A combinatorial approach to the power of 2 in the number of involutions

We provide a combinatorial approach to the largest power of $p$ in the number of permutations $\pi$ with $\pi^p=1$, for a fixed prime number $p$. With this approach, we find the largest power of $2$ in the number of involutions, in the signed sum of involutions and in the numbers of even or odd involutions.Comment: 13 page

2-adic properties for the numbers of involutions in the alternating groups

We study the 2-adic properties for the numbers of involutions in the alternative groups, and give an affirmative answer to a conjecture of Kim and Kim [A combinatorial approach to the power of 2 in the number of involutions, J. Combin. Theory Ser. A117 (2010) 1082–1094]. Some analogous and general results are also presented

On some properties of the number of permutations being products of pairwise disjoint d-cycles

Let d≥2 be an integer. In this paper we study arithmetic properties of the sequence (Hd(n))n∈N, where Hd(n) is the number of permutations in Sn being products of pairwise disjoint cycles of a fixed length d. In particular we deal with periodicity modulo a given positive integer, behaviour of the p-adic valuations and various divisibility properties. Moreover, we introduce some related families of polynomials and study their properties. Among many results we obtain qualitative description of the p-adic valuation of the number Hd(n) extending in this way earlier results of Ochiai and Ishihara, Ochiai, Takegehara and Yoshida