Riordan pattern's quest within simplicial complexes

TL;DR

This paper explores the use of Riordan matrices to connect combinatorial invariants in simplicial complexes and investigates the properties of topological joins and their associated Riordan matrices.

Contribution

It introduces a novel application of Riordan matrices to geometric combinatorics and analyzes the combinatorial and topological properties of joins in simplicial complexes.

Findings

Riordan matrices relate to $f$-vectors, $h$-vectors, and $ ext{ extgamma}$-vectors.

Properties of topological joins are characterized using Riordan matrices.

Iterated joins produce specific Riordan matrix patterns.

Abstract

The aim of this paper is twofold. First, we demonstrate how Riordan matrices can be employed to connect well-known concepts in geometric combinatorics, such as $f$-vectors, $h$-vectors $\gamma$-vectors, in a similar fashion to the McMullen Correspondence, and the Dehn-Sommerville equations, among others. Second, we investigate the combinatorial properties of the topological join operation, both for simplicial complexes and for Alexandroff spaces. Finally, we explore the Riordan matrices arising from the iteration of this topological operation and analyze their properties.