George Andrews introduced the smallest parts partition function in [A08], where it was denoted by $spt(n)$ and defined to be the number of occurrences of the smallest part in all the partitions of $n$. For example, all occurrences of the smallest part in the partitions of 5 are marked below with a "." over them, \[ \dot 5, \quad 4+ \dot 1, \quad 3+\dot 2, \quad 3+\dot 1+ \dot 1, \quad 2+ 2+ \dot 1, \quad 2+ \dot 1+ \dot 1+ \dot 1, \quad \dot 1+ \dot 1+ \dot 1+ \dot 1+ \dot 1. \] Thus, counting the parts with a "." over them, it can be seen that $spt(5)= 14$.

One result given by Andrews in [A08] was a form for the generating function: \begin{equation}\label{spteq1} \sum _{n=1}^{\infty } spt(n)q^n=\frac{1}{(q;q)_{\infty }} \sum _{n=1}^{\infty } \frac{n q^n}{1-q^n} +\frac{1}{(q;q)_{\infty }} \sum _{n=1}^{\infty } \frac{(-1)^n q^{\frac{1}{2} n (3 n+1)} \left(1+q^n\right)}{\left(1-q^n\right)^2}. \end{equation} Implicit in the proof in [A08] was an alternative form of the generating function, \begin{equation}\label{spteq2} \sum _{n=1}^{\infty } spt(n)q^n= \frac{1}{(q;q)_{\infty }} \sum _{n=1}^{\infty } \frac{(q;q)_n q^{n} }{\left(1-q^n\right)^2}. \end{equation}

Similar to the congruences for the regular partition function $p(n)$ stated by Ramanujan, namely, that for all $k\geq 0$, \begin{align*} p(5k+4)&\equiv 0 (\mod 5),\\ p(7k+5)&\equiv 0 (\mod 7), \\ p(11k+6)&\equiv 0 (\mod 11), \end{align*} Andrews [A08] found three families of congruences for $spt(n)$: \begin{align}\label{spteq3} spt(5k+4)&\equiv 0 (\mod 5),\\ spt(7k+5)&\equiv 0 (\mod 7), \\ spt(13k+6)&\equiv 0 (\mod 13). \end{align} The modulo 13 congruence for $spt(n)$ is surprising, given that there is no similar elementary congruence modulo 13 for $p(n)$.

Recall that the rank of a Partition is defined to be the largest part minus the number of parts. The number of partitions of $n$ with rank $m$ is denoted $N(m,n)$. Atkin and Garvan introduced in [AG03] the moments of ranks: \[ N_j(n) = \sum_{m=-\infty}^{\infty}m^j N(m,n). \] A third result of Andrews in [A08] was to show that \begin{equation}\label{spteq4} spt(n) = np(n) −\frac{1}{2}N_2 (n). \end{equation} In [B08] Bringmann gave an asymptotic expansion for $N_2(n)$, which implies that \begin{equation}\label{spteq5} spt(n)\sim \frac{\sqrt{6n}p(n)}{\pi}. \end{equation} If the well-known asymptotic estimate for $p(n)$ is inserted in the last formula, it implies that \begin{equation}\label{spteq6} spt(n)\sim \frac{1}{\pi\sqrt{8n}}e^{\pi \sqrt{2n/3}}. \end{equation} Let \[ S(n)=\sum_{k=0}^{n}p(k). \] Hirschhorn made the conjecture, later proved by Eichhorn and Hirschhorn in [EH15], that \begin{equation}\label{spteq7} S(n − 1) < spt(n) < S(n). \end{equation}

[A08] G. E. Andrews, The number of smallest parts in the partitions of $n$, J. Reine Angew. Math. 624 (2008), 133–142.

[AG03] A. O. L. Atkin and F. G. Garvan, Relations between the ranks and cranks of partitions, Ramanujan J., 7 (2003), 343–366.

[B08] K. Bringmann, On the explicit construction of higher deformations of partition statistics. Duke Math. J. 144 (2008), no. 2, 195–233.

[EH15] D. A. Eichhorn, and M. D. Hirschhorn, Notes on the spt function of George E. Andrews. Ramanujan J. 38 (2015), no. 1, 17–34.

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