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NEW INFINITE FAMILIES OF CONGRUENCES MODULO 4 AND 8 FOR 1-SHELL TOTALLY SYMMETRIC PLANE PARTITIONS

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  02 April 2014

OLIVIA X. M. YAO
OLIVIA X. M. YAO*
Affiliation:
Department of Mathematics, Jiangsu University, Zhenjiang, Jiangsu 212013, PR China email [email protected]
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Abstract

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In 2012, Blecher [‘Geometry for totally symmetric plane partitions (TSPPs) with self-conjugate main diagonal’, Util. Math. 88 (2012), 223–235] introduced a special class of totally symmetric plane partitions, called $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1$-shell totally symmetric plane partitions. Let $f(n)$ denote the number of $1$-shell totally symmetric plane partitions of weight $n$. More recently, Hirschhorn and Sellers [‘Arithmetic properties of 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc. to appear. Published online 27 September 2013] discovered a number of arithmetic properties satisfied by $f(n)$. In this paper, employing some results due to Cui and Gu [‘Arithmetic properties of $l$-regular partitions’, Adv. Appl. Math. 51 (2013), 507–523], and Hirschhorn and Sellers, we prove several new infinite families of congruences modulo 4 and 8 for $1$-shell totally symmetric plane partitions. For example, we find that, for $n\geq 0$ and $\alpha \geq 1$,

$$\begin{equation*} f(8 \times 5^{2\alpha } n+39\times 5^{2\alpha -1})\equiv 0 \pmod 8. \end{equation*}$$

Keywords

congruence1-shell totally symmetric plane partitionTSPP

MSC classification

Secondary: 11P83: Partitions; congruences and congruential restrictions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 1 , August 2014 , pp. 37 - 46
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

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