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ARITHMETIC PROPERTIES OF 1-SHELL TOTALLY SYMMETRIC PLANE PARTITIONS

Part of: Additive number theory; partitions Combinatorics

Published online by Cambridge University Press:  27 September 2013

MICHAEL D. HIRSCHHORN  and
JAMES A. SELLERS
MICHAEL D. HIRSCHHORN
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia email [email protected]
JAMES A. SELLERS*
Affiliation:
Department of Mathematics, Penn State University, University Park, PA 16802, USA
*
✉[email protected]
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Abstract

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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

$$\begin{eqnarray*}\displaystyle \sum _{n\geq 0}f(n){q}^{n} = 1+ \sum _{n\geq 1}{q}^{3n- 2} \prod _{i= 0}^{n- 2} (1+ {q}^{6i+ 3} ).\end{eqnarray*}$$
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

partitiontotally symmetric plane partitionTSPPgenerating functioncongruence

MSC classification

Secondary: 05A17: Partitions of integers 11P83: Partitions; congruences and congruential restrictions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 3 , June 2014 , pp. 473 - 478
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

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