### Combinatorics of partial wreath power of finite inverse symmetric semigroup \(\mathcal{IS}_d\)

#### Abstract

We study some combinatorial properties of \(\wr_p^k \mathcal{IS}_d\). In particular, we calculate its order, the number of idempotents and the number of \(\mathcal D\)-classes. For a given based graph \(\Gamma\subset T\) we compute the number of elements in its \(\mathcal D\)-class \(D_\Gamma\) and the number of \(\mathcal R\)- and \(\mathcal L\)-classes in \(D_\Gamma\).

#### Keywords

Wreath product, finite inverse symmetric semigroup, rooted tree, partial automorphism

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