Séminaire Lotharingien de Combinatoire, 93B.79 (2025), 12 pp.
Sergi Elizalde
A Generalized Lalanne-Kreweras Involution for Rectangular Tableaux
Abstract.
We give a bijective proof of the symmetry of the distribution of the number of descents over standard Young tableaux of any given rectangular shape.
Our bijection can be viewed as a generalization of the Lalanne-Kreweras involution on Dyck paths (corresponding to the 2-row case), which proves the symmetry of the Narayana numbers.
Using certain arrow encodings, we also describe new statistic-preserving involutions on rectangular tableaux, which give simpler proofs of known results, and allow us to prove a conjecture of Sulanke about the distribution of certain statistics on three-row tableaux.
Finally, our construction provides a bijective proof of the symmetry of the number of descents on so-called canon permutations, and suggests a possible notion of rowmotion on standard Young tableaux of rectangular shape.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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