Question 8
For integers $m, n \geq 1$, let $A(n, m)$ be the number of sequences $(a_{1}, \cdots, a_{n m})$ of integers satisfying the following two properties:
(a) Each integer $k$ with $1 \leq k \leq n$ occurs exactly $m$ times in the sequence $(a_{1}, \cdots, a_{n m})$.
(b) If $i, j$, and $k$ are integers such that $1 \leq i \leq n m$ and $1 \leq j \leq k \leq n$, then $j$ occurs in the sequence $(a_{1}, \cdots, a_{i})$ at least as many times as $k$ does.
For example, if $n=2$ and $m=5$, a possible sequence is $(a_{1}, \cdots, a_{10})=(1,1,2,1,2,2,1,2,1,2)$. On the other hand, the sequence $(a_{1}, \cdots, a_{10})=(1,2,1,2,2,1,1,1,2,2)$ does not satisfy property (2) for $i=5, j=1$, and $k=2$.
Prove that $A(n, m)=A(m, n)$.
(a) Each integer $k$ with $1 \leq k \leq n$ occurs exactly $m$ times in the sequence $(a_{1}, \cdots, a_{n m})$.
(b) If $i, j$, and $k$ are integers such that $1 \leq i \leq n m$ and $1 \leq j \leq k \leq n$, then $j$ occurs in the sequence $(a_{1}, \cdots, a_{i})$ at least as many times as $k$ does.
For example, if $n=2$ and $m=5$, a possible sequence is $(a_{1}, \cdots, a_{10})=(1,1,2,1,2,2,1,2,1,2)$. On the other hand, the sequence $(a_{1}, \cdots, a_{10})=(1,2,1,2,2,1,1,1,2,2)$ does not satisfy property (2) for $i=5, j=1$, and $k=2$.
Prove that $A(n, m)=A(m, n)$.