Abstract
We develop the theory of copartitions, which are a generalization of partitions with connections to many classical topics in partition theory, including Rogers–Ramanujan partitions, theta functions, mock theta functions, partitions with parts separated by parity, and crank statistics. Using both analytic and combinatorial methods, we give two forms of the three-parameter generating function, and we study several special cases that demonstrate the potential broader impact the study of copartitions may have.
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References
George E. Andrews, The number of smallest parts in the partitions of\(n\), J. Reine. Angew. Math. 624 (2008), 133–142.
George E. Andrews, Integer partitions with even parts below odd parts and the mock theta functions, Ann. Comb. 22 (2018), no. 3, 433–445.
George E. Andrews,Partitions with parts separated by parity, Ann. Comb. 23 (2019), no. 2, 241–248.
Hannah E. Burson and Dennis Eichhorn, On the parity of the number of \((a,b,m)\)-copartitions of \(n\). https://arxiv.org/pdf/2201.04247 (2022).
Hannah E. Burson and Dennis Eichhorn, On the positivity of infinite products connected to partitions with even parts below odd parts and copartitions. https://arxiv.org/pdf/2209.13784 (2022).
Shane Chern, On a problem of George Andrews concerning partitions with even parts below odd parts, Afr. Mat. 30 (2019), no. 5-6, 691–695.
Shane Chern, Note on partitions with even parts below odd parts, Math. Notes 110 (2021), no. 3-4, 454–457.
Frank Garvan and Michael J. Schlosser, Combinatorial interpretations of Ramanujan’s tau function, Discrete Math. 341 (2018), no. 10, 2831–2840.
Valery Gritsenko, Nils-Peter Skoruppa, and Don Zagier, Theta blocks, preprint (2019).
G. H. Hardy, Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, The University Press, Cambridge [England], 1940.
Byungchan Kim, On the number of partitions of n into k different parts, J. Number Theory 132 (2012), no. 6, 1306–1313.
Arnold Knopfmacher and Neville Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9–18.
Neil J. A. Sloane and The OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, http://oeis.org, 2022.
NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.19 of 2018-06-22, F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V. Saunders, eds.
L.J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894), no. 1, 318–343.
L.J. Rogers and Srinivasa Ramanujan, Proof of certain identities in combinatory analysis, Proc. Camb. Phil. Soc. 19 (1919), 211–216.
Issai Schur, Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche, S.-B. Preuss. Akad. Wiss. Phys.-Math. Kl. 302-321 (1917), 117–136.
Armin Straub, Core partitions into distinct parts and an analog of Euler’s theorem, Eur. J. Comb. 57 (2016), 40–49.
Acknowledgements
The authors would like to thank George Andrews for suggesting taking a deeper look at \(\mathcal{EO}\mathcal{}^*(n)\). The authors would also like to thank Frank Garvan for bringing the concept of capsids and [8] to our attention. Finally, the authors would like to thank the anonymous referees for their very thoughtful and helpful comments.
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Communicated by Jang Soo Kim.
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Burson, H.E., Eichhorn, D. Copartitions. Ann. Comb. 27, 519–537 (2023). https://doi.org/10.1007/s00026-022-00607-1
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DOI: https://doi.org/10.1007/s00026-022-00607-1