Simplicial Complexes of Triangular Ferrers Boards

TL;DR

This paper explores the topological and enumerative properties of simplicial complexes derived from non-attacking rook placements on a specific class of Ferrers boards, focusing on facets, homotopy type, and homology.

Contribution

It introduces a detailed analysis of the simplicial complexes associated with triangular Ferrers boards, highlighting their combinatorial and topological characteristics.

Findings

Enumeration of facets for the simplicial complexes

Determination of homotopy types and homology groups

Insights into the combinatorial structure of rook placements

Abstract

We study the simplicial complex that arises from non-attacking rook placements on a subclass of Ferrers boards that have $a_i$ rows of length $i$ where $a_i>0$ and $i\leq n$ for some positive integer $n$. In particular, we will investigate enumerative properties of their facets, their homotopy type, and homology.