Cycle Index for Symmetric Group Acting on Cartesian Product of Two, Three and Four Sets
Grace Wakesho Kivunga *
Department of Mathematical Sciences, University of South Africa, P. O. Box 392, 0003, Pretoria, South Africa.
*Author to whom correspondence should be addressed.
Abstract
This study derives cycle-index expressions for the symmetric group acting on the Cartesian products of two, three, and four copies of a finite set. The analysis extends established results for ordered pairs, triples, and higher-order ordered subsets by incorporating contributions that arise when entries are repeated or are drawn from different cycles of a permutation. For the Cartesian product of two sets, the derivation separates ordered pairs into three cases, including the additional contribution from pairs of the form (a, a). For the Cartesian product of three sets, six contributions are considered, with the first three accounting for triples containing repeated entries and the remaining cases corresponding to previously established ordered-triple configurations. For the Cartesian product of four sets, nine cases are examined. Four of these cases provide additional contributions associated with repeated entries, while the remaining cases correspond to established configurations for ordered four-element subsets. In each setting, the relevant monomial contributions are combined over the applicable cycle structures to obtain the corresponding cycle-index formula. Illustrative examples for S6 and S7 are provided for the Cartesian products of two, three, and four sets. The results organise the contributions required for these induced actions and show how repeated entries alter the cycle-index calculations within the framework developed in the manuscript.
Keywords: Cycle index, symmetric group, Cartesian product, cycle type, monomial, permutation group, ordered tuples, repeated entries, induced permutation, cycle structure