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.