Lattice Paths and Order-preserving Partial Transformations

TL;DR

This paper establishes bijections between specific lattice paths and subsemigroups of order-preserving partial transformations, revealing new combinatorial structures and relationships within the semigroup ${ m PO}_n$.

Contribution

It introduces novel bijections linking lattice paths to subsemigroups of ${ m PO}_n$, enhancing understanding of their combinatorial and algebraic properties.

Findings

Bijections between lattice paths and subsemigroups of ${ m PO}_n$

New combinatorial interpretations of order-preserving transformations

Implications for the structure and enumeration of these semigroups

Abstract

Let ${\cal PO}_n$ be the semigroup of all order-preserving partial transformations of a finite chain. It is shown that there exist bijections between the set of certain lattice paths in the Cartesian plane that start at $(0,0)$, end at $(n-1,n-1)$, and certain subsemigroups of ${\cal PO}_n$. Several consequences of these bijections were discussed.