The asymptotic number of $\mathrm{12..}d$-Avoiding Words with $r$ occurrences of each letter $1,2,...,n$

Abstract

Following Ekhad and Zeilberger (The Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, Dec 5 2014; see also arXiv:1412.2035), we study the asymptotics for large $n$ of the number $A_{d,r}(n)$ of words of length $rn$ having $r$ letters $i$ for $i=\mathrm{1..}n$, and having no increasing subsequence of length $d$. We prove an asymptotic formula conjectured by these authors, and we give explicitly the multiplicative constant appearing in the result, answering a question they asked. These two results should make the OEIS richer by 100+25=125 dollars. In the case $r=1$ we recover Regev's result for permutations. Our proof goes as follows: expressing $A_{d,r}(n)$ as a sum over tableaux via the RSK correspondence, we show that the only tableaux contributing to the sum are "almost" rectangular (in the scale $\sqrt{n}$). This relies on asymptotic estimates for the Kotska numbers $K_{\lambda ,r^n}$ when $\lambda$ has a fixed number of parts. Contrarily to the case $r=1$ where these numbers are given by the hook-length formula, we don't have closed form expressions here, so to get our asymptotic estimates we rely on more delicate computations, via the Jacobi-Trudi identity and saddle-point estimates.

v2: we correct some typos (including an unfortunate exchange of r and n in the abstract), improve the exposition, and correct the saddle-point set-up to avoid problems with multiple singularities

References in corpus (1)

• The Generating Functions Enumerating 12..d-Avoiding Words with r occurrences of each of 1,2, ... , n are D-finite for all d and all r