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Unformatted text preview: MAD 6206 Combinatorics I, Fall 2011 CRN: 87401 Assignment 5 Question 1 . (Eulerian numbers and their generating function) Recall that the the Eulerian numbers ( A ( n,k ) = A n,k ) are defined as the number of permutations on [ n ] with k runs. We proved in class that they satisfy the following recurrence A ( n,k ) = ( n k + 1) A ( n 1 ,k 1) + kA ( n 1 ,k ) , n 1 ,k 1 , with boundary conditions A ( n, 0) = ,n , A (0 ,k ) = ,k . Let G ( z,x ) = summationdisplay n summationdisplay k A ( n,k ) z n x k n ! . (a) Prove that (1 zx ) z G ( z,x ) x (1 x ) x G ( z,x ) = xG ( z,x ) . (b) To solve the pde, we change the variable z as follows. Let u = x exp( z (1 x )) (or z = 1 1 x log( u/x )), and by replacing z in G ( z,x ), we obtain a function Y ( u,x ) so that Y ( u,x ) = G ( z,x ). Prove that x Y ( u,x ) = Y/ (1 x ) , and hence solve for G ( z,x ). (c) Prove that A ( n,k ) = k summationdisplay j =0 ( 1) j parenleftbigg n + 1 j parenrightbigg ( k j ) n . Note. The end of the assignment contains a table of Eulerian numbers. Question 2 . (Eulerian numbers, Stirling numbers, and finite difference) Use the recurrence in Q1 and induction to prove that for any...
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- Spring '11