator of F F F 2 is x s y r ( x r y) d r r ( x r y) dr, since d s d s and d r d r if r s. Therefore the theorem holds for families consisting of q cards and by induction, it holds for all finite families. Step 3. Now F can be arbitrary, possibly infinite. Let s fix n and k. A hand of weight n can only contain cards of weight at most n, therefore h(n, k) is the coefficient of x n y k in the power series expansion of the 2-variable hand n enumerator of D D 2... D n, which is by Step 2. ( x r dr y) rn+ ( x r y) dr has constant term, all the other terms have degree at least n + in x, therefore h(n, k) is also equal to the coefficient of x n y k in 25 r
6 the power series expansion of n ( x r y) dr r rn+ ( x r y) dr ( x r y). dr r As this is true for all n, k, we have H(x, y), as claimed. As ( x r dr y) r we remarked at the beginning, the result for the -variable hand enumerator follows by substituting y. Examples: See pages 4 of teaching/math3900/cards.pdf and uk/~gm/teaching/math3900/excursion_ticket.pdf. 4.2 Cards, decks and hands: the restricted problem The family, cards, weight and decks are defined as before, but now there is a set of non-negative integers W such that 0 W and the multiplicity of each card in a hand must be an element of W. Let h(n, k) be the the number of hands of weight n consisting of k cards which satisfy the restriction on the multiplicities and let h(n) h(n, k) be the total number of hands of weight n satisfying the restriction. k0 The restricted 2-variable hand enumerator of F is H(x, y) k0 h(n, k)x n y k, and the restricted -variable hand enumerator is H(x) h(n)x n H(x, ). Theorem 4.3 (Not examinable) (Wilf, Theorem 3.4.2) Let F be a family of cards and let d r be the number of cards of weight r in F, as usual. Let W be a subset of non-negative integers containing 0. Then the restricted hand enumerators of F are ( ) dr ( ) dr H(x, y) (x r y) m and H(x) x mr. r 26 r
7 Sketch of proof. Prove the analogue of Lemma 4. for the restricted case and then follow the steps of the proof of Theorem 4.2. Note that for a family consisting of a single card of weight s in Step, H(x, y) (x s y) m by direct calculation. Remarks.. If W Z 0, (x r y) m (x r y) m x r y and x mr x mr, so we obtain Theorem 4.2 as a special case. xr 2. If the restriction is that each card can be used at most l times for a fixed l, then W {0,, 2,..., l} and (x l l y) m (x r y) m (xr y) l+. x r y Examples: See pages 5 6 of teaching/math3900/cards.pdf. 27
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