Poset structures on (m + 2)-angulations and polynomial bases of the quotient by G^m -quasisymmetric functions

TL;DR

This paper establishes a bijection between polygon dissections and a new basis of a polynomial quotient, revealing a poset structure that links combinatorial dissections with algebraic bases in higher quasi-symmetric functions.

Contribution

It introduces a novel bijection connecting polygon dissections to a basis of a polynomial quotient, and explores the associated poset structure.

Findings

Divisibility of basis elements corresponds to a new partial order on dissections.

The bijection links combinatorial structures with algebraic bases in higher quasi-symmetric functions.

The poset structure provides new insights into the algebraic and combinatorial interplay.

Abstract

For integers m, n $\ge$ 1, we describe a bijection sending dissections of the (mn + 2)-regular polygon into (m + 2)-sided polygons to a new basis of the quotient of the polynomial algebra in mn variables by an ideal generated by some kind of higher quasi-symmetric functions. We show that divisibility of the basis elements corresponds to a new partial order on dissections, which is studied in some detail.