Arithmetic Properties of 1-Shell Totally Symmetric Plane Partitions

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 eqnarray* ∑ n≥ 0f(n)qⁿ = 1+ ∑ n≥ 1q3n- 2 ∏ i= 0n- 2 (1+ q6i+ 3 ).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.