Refined enumeration of two-rowed set-valued standard tableaux via two-coloured Motzkin paths

TL;DR

This paper develops explicit formulas for counting two-rowed set-valued standard tableaux using bijections with two-coloured Motzkin paths, combining combinatorial bijections and generating function techniques.

Contribution

It introduces a novel enumeration method for two-rowed set-valued tableaux via bijections with two-coloured Motzkin paths and generating function analysis.

Findings

Derived explicit formulas for counting set-valued tableaux

Established a bijection with two-coloured Motzkin paths

Applied generating functions and Lagrange inversion for enumeration

Abstract

We derive formulae for the number of set-valued standard tableaux of two-rowed shapes, keeping track of the total number of entries, the number of entries in the first row, and the number of entries in the second row. Key in the proofs is a bijection with two-coloured Motzkin paths followed by generating function computations and coefficient extraction helped by the Lagrange inversion formula.