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 \[ 3,\quad \bar{3},\quad 2 + 1,\quad \bar{2}+1, \quad 2+\bar{1},\quad \bar{2}+\bar{1},\quad 1+1+1, \quad\bar{1}+1+1. \] Let $\bar{p}(n)$ denote the number of overpartitions of the integer $n$, so that, for example, $\bar{p}(3)=8$ from above.
Then the generating function for the sequence $\{\bar{p}(n)\}_{n=0}^{\infty}$ (as usual, $\bar{p}(0)$ is defined to have the value 1) has closed form given by \begin{equation}\label{overparteq1} \sum_{n=0}^{\infty}\bar{p}(n)q^n = \prod_{j=1}^{\infty}\frac{1+q^j}{1-q^j}. \end{equation}

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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