Combinatorial results for order-preserving partial injective contraction mappings

TL;DR

This paper explores the combinatorial structure and cardinalities of specific semigroups of order-preserving partial injective contraction mappings, revealing connections to Fibonacci numbers and extending to order-reversing cases.

Contribution

It provides new formulas for counting elements in these semigroups and relates these counts to Fibonacci numbers, advancing understanding of their algebraic structure.

Findings

Cardinalities of semigroups are expressed in terms of Fibonacci numbers.

Formulas for order-preserving and order-decreasing mappings are derived.

Results extend to order-reversing mappings, broadening the scope of combinatorial analysis.

Abstract

Let $ \mathcal{I}_n$ be the symmetric inverse semigroup on $X_n = \{1, 2, \ldots , n\}$. Let $\mathcal{OCI}_n$ be the subsemigroup of $\mathcal{I}_n$ consisting of all order-preserving injective partial contraction mappings, and let $\mathcal{ODCI}_n$ be the subsemigroup of $\mathcal{I}_n$ consisting of all order-preserving and order-decreasing injective partial contraction mappings of $X_n$. In this paper, we investigate the cardinalities of some equivalences on $\mathcal{OCI}_n$ and $\mathcal{ODCI}_n$ which lead naturally to obtaining the order of these semigroups. Then, we relate the formulae obtained to Fibonacci numbers. Similar results about $\mathcal{ORCI}_n$, the semigroup of order-preserving or order-reversing injective partial contraction mappings, are deduced.