Combinatorics of $\phi$-deformed stuffle Hopf algebras

TL;DR

This paper explores the combinatorial structure of $$-deformed stuffle Hopf algebras to extend Schfctzenberger's factorization to broader perturbations, advancing algebraic understanding.

Contribution

It systematically develops the combinatorial aspects of $$-deformed stuffle Hopf algebras, paralleling the shuffle product, to generalize factorization methods.

Findings

Systematic development of combinatorial structure of $$-deformed stuffle Hopf algebras.

Extension of Schfctzenberger's factorization to general perturbations.

Parallel analysis with the shuffle product.

Abstract

In order to extend the Sch\"utzenberger's factorization to general perturbations, the combinatorial aspects of the Hopf algebra of the $\phi$-deformed stuffle product is developed systematically in a parallel way with those of the shuffle product.