Rank, Subdegrees, and Primitivity in Product-Action Wreath Products Acting on the Cartesian Power of a Base Set
DOI:
https://doi.org/10.62054/ijdm/0301.05Abstract
We study the product action of the wreath product on the Cartesian power , where is a finite transitive permutation group of . A complete combinatorial description of the suborbit structure is obtained by showing that suborbits correspond bijectively to weak compositions of determined by the orbit decomposition of . This yields the explicit rank formula. together with closed subdegrees expressions given by multinomial coefficients weighted by stabilizer orbit sizes. Consequently, orbit enumeration for product action wreath products reduces to classical stars-and-bars counting and multinomial expansion. A unified primitivity criterion is established. acting in product action is primitive if and only if is primitive and non-regular (with ). Specializations to symmetric, alternating, cyclic, and dihedral groups produce explicit rank polynomials, concrete subdegrees formulas, and a complete block classification in each case. These results provide a unified enumerative and structural framework for wreath product actions, connecting permutation group theory, multinomial combinatorics, and algebraic graph theory
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Copyright (c) 2026 Salihu L. Aliyu, Shuaibu G. Ngulde, Babagana A. Madu, Bukar G. Ahmadu (Author)
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