  Recently, Schneider and Schneider defined a new class of partitions called
sequentially congruent partitions, in which each part is congruent to the next
part modulo its index, and they proved two partition bijections involving these
partitions. We introduce a new partition notation specific to sequentially
congruent partitions which allows us to more easily study these bijections and
their compositions, and we reinterpret them in terms of Young diagram
transformations. We also define a generalization of sequentially congruent
partitions, and we provide several new partition bijections for these
generalized sequentially congruent partitions. Finally, we investigate a
question of Schneider--Schneider regarding how sequentially congruent
partitions fit into Andrews' theory of partition ideals. We prove that the
maximal partition ideal of sequentially congruent partitions has infinite order
and is therefore not linked, and we identify its order 1 subideals.

Combinatorics