Residue periodicity in subgroup counting functions

Grady, Michael; Newman, Morris

doi:10.1090/conm/166/01634

Cited by 13 publications

(20 citation statements)

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“…We give a new proof of this known fact in order to obtain some results on periodicity of sequences Hp(n) p βn (mod p r ) n∈N , where r ∈ N + and β n is mentioned lower bound of ν p (H p (n)) or β n = ν p (H p (n)). Moreover, we show that for each prime number p and j ∈ {0, ..., p − 1} there exists a b j ∈ {0, ..., p − 1} such that H p (kp 2 + jp + b j ) = k(p − 1) + j for any k ∈ N. These results can be seen as a complement of the series of papers [5,10,6,7]. In Section 5 we describe the p-adic valuations of the numbers H d (n), where p > d. In order to do this, we show that the sequence (H d (n)) n∈N is a restriction of a differentiable function f d : Z p → Z p , where Z p is a ring of p-adic integers.…”

Section: Introductionmentioning

confidence: 59%

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

Miska

Ulas

2020

Monatsh Math

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Let d ≥ 2 be an integer. In this paper we study arithmetic properties of the sequence (H d (n)) n∈N , where H d (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 they 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.2010 Mathematics Subject Classification. 11B50, 11B83.

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Section: Introductionmentioning

confidence: 59%

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

Miska

Ulas

2020

Monatsh Math

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“…We will give a proof of the fact that ord p (a(n)) ≥ γ(n) (Corollary 3.3 in Section 3). This inequality is shown in [5,6,8,10] and is also a consequence of [7, equation (3)]. If p = 2, this result is equivalent to [3,Theorem 10].…”

Section: Introductionmentioning

confidence: 91%

p-Divisibility of the Number of Solutions of xp = 1 in a Symmetric Group

Ishihara¹,

Ochiai²,

Takegahara³

et al. 2001

Annals of Combinatorics

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For a prime p and for the number a(n) of solutions of x p = 1 in the symmetric group on n letters, ord p (a(n)) ≥ [n/p] − [n/p 2 ], and especially, ord p (a(n)) = [n/p] − [n/p 2 ] provided n ≡ 0 mod p 2 . Let r be an integer with 1 ≤ r ≤ p 2 − 1. If ord p (a(r)) ≤ [r/p] + 1, then, for each positive integer m, ord p (a(mp 2 + r)) = m(p − 1) + ord p (a(r)). Assume that ord p (a(r)) = [r/p] + 2. If a(p 2 + r) ≡ −p p−1 a(r) mod p p+[r/p]+2 , then ord p (a(mp 2 + r)) = m(p − 1) + [r/p] + 2; otherwise, there exists a p-adic integer b such that ord p (a(mp 2 + r)) = m(p − 1) + [r/p] + 2 + ord p (m − b).

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“…Example 2.1. Let π ∈ S 29,3 be the following permutation in cycle notation: Then f 3 (π ) = {(1, 2, 3) 3 , (4, 5, 6), (4), (4,6,5), (5), (6), (7,7,7), (8) 3 , (9) 2 , (9, 10, 10)}. The permutation π belongs to an equivalence class (7,7,7); (1, 2, 3) 3 , (4, 5, 6), (4, 6, 5), (9, 10, 10); (4), (5), (6), (9) (4,5,6), (4, 6, 5), (9, 10, 10), (4), (5), (6), (9) 2 .…”

Section: A Combinatorial Proofmentioning

confidence: 99%

“…Using generating function, Grady and Newman [6] obtained, for any prime p, ord p τ p (n) Using p-adic analysis, Ochiai [10] found the exact value of ord p (τ p (n)) for prime numbers p 23. Let .…”

Section: Introductionmentioning

confidence: 99%