Combinatorial and arithmetic properties of nr-partitions: A unified analytic generalization

Mateus Alegri, A. David Christopher and Elen Viviani Pereira Spreafico
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 3, Pages 497–513
DOI: 10.7546/nntdm.2026.32.3.497-513
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Authors and affiliations

Mateus Alegri
Department of Mathematics, Federal University of Sergipe
Itabaiana, Sergipe, Brazil

A. David Christopher
Department of Mathematics, The American College
Madurai, Tamil Nadu, India

Elen Viviani Pereira Spreafico
Department of Mathematics, Federal University of Mato Grosso do Sul
Campo Grande, Mato Grosso do Sul, Brazil

Abstract

This paper presents a comprehensive study of n^r-partitions, defined by the generating function \prod_{n=1}^{\infty} (1 - q^n)^{-n^r}, which interpolates analytically between ordinary partitions (r=0), plane partitions (r=1), and higher-dimensional combinatorial structures (r\geq2).

Our main contributions include: (1) explicit formulas relating p_{(r)}(n) to divisor sums \sigma_{r+1}(n) and integer compositions; (2) novel recurrence relations and congruence properties modulo primes and composite numbers; (3) complete Bell polynomial representations; and (4) combinatorial interpretations through generating function techniques. For weighted restricted n^r-partitions into three parts, we establish exact quasi-polynomial formulas.

The results extend classical partition theory while providing new connections between combinatorics and number theory. The methods developed here offer a unified framework for investigating partition-like structures across dimensions, with applications in enumerative combinatorics.

Keywords

  • n^r-partitions
  • Plane partitions
  • Bell polynomials
  • Sum of divisors functions
  • Partition congruences
  • Ehrhart theory

2020 Mathematics Subject Classification

  • 05A17
  • 11P81
  • 11P83
  • 05A19

References

  1. Agarwal, A. K., & Andrews, G. E. (1987). Rogers–Ramanujan identities for partitions with “n copies of n”. Journal of Combinatorial Theory, Series A, 45(1), 40–49.
  2. Ahlgren, S., & Ono, K. (2001). Congruence properties for the partition function. Proceedings of the National Academy of Sciences of the United States of America, 98(23), 12882–12884.
  3. Alegri, M. (2024). Identities involving sum of divisors, compositions and integer partitions hiding in the q-binomial theorem. Sao Paulo Journal of Mathematical Sciences, 18, 1–13.
  4. Alegri, M. (2022). Identities involving sum of divisors, integer partitions and compositions. Online Journal of Analytic Combinatorics, 17, Article ID 3.
  5. Andrews, G. E. (2008). A survey of multipartitions: Congruences and identities. In: Alladi, K. (Ed.). Surveys in Number Theory, 1–19, Springer.
  6. Andrews, G. E., Kumar Jha, S., & Lopez-Bonilla, J. (2023). Sums of squares, triangular numbers, and divisor sums. Journal of Integer Sequences, 26, Article ID 23.2.5.
  7. Beck, M., & Robins, S. (2015). Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra. Springer, New York, NY.
  8. Berline, N., & Vergne, M. (2007). Local Euler–Maclaurin formula for polytopes. Moscow Mathematical Journal, 7 (3), 355–383.
  9. Brion, M. (1988). Points entiers dans les polyèdres convexes. Annales scientifiques de l’École Normale Supérieure, 21 (4), 653–663.
  10. Dousse, J. (2016). A generalisation of a partition theorem of Andrews. Monatshefte fur Mathematik, 179, 227–251.
  11. Euler, L. (1911). De partitione numerorum. In: Opera Omnia. Teubner, 2, 254–294.
  12. Guru, G. S., & Jamudulia, Y. (2026). On congruences modulo powers of 5 for k-tuple ℓ-regular partitions. Annali dell’Universita di Ferrara, 72, Article ID 51.
  13. MacMahon, P. A. (1896). Memoir on the theory of the partition of numbers. Part I. Philosophical Transactions of the Royal Society London Series A, 187, 619–673.
  14. MacMahon, P. A. (1916). Combinatory Analysis, Vol. 2. Cambridge University Press.
  15. Nadji, M. L., & Ahmia, M. (2024). Congruences for ℓ-regular tripartitions for ℓ ∈ {2, 3}. Integers, 24, Article ID #A86.
  16. Nath, H., Saikia, M. P., & Sarma, A. (2025). Arithmetic properties of k-tuple ℓ-regular partitions. Journal of Mathematical Analysis and Applications, 551(2), Article ID 129688.
  17. OEIS Foundation Inc. (2025). Entry A000219. The On-Line Encyclopedia of Integer Sequences. Available online at: https://oeis.org/A000219.
  18. OEIS Foundation Inc. (2025). Entry A000041. The On-Line Encyclopedia of Integer Sequences. Available online at: https://oeis.org/A000041.
  19. Ono, K. (2000). Distribution of the partition function modulo m. Annals of Mathematics, 151(1), 293–307.
  20. Ramanujan, S. (1916). On certain arithmetical functions. Transactions of the Cambridge Philosophical Society, 22(9), 159–184.
  21. Ranganatha, D. (2016). On a Ramanujan-type congruence for bipartitions with 5-cores. Journal of Integer Sequences, 19, Article ID 16.8.1.
  22. Saikia, M. P., Sarma, A., & Sellers, J. A. (2025). Arithmetic properties modulo powers of 2 for overpartition k-tuples with odd parts. Journal of the Ramanujan Mathematical Society, 40(2), 107–123.
  23. Slater, L. J. (1952). Further identities of the Rogers–Ramanujan type. Proceedings of the London Mathematical Society, Series 2, 54, 147–167.
  24. Sriram, S., & Christopher, A. D. (2024). Congruences relating regular partition functions, a generalized tau function, and partition function weighted composition sums. Integers, 24, Article ID #A39.
  25. Sriram, S., & Christopher, A. D. (2024). Congruences and integral roots of D’Arcais polynomials. Research in Number Theory, 10, Article ID 9.

Manuscript history

  • Received: 10 November 2025
  • Revised: 15 September 2026
  • Accepted: 6 October 2026
  • Online First: 15 October 2026

Copyright information

Ⓒ 2026 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Alegri, M., David Christopher, A., Pereira Spreafico, E. V. (2026). Combinatorial and arithmetic properties of nr-partitions: A unified analytic generalization. Notes on Number Theory and Discrete Mathematics, 32(3), 497-513, DOI: 10.7546/nntdm.2026.32.3.497-513.

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