Asymptotics for the number of standard tableaux of skew shape and for weighted lozenge tilings

TL;DR

This paper establishes asymptotic formulas for the count of standard Young tableaux of skew shape, linking combinatorial enumeration to weighted lozenge tilings and their limit shapes.

Contribution

It proves and generalizes a conjecture relating skew shape tableaux asymptotics to weighted lozenge tilings using a variational principle.

Findings

Asymptotic formulas for skew shape tableaux counts

Connection between tableaux asymptotics and lozenge tilings

Validation of the stable limit shape under scaling

Abstract

We prove and generalize a conjecture in arXiv:1610.0474(4) about the asymptotics of $\frac{1}{\sqrt{n!}} f^{\lambda/\mu}$, where $f^{\lambda/\mu}$ is the number of standard Young tableaux of skew shape $\lambda/\mu$ which have stable limit shape under the $1/\sqrt{n}$ scaling. The proof is based on the variational principle on the partition function of certain weighted lozenge tilings.