Partitions with constrained ranks and lattice paths

TL;DR

This paper studies partitions with constrained successive ranks, providing enumeration formulas and a bijective proof of a known partition identity, while tracking various partition statistics.

Contribution

It introduces a new bijective proof of a partition identity and extends enumeration techniques to partitions with bounded successive ranks.

Findings

Derived enumeration formulas for constrained partitions.

Provided a bijective proof of Andrews and Bressoud's result.

Refined partition counts with additional statistics.

Abstract

In this paper we study partitions whose successive ranks belong to a given set. We enumerate such partitions while keeping track of the number of parts, the largest part, the side of the Durfee square, and the height of the Durfee rectangle. We also obtain a new bijective proof of a result of Andrews and Bressoud that the number of partitions of $N$ with all ranks at least $1-\ell$ equals the number of partitions of $N$ with no parts equal to $\ell+1$, for $\ell\ge0$, which allows us to refine it by the above statistics. Combining Foata's second fundamental transformation for words with Greene and Kleitman's mapping for subsets, interpreted in terms of lattice paths, we obtain enumeration formulas for partitions whose successive ranks satisfy certain constraints, such as being bounded by a constant.