Proofs of some conjectures of Andrews and Paule on 2-elongated plane partitions

Abstract:

      Recently, Andrews and Paule established the generating functions for the $k$-elongated plane partition function $d_k(n)$ and proved a large number of results on $d_k(n)$ with $k=2,3$. In particular, they posed some conjectures on congruences modulo powers of 3 for $d_2(n)$. Their work has attracted the attention of Hirschhorn, Sellers, Silva and Smoot. Very recently, Smoot proved a congruence family for $d_2(n)$ which implies one conjecture due to Andrews and Paule by using the localization method. In this paper, we prove the rest two conjectures given by Andrew and Paule.