Prime power divisors of the number of n × n alternating sign matrices

Darrin D. Frey, James Allen Sellers

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

We let A(n) equal the number of n × n alternating sign matrices. From the work of a variety of sources, we know that A(n) =∏ l=0 n-1(3l+1)!/(n+l)! We find an efficient method of determining ord p(A(n)), the highest power of p which divides A(n), for a given prime p and positive integer n, which allows us to efficiently compute the prime factorization of A(n). We then use our method to show that for any nonnegative integer k, and for any prime p > 3, there are infinitely many positive integers n such that ord p(A(n)) =k. We show a similar but weaker theorem for the prime p = 3, and note that the opposite is true for p = 2.

Original languageEnglish (US)
Pages (from-to)139-147
Number of pages9
JournalArs Combinatoria
Volume71
StatePublished - Apr 2004

Fingerprint

Alternating Sign Matrices
Divisor
Integer
L'Hôpital's Rule
High Power
Divides
Non-negative
Theorem

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Frey, Darrin D. ; Sellers, James Allen. / Prime power divisors of the number of n × n alternating sign matrices. In: Ars Combinatoria. 2004 ; Vol. 71. pp. 139-147.
@article{53b6cf65fc6d41ff8950c9109243148f,
title = "Prime power divisors of the number of n × n alternating sign matrices",
abstract = "We let A(n) equal the number of n × n alternating sign matrices. From the work of a variety of sources, we know that A(n) =∏ l=0 n-1(3l+1)!/(n+l)! We find an efficient method of determining ord p(A(n)), the highest power of p which divides A(n), for a given prime p and positive integer n, which allows us to efficiently compute the prime factorization of A(n). We then use our method to show that for any nonnegative integer k, and for any prime p > 3, there are infinitely many positive integers n such that ord p(A(n)) =k. We show a similar but weaker theorem for the prime p = 3, and note that the opposite is true for p = 2.",
author = "Frey, {Darrin D.} and Sellers, {James Allen}",
year = "2004",
month = "4",
language = "English (US)",
volume = "71",
pages = "139--147",
journal = "Ars Combinatoria",
issn = "0381-7032",
publisher = "Charles Babbage Research Centre",

}

Prime power divisors of the number of n × n alternating sign matrices. / Frey, Darrin D.; Sellers, James Allen.

In: Ars Combinatoria, Vol. 71, 04.2004, p. 139-147.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Prime power divisors of the number of n × n alternating sign matrices

AU - Frey, Darrin D.

AU - Sellers, James Allen

PY - 2004/4

Y1 - 2004/4

N2 - We let A(n) equal the number of n × n alternating sign matrices. From the work of a variety of sources, we know that A(n) =∏ l=0 n-1(3l+1)!/(n+l)! We find an efficient method of determining ord p(A(n)), the highest power of p which divides A(n), for a given prime p and positive integer n, which allows us to efficiently compute the prime factorization of A(n). We then use our method to show that for any nonnegative integer k, and for any prime p > 3, there are infinitely many positive integers n such that ord p(A(n)) =k. We show a similar but weaker theorem for the prime p = 3, and note that the opposite is true for p = 2.

AB - We let A(n) equal the number of n × n alternating sign matrices. From the work of a variety of sources, we know that A(n) =∏ l=0 n-1(3l+1)!/(n+l)! We find an efficient method of determining ord p(A(n)), the highest power of p which divides A(n), for a given prime p and positive integer n, which allows us to efficiently compute the prime factorization of A(n). We then use our method to show that for any nonnegative integer k, and for any prime p > 3, there are infinitely many positive integers n such that ord p(A(n)) =k. We show a similar but weaker theorem for the prime p = 3, and note that the opposite is true for p = 2.

UR - http://www.scopus.com/inward/record.url?scp=26844456482&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=26844456482&partnerID=8YFLogxK

M3 - Article

VL - 71

SP - 139

EP - 147

JO - Ars Combinatoria

JF - Ars Combinatoria

SN - 0381-7032

ER -