Arithmetic properties of 1-shell totally symmetric plane partitions

Michael D. Hirschhorn; James A. Sellers

doi:10.1017/S0004972713000865

Arithmetic properties of 1-shell totally symmetric plane partitions

Michael D. Hirschhorn, James A. Sellers

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

Blecher ['Geometry for totally symmetric plane partitions (TSPPs) with self-conjugate main diagonal', Util. Math. 88 (2012), 223-235] defined the combinatorial objects known as 1-shell totally symmetric plane partitions of weight n. He also proved that the generating function for f(n), the number of 1-shell totally symmetric plane partitions of weight n, is given by ∑ _n≥0 f(n)qⁿ = 1+ ∑_n≥1q^3n-2 ∏_i=0^n-2(1+q⁶ⁱ⁺³). In this brief note, we prove a number of arithmetic properties satisfied by f(n) using elementary generating function manipulations and well-known results of Ramanujan and Watson.

Original language	English (US)
Pages (from-to)	473-478
Number of pages	6
Journal	Bulletin of the Australian Mathematical Society
Volume	89
Issue number	3
DOIs	https://doi.org/10.1017/S0004972713000865
State	Published - 2014
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2013 Australian Mathematical Publishing Association Inc.

Keywords

TSPP
congruence
generating function
partition
totally symmetric plane partition

Access

10.1017/S0004972713000865

OpenUrl availability

Full text

Cite this

@article{81b1944f2b1f4df19a4a2d2efac7f7da,

title = "Arithmetic properties of 1-shell totally symmetric plane partitions",

abstract = "Blecher ['Geometry for totally symmetric plane partitions (TSPPs) with self-conjugate main diagonal', Util. Math. 88 (2012), 223-235] defined the combinatorial objects known as 1-shell totally symmetric plane partitions of weight n. He also proved that the generating function for f(n), the number of 1-shell totally symmetric plane partitions of weight n, is given by ∑ n≥0 f(n)qn = 1+ ∑n≥1q3n-2 ∏i=0n-2(1+q6i+3). In this brief note, we prove a number of arithmetic properties satisfied by f(n) using elementary generating function manipulations and well-known results of Ramanujan and Watson.",

keywords = "TSPP, congruence, generating function, partition, totally symmetric plane partition",

author = "Hirschhorn, {Michael D.} and Sellers, {James A.}",

note = "Publisher Copyright: {\textcopyright} 2013 Australian Mathematical Publishing Association Inc.",

year = "2014",

doi = "10.1017/S0004972713000865",

language = "English (US)",

volume = "89",

pages = "473--478",

journal = "Bulletin of the Australian Mathematical Society",

issn = "0004-9727",

publisher = "Cambridge University Press",

number = "3",

}

TY - JOUR

T1 - Arithmetic properties of 1-shell totally symmetric plane partitions

AU - Hirschhorn, Michael D.

AU - Sellers, James A.

PY - 2014

Y1 - 2014

N2 - Blecher ['Geometry for totally symmetric plane partitions (TSPPs) with self-conjugate main diagonal', Util. Math. 88 (2012), 223-235] defined the combinatorial objects known as 1-shell totally symmetric plane partitions of weight n. He also proved that the generating function for f(n), the number of 1-shell totally symmetric plane partitions of weight n, is given by ∑ n≥0 f(n)qn = 1+ ∑n≥1q3n-2 ∏i=0n-2(1+q6i+3). In this brief note, we prove a number of arithmetic properties satisfied by f(n) using elementary generating function manipulations and well-known results of Ramanujan and Watson.

AB - Blecher ['Geometry for totally symmetric plane partitions (TSPPs) with self-conjugate main diagonal', Util. Math. 88 (2012), 223-235] defined the combinatorial objects known as 1-shell totally symmetric plane partitions of weight n. He also proved that the generating function for f(n), the number of 1-shell totally symmetric plane partitions of weight n, is given by ∑ n≥0 f(n)qn = 1+ ∑n≥1q3n-2 ∏i=0n-2(1+q6i+3). In this brief note, we prove a number of arithmetic properties satisfied by f(n) using elementary generating function manipulations and well-known results of Ramanujan and Watson.

KW - TSPP

KW - congruence

KW - generating function

KW - partition

KW - totally symmetric plane partition

UR - http://www.scopus.com/inward/record.url?scp=84912087457&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84912087457&partnerID=8YFLogxK

U2 - 10.1017/S0004972713000865

DO - 10.1017/S0004972713000865

M3 - Article

AN - SCOPUS:84912087457

SN - 0004-9727

VL - 89

SP - 473

EP - 478

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

IS - 3

ER -

Arithmetic properties of 1-shell totally symmetric plane partitions

Abstract

Bibliographical note

Keywords

Access

OpenUrl availability

Other files and links

Fingerprint

Cite this