Combinatorics of patience sorting monoids

TL;DR

This paper explores the combinatorics of patience sorting monoids and tableaux, establishing bijections, counting formulas, and connections to Bell numbers and hook length formulas for standard tableaux.

Contribution

It introduces new Robinson--Schensted--Knuth-type correspondences for generalized tableaux and derives formulas for counting tableaux and relating them to Bell numbers.

Findings

Established bijections between words and tableaux of the same shape.

Derived formulas for counting tableaux with specified evaluations.

Connected standard tableaux counts to Bell numbers and provided a hook length formula.

Abstract

This paper makes a combinatorial study of the two monoids and the two types of tableaux that arise from the two possible generalizations of the Patience Sorting algorithm from permutations (or standard words) to words. For both types of tableaux, we present Robinson--Schensted--Knuth-type correspondences (that is, bijective correspondences between word arrays and certain pairs of semistandard tableaux of the same shape), generalizing two known correspondences: a bijective correspondence between standard words and certain pairs of standard tableaux, and an injective correspondence between words and pairs of tableaux. We also exhibit formulas to count both the number of each type of tableaux with given evaluations (that is, containing a given number of each symbol). Observing that for any natural number $n$, the $n$-th Bell number is given by the number of standard tableaux containing $n$ symbols, we restrict the previous formulas to standard words and extract a formula for the Bell numbers. Finally, we present a `hook length formula' that gives the number of standard tableaux of a given shape and deduce some consequences.