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Overpartitions

An overpartition of a positive integer n is a non-increasing sequence of positive integers whose sum is n, in which the first occurrence (or equivalently, the final occurrence) of a part may be overlined.

As an example, the overpartitions of 3 are 3,ˉ3,2+1,ˉ2+1,2+ˉ1,ˉ2+ˉ1,1+1+1,ˉ1+1+1. Let ˉp(n) denote the number of overpartitions of the integer n, so that, for example, ˉp(3)=8 from above.
Then the generating function for the sequence {ˉp(n)}n=0 (as usual, ˉp(0) is defined to have the value 1) has closed form given by n=0ˉp(n)qn=j=11+qj1qj.

The modern study of overpartitions began with the paper [2] by Corteel and Lovejoy, in which, amongst other results, they gave overpartition bijective proofs of a number of basic hypergeometric identities, including the Rogers-Fine identity.

Congruences for overpartitions, similar to the Ramanujan congruences for the regular partition function p(n) (see the page on partition pongruences) have been proved. One example is the congruence \bar{p}(40n + 35) \equiv 0 (\mod 40),\,\, \forall \,n \geq 0, which was conjectured by Hirschhorn and Sellers in [3], and proved by Chen and Xia in [1]

[1] W. Y. Chen and E. X. W. Xia, Proof of a conjecture of Hirschhorn and Sellers on overpartitions. Acta Arith. 163 (2014), no. 1, 59–69.

[2] S. Corteel and J. Lovejoy, Overpartitions. Trans. Amer. Math. Soc. 356 (2004), no. 4, 1623–1635.

[3] M.D. Hirschhorn and J.A. Sellers, Arithmetic relations for overpartitions, J. Comb. Math. Comb. Comp. 53 (2005) 65–73.

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