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A Combinatorial Proof of a Recursive Formula for Multipartitions

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Veröffentlicht/Copyright: 24. Januar 2012
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Integers
Aus der Zeitschrift Band 12 Heft 1

Abstract.

For k≥1, let pk(n) count the number of k-component multipartitions of a nonnegative integer n, and let σ(n)=∑d∣nd be the usual divisor function. In this paper, we give a combinatorial proof of the recursive formula

pk(n)=kn∑r=1npk(n-r)σ(r),

both for k≥1, where pk(n) is defined as above, and also for k<0, which requires a subtler approach. This formula was used by Gandhi in 1963 to prove several theorems, which yield numerous Ramanujan type congruences for pk(n), including some well-known congruences for Ramanujan's τ-function.

Keywords.: Multipartitions; Bijections; Congruences
Received: 2011-04-27
Revised: 2011-06-25
Accepted: 2011-08-22
Published Online: 2012-01-24
Published in Print: 2012-February

© 2012 by Walter de Gruyter Berlin Boston

Artikel in diesem Heft

  1. Prelims
  2. Selmer's Multiplicative Algorithm
  3. Lucas Numbers and Determinants
  4. A Combinatorial Proof of Guo's Multi-Generalization of Munarini's Identity
  5. Enumeration of Triangles in Quartic Residue Graphs
  6. A remark on the Boros–Moll Sequence
  7. A Relation Between Triangular Numbers and Prime Numbers
  8. Some Remarks on a Paper of V. A. Liskovets
  9. A Combinatorial Proof of a Recursive Formula for Multipartitions
  10. Aliquot Cycles of Repdigits
  11. Digital Sums and Functional Equations
https://doi.org/10.1515/integ.2011.089
Schlagwörter für diesen Artikel
Multipartitions; Bijections; Congruences

Artikel in diesem Heft

  1. Prelims
  2. Selmer's Multiplicative Algorithm
  3. Lucas Numbers and Determinants
  4. A Combinatorial Proof of Guo's Multi-Generalization of Munarini's Identity
  5. Enumeration of Triangles in Quartic Residue Graphs
  6. A remark on the Boros–Moll Sequence
  7. A Relation Between Triangular Numbers and Prime Numbers
  8. Some Remarks on a Paper of V. A. Liskovets
  9. A Combinatorial Proof of a Recursive Formula for Multipartitions
  10. Aliquot Cycles of Repdigits
  11. Digital Sums and Functional Equations
Heruntergeladen am 29.4.2026 von https://www.degruyterbrill.com/document/doi/10.1515/integ.2011.089/html?lang=de
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