Congruences for Stirling numbers of the second kind

Chan, O-Yeat; Manna, Dante

doi:10.1090/conm/517/10135

Cited by 15 publications

(12 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another special case can be treated by the following theorem proved by Chan and Manna [2] in a recent paper.…”

Section: Theorem 6 ([4]mentioning

confidence: 99%

Alternative Proofs on the 2-adic Order of Stirling Numbers of the Second Kind

Lengyel¹

2010

Integers

View full text Add to dashboard Cite

An interesting 2-adic property of the Stirling numbers of the second kind S.n; k/ was conjectured by the author in 1994 and proved by De Wannemacker in 2005: 2 .S.2 n ; k// D d 2 .k/ 1, 1 Ä k Ä 2 n . It was later generalized to 2 .S.c2 n ; k// D d 2 .k/ 1, 1 Ä k Ä 2 n , c 1 by the author in 2009. Here we provide full and two partial alternative proofs of the generalized version. The proofs are based on non-standard recurrence relations for S.n; k/ in the second parameter and congruential identities.

show abstract

“…Another special case can be treated by the following theorem proved by Chan and Manna [2] in a recent paper.…”

Section: Theorem 6 ([4]mentioning

confidence: 99%

Alternative Proofs on the 2-adic Order of Stirling Numbers of the Second Kind

Lengyel¹

2010

Integers

View full text Add to dashboard Cite

show abstract

“…Carlitz [3] and Kwong [14] have studied the length of this period, respectively. Chan and Manna [4] characterized S(n, k) modulo prime powers in terms of binomial coefficients when k is a multiple of prime powers. Various congruences involving sums of S(n, k) are also known [20].…”

Section: Introduction and The Statements Of Main Resultsmentioning

confidence: 99%

The 2-adic valuations of differences of Stirling numbers of the second kind

Zhao

Hong

2015

Journal of Number Theory

View full text Add to dashboard Cite

Abstract. Let m, n, k and c be positive integers. Let ν 2 (k) be the 2-adic valuation of k. By S(n, k) we denote the Stirling numbers of the second kind. In this paper, we first establish a convolution identity of the Stirling numbers of the second kind and provide a detailed 2-adic analysis to the Stirling numbers of the second kind. Consequently, we show that if 2 ≤ m ≤ n and c is odd, then ν 2 (S(c2 n+1 , 2 m − 1) − S(c2 n , 2 m − 1)) = n + 1 except when n = m = 2 and c = 1, in which case ν 2 (S(8, 3) − S(4, 3)) = 6. This solves a conjecture of Lengyel proposed in 2009.

show abstract

“…For s ≥ 11, ψ(s) ≥ 7 + s by 2.1. This information makes it easy to check that the minimum value of ψ(k + j) − ν(j) + ν 8x+2 k is indeed 14, and this value occurs exactly when (k, j) = (0, 8), (2,8), or (1,8). This completes the proof that for all integers x we have ν(T 19 (8x + 2)) = ν(x − x 0 ) + 17 for some 2-adic integer x 0 .…”

Section: Moreovermentioning

confidence: 95%

Divisibility by 2 of partial Stirling numbers

Davis¹

2013

Funct. Approx. Comment. Math.

View full text Add to dashboard Cite

Abstract. The partial Stirling numbers T n (k) used here are defined as i odd n i i k . Their 2-exponents ν(T n (k)) are important in algebraic topology. We provide many specific results, applying to all values of n, stating that, for all k in a certain congruence class mod 2 t , ν(T n (k)) = ν(k − k 0 ) + c 0 , where k 0 is a 2-adic integer and c 0 a positive integer. Our analysis involves several new general results for ν( n 2i+1 i j ), the proofs of which involve a new family of polynomials. Following Clarke ([3]), we interpret T n as a function on the 2-adic integers, and the 2-adic integers k 0 described above as the zeros of these functions.

show abstract

Congruences for Stirling numbers of the second kind

Abstract: We characterize the Stirling numbers of the second kind S(n, k) modulo prime powers in terms of binomial coefficients. Our results are surprisingly simple when k is a multiple of the modulus.

Cited by 15 publications

References 7 publications

Alternative Proofs on the 2-adic Order of Stirling Numbers of the Second Kind

Alternative Proofs on the 2-adic Order of Stirling Numbers of the Second Kind

The 2-adic valuations of differences of Stirling numbers of the second kind

Divisibility by 2 of partial Stirling numbers

Contact Info

Product

Resources

About