Problem 2. Intk (a) Show that counts the number of integer partitions whose Ferrers diagram fits k inside an n xk box by
Posted: Thu May 12, 2022 8:58 am
- Problem 2 Intk A Show That Counts The Number Of Integer Partitions Whose Ferrers Diagram Fits K Inside An N Xk Box By 1 (323.99 KiB) Viewed 26 times
Problem 2. Intk (a) Show that counts the number of integer partitions whose Ferrers diagram fits k inside an n xk box by the size of the partition. More precisely, for 1 = (41, 42, ..., ts), we say that de B(n,k) if X1 <n and s <k and we write 11 = 11 + 12+...+ 1s. Then show that Σ qΑΙ. LEB(n,k) For example, there are 10 partitions that fit in a 2 x 3 box: Ε. - . and () +3 2 = 1+q + 2q2 + 2q2 + 204 + q*+q. 9 Hint: show such paths satisfy the same recurrence as in Problem 1(b). n (b) Use part (c) to give a combinatorial explanation for why = [] = [n=1] n (c) Use Durfee squares to generalize the identity (mit") i-o (1) (,^.) to i=0 n m m+n n n [7], [-=-1; () 22 i=0