Linear algebra and congruences for MacMahon’s k-rowed plane partitions

In this paper, we provide an algorithm to detect linear congruences of Formula: see text, the number of MacMahon’s Formula: see text-rowed plane partitions, and give a quantitative result on the nonexistence of Ramanujan-type congruences of the Formula: see text-rowed plane partition functions. We also show Formula: see text that the number of partitions at most Formula: see text parts always admits linear congruences.