Partitions generated by Mock Theta Functions ρ(q), σ(q) and ν(q) and relations with partitions into distinct parts

Alessandro Bagatini, Marília Luiza Matte and Adriana Wagner
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 1, Pages 54–74
DOI: 10.7546/nntdm.2019.25.1.54-74
Full paper (PDF, 252 Kb)

Details

Authors and affiliations

Alessandro Bagatini
IME, Universidade Federal do Rio Grande do Sul
Av. Bento Gonçalves, 9500 – 90509-900, Porto Alegre-RS, Brazil
Present address: Instituto Federal Catarinense–IFC
88960-000, Sombrio-SC, Brazil

Marília Luiza Matte
IME, Universidade Federal do Rio Grande do Sul
Av. Bento Gonçalves, 9500 – 90509-900, Porto Alegre-RS, Brazil
Present address: Colégio Militar de Porto Alegre–CMPA
90040-130, Porto Alegre-RS, Brazil

Adriana Wagner
IMECC–Universidade Estadual de Campinas
Rua Sérgio Buarque de Holanda, 651 – 13083-859, Campinas-SP, Brazil
Present address: Campus de Aquidauana–UFMS
79200-000, Aquidauana-MS, Brazil

Abstract

From two-line matrix interpretations of Mock Theta Functions ρ(q), σ(q) and ν(q) introduced in [5], we have obtained identities for the partitions generated by their respective general terms, whose proofs are done in a completely combinatorial way. We have also obtained relations between partitions into two colours generated by ρ(q) and σ(q), and also by ν(q).

Keywords

• Mock Theta Function
• Integer partition
• Combinatorial interpretation
• Partition enumeration

2010 Mathematics Subject Classification

• 11P81
• 05A19

References

1. Andrade, C. P., da Silva, R., Santos, J. P. O., Silva, K. C. P., & Spreafico, E. V. P. Some consequences of the two-line matrix representations for partitions. Preprint.
2. Andrews, G. E. (1998). The Theory of Partitions, Cambridge University Press.
3. Bagatini, A., Matte, M. L., & Wagner, A. (2017). Identities for partitions generated by the unsigned versions of some mock theta functions. Bulletin of the Brazilian Mathematical Society, New Series, Springer, 48 (3), 413–437.
4. Brietzke, E. H. M., Santos, J. P. O., & da Silva, R. (2010). Bijective proofs using two-line matrix representations for partitions, The Ramanujan Journal, Springer, 23 (1–3), 265–295.
5. Brietzke, E. H. M., Santos, J. P. O., & da Silva, R. (2013). Combinatorial interpretations as two-line array for the mock theta functions, Bulletin of the Brazilian Mathematical Society, New Series, Springer, 44 (2), 233–253.
6. Santos, J. P. O., Mondek, P., & Ribeiro, A. C. (2011). New two-line arrays representing partitions, Annals of Combinatorics, Springer, 15 (2), 341–354.
7. Wagner, A., Bagatini, A., & Matte, M. (2017). On new results about partitions into parts congruent to ±1 (mod 5), Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, 5 (1), Article 0226, 7 pages, DOI: 10.5540/03.2017.005.01. 0226.