Séminaire Lotharingien de Combinatoire, 91B.80 (2024), 12 pp.
Alexander Lazar and Svante Linusson
Two-Row Set-Valued Tableaux: Catalan+k Combinatorics
Abstract.
Set-valued standard Young tableaux are a generalization of standard Young tableaux due to Buch (2002) with applications in algebraic geometry. The enumeration of set-valued SYT is significantly more complicated than in the ordinary case, although product formulas are known in certain special cases. In this work we study the case of two-rowed set-valued SYT with a fixed number of entries. These tableaux are a new combinatorial model for the Catalan, Narayana, and Kreweras numbers, and can be shown to be in correspondence with both 321-avoiding permutations and a certain class of bicolored Motzkin paths. We also introduce a generalization of the set-valued comajor index studied by Hopkins, Lazar, and Linusson (2023), and use this statistic to find seemingly new q-analogs of the Catalan and Narayana numbers.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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