# Sequentially congruent partitions and related bijections

## TL;DR

This paper introduces and explores a new class of partitions called sequentially congruent partitions, establishing their deep connections with classical partition theory through bijections and extending the theory of partition ideals.

## Contribution

It defines sequentially congruent partitions, proves their bijection with standard partitions, and links them to frequency conditions and partition ideals, enriching partition theory.

## Key findings

- Sequentially congruent partitions are in bijection with all partitions of n.
- They induce a bijection between partitions of n and partitions of length n with frequency divisibility conditions.
- The work connects these concepts with G. E. Andrews's theory of partition ideals.

## Abstract

We study a curious class of partitions, the parts of which obey an exceedingly strict congruence condition we refer to as "sequential congruence": the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to zero modulo the length of the partition. It turns out these obscure-seeming objects are embedded in a natural way in partition theory. We show that sequentially congruent partitions with largest part $n$ are in bijection with the partitions of $n$. Moreover, we show sequentially congruent partitions induce a bijection between partitions of $n$ and partitions of length $n$ whose parts obey a strict "frequency congruence" condition -- the frequency (or multiplicity) of each part is divisible by that part -- and prove families of similar bijections, connecting with G. E. Andrews's theory of partition ideals.