2023/06/07 by Robin D. P. Zhou, Zhou, Robin D. P., Sherry H.F. Yan +1 · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2306.04382
openalex publication_date 2023/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
As a natural generalization of permutations, transversals of Young diagrams play an important role in the study of pattern avoiding permutations. Let Tλ(τ) and STλ(τ) 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, 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 λ/μ for any skew diagram λ/μ. The equidistribution enables us to show that the peak set is equidistributed over Tλ(12⋯ kτ) (resp. STλ(12⋯ k τ)) and Tλ(k⋯ 21τ) (resp. STλ(k⋯ 21τ)) for any Young diagram λ and any permutation τ of \k+1, k+2, …, k+m\ with k,m≥ 1. Our results are refinements of the result of Backelin-West-Xin which states that |Tλ(12⋯ kτ)|=|Tλ(k⋯ 21 τ)| and the result of Bousquet-Mélou and Steingrímsson which states that |STλ(12⋯ k τ)|=|STλ(k⋯ 21 τ)|.