Combinatorial Hopf algebras, noncommutative Hall-Littlewood functions, and permutation tableaux

TL;DR

This paper introduces noncommutative Hall-Littlewood functions linked to permutation tableaux, providing explicit formulas for their q-enumeration and applications to PASEP steady states and permutation statistics.

Contribution

It develops a new family of noncommutative Hall-Littlewood functions using Tevlin's bases and q-deformations, connecting them to permutation tableaux and combinatorial enumeration.

Findings

Explicit q-enumeration formulas for permutation tableaux

Exact formulas for PASEP steady state probabilities

Polynomials for permutation statistics with fixed features

Abstract

We introduce a new family of noncommutative analogues of the Hall-Littlewood symmetric functions. Our construction relies upon Tevlin's bases and simple q-deformations of the classical combinatorial Hopf algebras. We connect our new Hall-Littlewood functions to permutation tableaux, and also give an exact formula for the q-enumeration of permutation tableaux of a fixed shape. This gives an explicit formula for: the steady state probability of each state in the partially asymmetric exclusion process (PASEP); the polynomial enumerating permutations with a fixed set of weak excedances according to crossings; the polynomial enumerating permutations with a fixed set of descent bottoms according to occurrences of the generalized pattern 2-31.