A generalization of Bressoud's beautiful bijection

TL;DR

This paper explores two Young diagram-based bijections, including a generalization that maintains structure under parameter changes and reveals symmetry in combinatorial mappings.

Contribution

It introduces a generalized bijection extending Bressoud's work, demonstrating stability and symmetry in combinatorial structures across parameters.

Findings

Established a bijective correspondence with explicit constructive proof.

Generalized the bijection to any natural d, preserving bijectivity.

Revealed symmetry in the combinatorial construction.

Abstract

Within this research, two combinatorial bijections using Young diagrams were studied. The first is a special case of a bijective correspondence between two classes of combinatorial objects. Its proof, based on Young diagrams, establishes equinumerosity and provides an explicit constructive mapping. The second is a generalization to any natural d, preserving bijectivity. It shows the combinatorial structure remains stable under changes in the parameter, with Young diagrams serving as a universal language. A notable and non-obvious aspect of this generalization is the symmetry revealed in the construction. Intuitively, it was not evident that one could consider not only the natural order of residues but also any permutation of them.