Equidistribution of set-valued statistics on standard Young tableaux and transversals

Abstract

As a natural generalization of permutations, transversals of Young diagrams play an important role in the study of pattern avoiding permutations. Let $\mathcal{T}_{\lambda}(\tau)$ and $\mathcal{ST}_{\lambda}(\tau)$ denote the set of $\tau$-avoiding transversals and $\tau$-avoiding symmetric transversals of a Young diagram $\lambda$, respectively. In this paper, we are mainly concerned with the distribution of the peak set and the valley set on standard Young tableaux and pattern avoiding transversals. In particular, by introducing Knuth transformations on standard Young tableaux, we prove that the peak set and the valley set are equidistributed on the standard Young tableaux of shape $\lambda/\mu$ for any skew diagram $\lambda/\mu$. The equidistribution enables us to show that the peak set is equidistributed over $\mathcal{T}_{\lambda}(12\cdots k\tau)$ (resp. $\mathcal{ST}_{\lambda}(12\cdots k \tau)$) and $\mathcal{T}_{\lambda}(k\cdots 21\tau) $ (resp. $\mathcal{ST}_{\lambda}(k\cdots 21\tau)$) for any Young diagram $\lambda$ and any permutation $\tau$ of $\{k+1, k+2, \ldots, k+m\}$ with $k,m\geq 1$. Our results are refinements of the result of Backelin-West-Xin which states that $|\mathcal{T}_{\lambda}(12\cdots k\tau)|=|\mathcal{T}_{\lambda}(k\cdots 21 \tau)|$ and the result of Bousquet-M\'elou and Steingr\'imsson which states that $|\mathcal{ST}_{\lambda}(12\cdots k \tau)|=|\mathcal{ST}_{\lambda}(k\cdots 21 \tau)|$.