Bijections, generalizations, and other properties of sequentially congruent partitions

TL;DR

This paper introduces a new notation and generalizations for sequentially congruent partitions, explores their properties through bijections and Young diagram transformations, and situates them within partition ideal theory.

Contribution

It presents a novel notation for sequentially congruent partitions, extends the concept to generalized forms, and analyzes their structure within partition ideals.

Findings

New notation simplifies study of bijections

Generalized sequentially congruent partitions are characterized

Maximal partition ideal of these partitions has infinite order

Abstract

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.