  In the eleventh paper in the series on MacMahons partition analysis, Andrews
and Paule [1] introduced the $k$ elongated partition diamonds. Recently, they
[2] revisited the topic. Let $d_k(n)$ count the partitions obtained by adding
the links of the $k$ elongated plane partition diamonds of length $n$. Andrews
and Paule [2] obtained several generating functions and congruences for
$d_1(n)$, $d_2(n)$, and $d_3(n)$. They also posed some conjectures, among which
the most difficult one was recently proved by Smoot [11]. Da Silva, Hirschhorn,
and Sellers [5] further found many congruences modulo certain primes for
$d_k(n)$ whereas Li and Yee [8] studied the combinatorics of Schmidt type
partitions, which can be viewed as partition diamonds. In this article, we give
elementary proofs of the remaining conjectures of Andrews and Paule [2], extend
some individual congruences found by Andrews and Paule [2] and da Silva,
Hirschhorn, and Sellers [5] to their respective families as well as find new
families of congruences for $d_k(n)$, present a refinement in an existence
result for congruences of $d_k(n)$ found by da Silva, Hirschhorn, and Sellers
[5], and prove some new individual as well as a few families of congruences
modulo 5, 7, 8, 11, 13, 16, 17, 19, 23, 25, 32, 49, 64 and 128.
