hw7sol - Math 146: Algebraic Combinatorics Homework 7...

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Unformatted text preview: Math 146: Algebraic Combinatorics Homework 7 Solutions 3.11 Let S , T be fixed sets of positive integers. Let f ( n ; S , T ) be the number of partitions of [ n ] whose class sizes all lie in S and whose number of classes lies in T . Show that { f ( n ; S , T ) } n has egf e T ( e S ( x )) , where e S ( x ) = s S x s s ! . Solution. This is a direct application of the exponential formula (Theorem 3.11). More pre- cisely we use Corollary 3.13. Define the the deck D i in the following way: D i will contain zero cards if i / S and one card if i S . In the latter case the card will correspond to the partition of [ i ] that places everything in one class: { 1 , 2 ,..., i } . Thus the deck enumerator is D ( x ) = n d n x n n ! = s S x s s ! = e S ( x ) . By making hands that draw from the decks D i we ensure the partition of [ n ] has class sizes all in S . To ensure the number of classes (that is, the number of cards in the hand) is in T we apply the exponential formula with the condition that h ( n , k ) = 0 if k / T and h ( n , k ) = x n n ! D ( x ) k k ! if k T , where h ( n , k ) is the number of partitions of [ n ] with class sizes in S with k classes where k T . This is exactly the coefficient of x n n ! in e T ( e S ( x )) . 3.12 Fix k > 0. Let f ( n , k ) be the number of permutations of n letters whose longest cycle has length k . Find the egf of { f ( n , k ) } n for k fixed. Solution. Let the deck D i have a card for each cyclic permutation of i letters. Thus | D i | = ( i- 1 ) !. If we restrict to cyclic permutations of length at most k , then the deck enumerator is x + x 2 2 + + x k k . Let F ( n , k ) be the number of permutations of n letters whose cycles have lengths at most k . By the exponential formula F has the egf exp ( x + + x k / k ) . But f ( n , k ) counts those whose longest cycle has length exactly equal to k . So if k 1 then f ( n , k ) = F ( n , k )- F ( n , k- 1 ) , and the required egf is e x + + x k- 1 k- 1 e x k k- 1 . GK7.1. The aim of this exercise is yet another proof of the exponential formula. Let A ( x ) = k = 1 a k x k be the exponential generating function for some types of animals of positive sizes. Youre suppose to prove everything in this exercise directly, rather than by using theorems in chapter...
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This note was uploaded on 05/19/2010 for the course MATH mat 146 taught by Professor Gregkuperberg during the Spring '10 term at UC Davis.

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hw7sol - Math 146: Algebraic Combinatorics Homework 7...

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