The asymptotic number of $12..d$-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 $i=1..n$, 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.