Spt function

The spt function (smallest parts function) is a function in number theory that counts the sum of the number of smallest parts in each integer partition of a positive integer. It is related to the partition function.^[1]

The first few values of spt(n) are:

1, 3, 5, 10, 14, 26, 35, 57, 80, 119, 161, 238, 315, 440, 589 ... (sequence A092269 in the OEIS)

For example, there are five partitions of 4 (with smallest parts underlined):

4

3 + 1

2 + 2

2 + 1 + 1

1 + 1 + 1 + 1

These partitions have 1, 1, 2, 2, and 4 smallest parts, respectively. So spt(4) = 1 + 1 + 2 + 2 + 4 = 10.

Like the partition function, spt(n) has a generating function. It is given by

${\displaystyle S(q)=\sum _{n=1}^{\infty }\mathrm {spt} (n)q^{n}={\frac {1}{(q)_{\infty }}}\sum _{n=1}^{\infty }{\frac {q^{n}\prod _{m=1}^{n-1}(1-q^{m})}{1-q^{n}}}}$

where ${\displaystyle (q)_{\infty }=\prod _{n=1}^{\infty }(1-q^{n})}$.

The function ${\displaystyle S(q)}$ is related to a mock modular form. Let ${\displaystyle E_{2}(z)}$ denote the weight 2 quasi-modular Eisenstein series and let ${\displaystyle \eta (z)}$ denote the Dedekind eta function. Then for ${\displaystyle q=e^{2\pi iz}}$, the function

${\displaystyle {\tilde {S}}(z):=q^{-1/24}S(q)-{\frac {1}{12}}{\frac {E_{2}(z)}{\eta (z)}}}$

is a mock modular form of weight 3/2 on the full modular group ${\displaystyle SL_{2}(\mathbb {Z} )}$ with multiplier system ${\displaystyle \chi _{\eta }^{-1}}$, where ${\displaystyle \chi _{\eta }}$ is the multiplier system for ${\displaystyle \eta (z)}$.

While a closed formula is not known for spt(n), there are Ramanujan-like congruences including

${\displaystyle \mathrm {spt} (5n+4)\equiv 0\mod (5)}$

${\displaystyle \mathrm {spt} (7n+5)\equiv 0\mod (7)}$

${\displaystyle \mathrm {spt} (13n+6)\equiv 0\mod (13).}$