Abstract
Let dge 2 be an integer. In this paper we study arithmetic properties of the sequence (H_d(n))_{nin mathbb {N}}, where H_{d}(n) is the number of permutations in S_{n} 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 H_{d}(n) extending in this way earlier results of Ochiai and Ishihara, Ochiai, Takegehara and Yoshida.
Highlights
We let N denote the set of non-negative integers, N+ the set of positive integers, P the set of prime numbers and we write N≥k for the set {n ∈ N : n ≥ k}
We show that Wd,m(x) is a monic polynomial with integral coefficients and we give a formula for its coefficients in case of odd d
Proof We prove by induction on m ∈ N that
Summary
We let N denote the set of non-negative integers, N+ the set of positive integers, P the set of prime numbers and we write N≥k for the set {n ∈ N : n ≥ k}. In the paper [1] the authors initiated the study of arithmetic properties of the sequence (Hd (n))n∈N for d = 2 In this case, the number H2(n) is the number of involutions in the group Sn. In this case, the number H2(n) is the number of involutions in the group Sn They obtained several combinatorial identities, presented description of the 2-adic valuation of H2(n) and gave precise information about the rates of growth of H2(n). We show that if c is a power of a prime number not equal to d, c is a period of the sequence (Hd (n) (mod c))n∈N.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
