Omitting parentheses from the cyclic notation

TL;DR

This paper explores a new combinatorial structure derived from permutations by removing parentheses from their cyclic notation, revealing rich poset properties and topological characteristics.

Contribution

It introduces a novel class of permutation sets with specific order properties and determines their topological and combinatorial features.

Findings

The sets form bounded, graded, unimodal, rank-symmetric, EL-shellable posets.

The homotopy types of the order complexes are characterized.

The study initiates a new combinatorial framework for permutation analysis.

Abstract

The purpose of this article is to initiate a combinatorial study of the Bruhat-Chevalley ordering on certain sets of permutations obtained by omitting the parentheses from their standard cyclic notation. In particular, we show that these sets form a bounded, graded, unimodal, rank-symmetric and EL-shellable posets. Moreover, we determine the homotopy types of the associated order complexes.