A formula for the partition function that “counts”
نویسندگان
چکیده
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 D̃(n, k) to the quasipolynomial representation of D(n, k). Numerically, the sum ∑ 1≤k≤ √ n D̃(n, k) appears to be extremely close to the initial term of the Hardy–Ramanujan– Rademacher convergent series for p(n). Mathematics Subject Classification (2010). Primary 05A17 Secondary 11P81.
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