Set-Valued Catalan Combinatorics

TL;DR

This paper introduces set-valued standard Young tableaux as a new combinatorial model for Catalan-related numbers, establishing connections with permutations and Motzkin paths, and develops q-analogs of these numbers.

Contribution

It studies two-rowed set-valued SYT with fixed entries, linking them to Catalan numbers and related combinatorial objects, and introduces a generalized comajor index for q-analogs.

Findings

Set-valued SYT correspond to Catalan, Narayana, and Kreweras numbers.

Established bijections with 321-avoiding permutations and bicolored Motzkin paths.

Developed new q-analogs of Catalan and Narayana numbers.

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.