Superization and (q,t)-specialization in combinatorial Hopf algebras

TL;DR

This paper extends superization to combinatorial Hopf algebras, deriving (q,t)-analogues of hook-content and hook-length formulas for various combinatorial structures, using dendriform structures.

Contribution

It introduces a superization process for combinatorial Hopf algebras and derives new (q,t)-analogues of classical hook-length formulas.

Findings

Derived (q,t)-hook-content formulas for symmetric functions.

Obtained (q,t)-analogues of Bjorner-Wachs q-hook-length formulas for binary trees.

Extended formulas to plane trees using dendriform structures.

Abstract

We extend a classical construction on symmetric functions, the superization process, to several combinatorial Hopf algebras, and obtain analogs of the hook-content formula for the (q,t)-specializations of various bases. Exploiting the dendriform structures yields in particular (q,t)-analogs of the Bjorner-Wachs q-hook-length formulas for binary trees, and similar formulas for plane trees.