An Asymptotic Version of a Theorem of Knuth

TL;DR

This paper establishes an asymptotic equivalence between the number of permutations with bounded decreasing subsequences and the count of certain Young tableaux, extending combinatorial understanding in asymptotic regimes.

Contribution

It proves an asymptotic version of Knuth's theorem relating permutations and Young tableaux for fixed dimensions as size grows.

Findings

Asymptotic equality between permutation counts and Young tableaux counts.

Extension of Knuth's theorem to large size limits.

Provides new insights into the structure of permutations with restricted subsequences.

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.