Take the 2-minute tour ×  Let $S_{n,r}$ denote the Stirling
number of the second kind. Define
$A_{n,r}:=\frac{\binom{n+r-1}{n}(n+r)!}{S_{n+r,r}r!}$. I want to
prove: $A_{n,1}\ge A_{n,2}\ge..\ge A_{n,r}\ge \lim_{r\to\infty}
A_{n,r}=2^n$.  I can prove this for some small particular $r$ or
for some small particular $n$. And checked this numerically (by
computer) for a big scope of $(n,r)$.  I do not see any
immediate induction or recursion. :( (As I never learned
combinatorics, the only thing I know about $S_{n,r}$ is wiki-
page.) Probably this is some well known property? Any advice,
reference is welcome.  share|improve this question  1 Answer 1
up vote 6 down vote accepted  ($A_{n,r}$ has a combinatorial
interpretation: the denominator is the number of ways of
partitioning the numbers $\{1,\ldots,n+r\}$ into $r$ blocks,
where the list of blocks is ordered but the numbers within each
block are unordered, and the numerator is the same count, except
that the numbers within each block are now also ordered. $A_{n,r