Row-strict dual immaculate functions

TL;DR

This paper introduces the row-strict dual immaculate functions, a new basis of quasisymmetric functions, and explores their combinatorial properties, connections to involutions, and extensions to skew and hook variants.

Contribution

It defines the row-strict dual immaculate functions and shows their relation to existing dual immaculate functions via the $$ involution, along with combinatorial and operator-based characterizations.

Findings

Defined the row-strict dual immaculate functions as generating functions of specific tableaux.

Established the relation to dual immaculate functions via the $$ involution.

Extended the framework to skew and hook dual immaculate functions with combinatorial properties.

Abstract

We define a new basis of quasisymmetric functions, the row-strict dual immaculate functions, as the generating function of a particular set of tableaux. We establish that this definition gives a function that can also be obtained by applying the $\psi$ involution to the dual immaculate functions of Berg, Bergeron, Saliola, Serrano, and Zabrocki (2014) and establish numerous combinatorial properties for our functions. We give an equivalent formulation of our functions via Bernstein-like operators, in a similar fashion to Berg et. al (2014). We conclude the paper by defining skew dual immaculate functions and hook dual immaculate functions and establishing combinatorial properties for them.