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

As a natural generalization of permutations, transversals of Young diagrams play an important role in the study of pattern avoiding permutations. Let $T_{\lambda }\left(\tau \right)$ and ${\mathrm{ST}}_{\lambda }\left(\tau \right)$ denote the set of τ-avoiding transversals and τ-avoiding symmetric transversals of a Young diagram λ, 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, 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 $T_{\lambda }(12\cdots k\tau )$ (resp. ${\mathrm{ST}}_{\lambda }(12\cdots k\tau )$) and $T_{\lambda }(k\cdots 21\tau )$ (resp. ${\mathrm{ST}}_{\lambda }(k\cdots 21\tau )$) for any Young diagram λ and any permutation τ of $\{k+1,k+2,\dots ,k+m\}$ with $k,m\ge 1$. Our results are refinements of the result of Backelin-West-Xin which states that $\left|T_{\lambda }\right(12\cdots k\tau \left)\right|=\left|T_{\lambda }\right(k\cdots 21\tau \left)\right|$ and the result of Bousquet-Mélou and Steingrímsson which states that $\left|{\mathrm{ST}}_{\lambda }\right(12\cdots k\tau \left)\right|=\left|{\mathrm{ST}}_{\lambda }\right(k\cdots 21\tau \left)\right|$. As applications, we are able to

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  confirm a recent conjecture posed by Yan-Wang-Zhou which asserts that the peak set is equidistributed over $12\cdots k\tau$-avoiding involutions and $k\cdots 21\tau$-avoiding involutions;

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  prove that alternating involutions avoiding the pattern $12\cdots k\tau$ are equinumerous with alternating involutions avoiding the pattern $k\cdots 21\tau$, paralleling the result of Backelin-West-Xin for permutations, the result of Bousquet-Mélou and Steingrímsson for involutions, and the result of Yan for alternating permutations.