Annals of Combinatorics

, Volume 20, Issue 2, pp 301–316 | Cite as

A Formula for the Partition Function That “Counts”



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 \({\tilde{D}(n, k)}\) to the quasipolynomial representation of D(n, k). Numerically, the sum \({\sum_{1\leq k \leq \sqrt{n}} \tilde{D}(n, k)}\) appears to be extremely close to the initial term of the Hardy-Ramanujan-Rademacher convergent series for p(n).


integer partitions partition function Durfee square 

Mathematics Subject Classification

05A17 11P81 


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© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of Chemistry and Chemical BiologyRutgers University New Brunswick —Busch CampusPiscatawayUSA
  2. 2.Department of Mathematical SciencesGeorgia Southern UniversityStatesboroUSA

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