Generalization of a few results in integer partitions

Manosij Ghosh Dastidar and Sourav Sen Gupta
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 19, 2013, Number 2, Pages 69—76
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Authors and affiliations

Manosij Ghosh Dastidar
Ramakrishna Mission Vidyamandira
Belur, West Bengal, India

Sourav Sen Gupta
Indian Statistical Institute
Kolkata, India
* Corresponding author

Abstract

In this paper, we generalize a few important results in Integer Partitions; namely the results known as Stanley’s theorem and Elder’s theorem, and the congruence results proposed by Ramanujan for the partition function. We generalize the results of Stanley and Elder from a fixed integer to an array of subsequent integers, and propose an analogue of Ramanujan’s congruence relations for the ‘number of parts’ function instead of the partition function. We also deduce the generating function for the ‘number of parts’, use it to provide an alternative proof of Ramaunjan’s congruence relations, and relate the technical results with their graphical interpretations through a novel use of the Ferrer’s diagrams.

Keywords

  • Stanley’s theorem
  • Elder’s theorem
  • Ramanujan congruences
  • Ferrer’s diagram

AMS Classification

  • 05A17

References

  1. Ahlgren, S., K. Ono. Congruences and conjectures for the partition function. Proceedings of the Conference on q-series with Applications to Combinatorics, Number Theory and Physics, AMS Contemporary Mathematics, Vol. 291, 2001, 1–10.
  2. Sloane. N. J. A. Sum_{k = 0…n}p(k) where p(k) = number of partitions of k. The on-line encyclopedia of integer sequences, 2010. Available at http://www.research.att. com/˜njas/sequences/A000070.
  3. Sloane. N. J. A. Triangle T(n, k); n >= 1; 1 <= k <= n, giving number of k’s in all partitions of n. The on-line encyclopedia of integer sequences, 2010. Available at http://www.research.att.com/˜njas/sequences/A066633.
  4. Wolfram Mathworld. Elder’s Theorem. 2010. Available at http://mathworld.wolfram.com/EldersTheorem.html
  5. Wolfram Mathworld. Stanley’s Theorem. 2010. Available at http://mathworld.wolfram.com/StanleysTheorem.html

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Cite this paper

APA

Manosij, G. D., & Sourav, S. G. (2013). Generalization of a few results in integer partitions. Notes on Number Theory and Discrete Mathematics, 19(2), 69-76.

Chicago

Manosij, Ghosh Dastidar, and Sen Gupta Sourav. “Generalization of a Few Results in Integer Partitions.” Notes on Number Theory and Discrete Mathematics 19, no. 2 (2013): 69-76.

MLA

Manosij, Ghosh Dastidar, and Sen Gupta Sourav. “Generalization of a Few Results in Integer Partitions.” Notes on Number Theory and Discrete Mathematics 19.2 (2013): 69-76. Print.

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