A Formula for the Partition Function That "Counts"

Yuriy Choliy, Andrew V. Sills

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

We derive a combinatorial multisum expression for the number D(n, k) of partitions of n with Durfee square of order k. An immediate corollary is therefore a combinatorial formula for p(n), the number of partitions of n. We then study D(n, k) as a quasipolynomial. We consider the natural polynomial approximation D~ (n, k) to the quasipolynomial representation of D(n, k). Numerically, the sum ∑1≤k≤nD~(n,k) appears to be extremely close to the initial term of the Hardy-Ramanujan-Rademacher convergent series for p(n).

Original languageAmerican English
JournalAnnals of Combinatorics
Volume20
DOIs
StatePublished - Jun 1 2016

Keywords

  • Durfee square
  • Integer partitions
  • Partition function

DC Disciplines

  • Education
  • Mathematics

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