A note on rank complement of rational Dyck paths and conjugation of $(m,n)$-core partitions

Given a coprime pair $(m,n)$ of positive integers, rational Catalan numbers $\dfrac{1}{m+1} \begin{pmatrix}m+n \\ m,n \end{pmatrix}$ 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.

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This work was done during the author’s stay at UCSD. The author is very grateful to Professor Adriano Garsia for inspirations and encouraging conversations. This work was partially supported by the National Natural Science Foundation of China (11171231).

Received 11 October 2015

Published 17 July 2017