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

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10.1017/S0004972713000865

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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.",

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language = "English (US)",

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

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AU - Sellers, James A.

PY - 2014

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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

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Arithmetic properties of 1-shell totally symmetric plane partitions

Abstract

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