Rank complement of rational Dyck paths and conjugation of $(m,n)$-core partitions

TL;DR

This paper introduces the rank complement transformation on rational Dyck paths, linking it to conjugation of $(m,n)$-cores, and uses this to derive new characterizations, counting methods, and proofs of conjectured formulas related to rational Catalan numbers.

Contribution

It defines the rank complement transformation on rational Dyck paths and shows its correspondence to conjugation of $(m,n)$-cores, providing new insights and proofs in the combinatorics of rational Catalan objects.

Findings

Rank complement corresponds to conjugation of $(m,n)$-cores.

New characterization of $n$-cores using rank complement.

Proof of equivalence of two conjectured formulas for rational Catalan polynomials.

Abstract

Given a coprime pair $(m,n)$ of positive integers, rational Catalan numbers $\frac{1}{m+n} \binom{m+n}{m,n}$ counts two combinatorial objects:rational $(m,n)$-Dyck paths are lattice paths in the $m\times n$ rectangle that never go below the diagonal; $(m,n)$-cores are partitions with no hook length equal to $m$ or $n$.Anderson established a bijection between $(m,n)$-Dyck paths and $(m,n)$-cores. We define a new transformation, called rank complement, on rational Dyck paths. We show that rank complement corresponds to conjugation of $(m,n)$-cores under Anderson's bijection. This leads to: i) a new approach to characterizing $n$-cores; ii) a simple approach for counting the number of self-conjugate $(m,n)$-cores; iii) a proof of the equivalence of two conjectured combinatorial sum formulas, one over rational $(m,n)$-Dyck paths and the other over $(m,n)$-cores, for rational Catalan polynomials.