Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 097, 14 pages      arXiv:2406.12049      https://doi.org/10.3842/SIGMA.2024.097
Contribution to the Special Issue on Basic Hypergeometric Series Associated with Root Systems and Applications in honor of Stephen C. Milne

Combinatorial Interpretations of Cranks of Overpartitions and Partitions without Repeated Odd Parts

Frank G. Garvan and Rishabh Sarma
Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, FL 32611-8105, USA

Received June 19, 2024, in final form October 12, 2024; Published online October 26, 2024

Abstract
We give combinatorial interpretations of two residual cranks of overpartitions defined by Bringmann, Lovejoy and Osburn in 2009 analogous to the crank of partitions given by Andrews and the first author in 1988. As a consequence, we give new versions of their definitions without adjusted weights. Furthermore, we investigate the combinatorial interpretation of an $M_2$-crank of partitions without repeated odd parts and explore connections of these statistics with their companion rank counterparts and the tenth order mock theta functions of Ramanujan.

Key words: overpartitions; residual crank.

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