Combinatorics of partial wreath power of finite inverse symmetric semigroup $\mathcal{IS}_d$

TL;DR

This paper investigates the combinatorial structure of the partial wreath power of the finite inverse symmetric semigroup, including calculations of its order, idempotents, and D-classes.

Contribution

It provides explicit combinatorial formulas and properties for the partial wreath power of $ ext{IS}_d$, advancing understanding of its algebraic structure.

Findings

Calculated the order of the partial wreath power.

Counted the number of idempotents.

Determined the number of D-classes.

Abstract

We study some combinatorial properties of partial wreath $k$-th power of the semigroup $\mathcal{IS}_d$. In particular, we calculate its order, the number of idempotents and the number of D-classes.