A generalization of $(q,t)$-Catalan and nabla operators

Abstract

We introduce non-commutative analogs of $k$-Schur functions and prove that their images by the non-commutative nabla operator $\blacktriangledown$ is ribbon Schur positive, up to a global sign. Inspired by these results, we define new filtrations of the usual $(q,t)$-Catalan polynomials by computing the image of certain commutative $k$-Schur functions by the commutative nabla operator $\nabla$. In some particular cases, we give a combinatorial interpretation of these polynomials in terms of nested quantum Dick paths.

Keywords

• $k$-schur functions
• nabla operator
• $(q t)$-catalan numbers
• [math.math-co] mathematics [math]/combinatorics [math.co]
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]