Abstract
Andrews, Hirschhorn, and Sellers studied the partition function ped(n) which enumerates the number of partitions of n with even parts distinct, and obtained a number of interesting congruences. This paper aims to introduce a partition statistic to investigate the partition function ped(n). We give combinatorial interpretations for some properties of ped(n) including the infinite families of congruences given by Andrews et al.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrews, G.E.: Generalized Frobenius partitions. Mem. Amer. Math. Soc. 49, 301 (1984)
Andrews, G.E., Garvan, F.G.: Dyson’s crank of a partition. Bull. Amer. Math. Soc. 18(2), 167–171 (1988)
Andrews, G.E., Hirschhorn, M.D., Sellers, J.A.: Arithmetic properties of partitions with even parts distinct. Ramanujan J. 23(1), 169–181 (2010)
Berndt, B.C.: Ramanujan’s Notebooks Part III. Springer-Verlag, New York (1991)
Chern, S., Hao, L.J.: Congruences for partition functions related to mock theta functions. Ramanujan J. 48(2), 369–384 (2019)
Garvan, F.G.: New combinatorial interpretations of Ramanujan’s partition congruences mod \(5\), \(7\), \(11\). Trans. Am. Math. Soc. 305(1), 47–77 (1988)
Garvan, F.G.: The crank of partitions mod 8, 9 and 10. Trans. Amer. Math. Soc. 322(1), 79–94 (1990)
Hirschhorn, M.D., Sellers, J.A.: A congruence modulo \(3\) for partitions into distinct non-multiples of four. J. Integer Seq. 17(9), 14–19 (2014)
Merca, M.: New relations for the number of partitions with distinct even parts. J. Number Theory. 176, 1–12 (2017)
Seo, S., Yee, A.J.: Overpartitions and singular overpartitions. Analytic Number Theory, Modular Forms and \(q\)-Hypergeometric Series: In Honor of Krishna Alladi’s 60th Birthday. University of Florida, Gainesville, 693–711 (2016)
Sloane, N.J.A.: The on-line encyclopedia of integer sequences, published electronically at http://oeis.org
Xia, E.X.W.: New infinite families of congruences modulo 8 for partitions with even parts distinct. Electron J. Combin. 21(4), P4–P8 (2014)
Yao, O.X.M., Xia, E.X.W.: New Ramanujan-like congruences modulo powers of 2 and 3 for overpartitions. J. Number Theory. 133(6), 1932–1949 (2013)
Yee, A.J.: Combinatorial proofs of generating function identities for F-partitions. J. Combin. Theory A. 102(1), 217–228 (2003)
Acknowledgements
The author would like to thank the anonymous referee for making valuable comments and suggestions that significantly improved the quality of this paper. This work was supported by the National Natural Science Foundation of China (No. 12101307).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Ilse Fischer.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Hao, R.X.J. A partition statistic for partitions with even parts distinct. Monatsh Math 201, 1105–1123 (2023). https://doi.org/10.1007/s00605-022-01816-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-022-01816-9