# Combinatorial proofs for identities related to generalizations of the mock theta functions \(\omega (q)\) and \(\nu (q)\)

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## Abstract

The two partition functions \(p_\omega (n)\) and \(p_\nu (n)\) were introduced by Andrews, Dixit and Yee, which are related to the third-order mock theta functions \(\omega (q)\) and \(\nu (q)\), respectively. Recently, Andrews and Yee analytically studied two identities that connect the refinements of \(p_\omega (n)\) and \(p_\nu (n)\) with the generalized bivariate mock theta functions \(\omega (z;q)\) and \(\nu (z;q)\), respectively. However, they stated these identities begged for bijective proofs. In this paper, we first define the generalized trivariate mock theta functions \(\omega (y,z;q)\) and \(\nu (y,z;q)\). Then by utilizing odd Ferrers graph, we obtain certain identities concerning to \(\omega (y,z;q)\) and \(\nu (y,z;q)\), which extend some early results of Andrews that are related to \(\omega (z;q)\) and \(\nu (z;q)\). In virtue of the combinatorial interpretations that arise from the identities involving \(\omega (y,z;q)\) and \(\nu (y,z;q)\), we finally present bijective proofs for the two identities of Andrews–Yee.

## Keywords

Partitions Bijections Mock theta functions Odd Ferrers graph## Mathematics Subject Classification

05A17 05A19## Notes

### Acknowledgements

The authors appreciate the referee for his/her helpful comments which improved the quality of this manuscript.

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