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ARITHMETIC PROPERTIES OF 1-SHELL TOTALLY SYMMETRIC PLANE PARTITIONS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 89 / Issue 3 / June 2014
- Published online by Cambridge University Press:
- 27 September 2013, pp. 473-478
- Print publication:
- June 2014
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- Article
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- You have access
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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 In this brief note, we prove a number of arithmetic properties satisfied by
$$\begin{eqnarray*}\displaystyle \sum _{n\geq 0}f(n){q}^{n} = 1+ \sum _{n\geq 1}{q}^{3n- 2} \prod _{i= 0}^{n- 2} (1+ {q}^{6i+ 3} ).\end{eqnarray*}$$
$f(n)$ using elementary generating function manipulations and well-known results of Ramanujan and Watson.
