Combinatorial Properties, Invariants and Structures of the Action of Sn×An on X×Y

Abstract

The transitivity, primitivity, rank and subdegrees, as well as pairing of the suborbits associated with the action of the actions of the direct product𝑆𝑆𝑛𝑛×𝐴𝐴𝑛𝑛, of the symmetric group 𝑆𝑆𝑛𝑛by the alternating group 𝐴𝐴𝑛𝑛alternating on the Cartesian product𝑋𝑋×𝑌𝑌, where 𝑋𝑋={𝑥𝑥1,𝑥𝑥2, . . . ,𝑥𝑥𝑛𝑛} and 𝑌𝑌={𝑦𝑦1,𝑦𝑦2, . . . ,𝑦𝑦𝑛𝑛} are disjoint sets each containing n elements is an area that has never received attention from researchers for a very long time. In this paper, we prove that the action is both transitive and imprimitive when𝑛𝑛≥3. Also, we establish that that the rank is 6 if𝑛𝑛=3, but is 4 for all 𝑛𝑛≥3. In addition, we show in this paper that the subdegrees associated with the action are 1,(𝑛𝑛−1), (𝑛𝑛−1), (𝑛𝑛−1)2. Lastly, we show that all the suborbits corresponding to the action, are self-paired when 𝑛𝑛≥ 4.