An Asymptotic Version of a Theorem of Knuth

Abstract

Let $S(d,N)$ denote the number of permutations in the symmetric group on $[N]$ which have no decreasing subsequence of length $d+1.$ We prove that $S(d,dn)$ is asymptotically equal to the number of standard Young tableaux of rectangular shape $R(d,2n)$ in the limit $n\to \infty ,$ with $d$ fixed.

9 pages