Combinatorial Properties of Order Decreasing and Order Reversing Partial One– to–One Transformation Semigroup
DOI:
https://doi.org/10.62054/ijdm/0302.06Abstract
We study the semigroup of partial one-to-one transformations on a finite chain that are simultaneously order-decreasing and order-reversing. This class arises naturally as the intersection of two well-studied transformation semigroups. For a finite chain , we give a detailed combinatorial analysis with respect to fixed points, height, and image changes. Explicit counting formulas, generating polynomials, and recurrence relations are obtained and justified combinatorially. Our results unify and extend earlier enumerations for order-decreasing and order-reversing partial one-to-one transformation semigroups and provide refined distributions via univariate and bivariate generating functions.
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