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.

The first few values of spt(n) are:

1, 3, 5, 10, 14, 26, 35, 57, 80, 119, 161, 238, 315, 440, 589 ... Template:OEIS

Example

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

Template:Underline

3 + Template:Underline

Template:Underline + Template:Underline

2 + Template:Underline + Template:Underline

Template:Underline + Template:Underline + Template:Underline + Template:Underline

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

Properties

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

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

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

The function $S(q)$ is related to a mock modular form. Let $E_2(z)$ denote the weight 2 quasi-modular Eisenstein series and let $\eta (z)$ denote the Dedekind eta function. Then for $q=e^{2\pi iz}$, the function

$\tilde{S}(z):=q^{-1/24}S(q)-\frac{1}{12}\frac{E_2(z)}{\eta (z)}$

is a mock modular form of weight 3/2 on the full modular group $S{L}_2(\mathbb{Z})$ with multiplier system ${\chi }_{\eta }^{-1}$, where ${\chi }_{\eta }$ is the multiplier system for $\eta (z)$.

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

$\mathrm{s}\mathrm{p}\mathrm{t}(5n+4)\equiv 0mod(5)$

$\mathrm{s}\mathrm{p}\mathrm{t}(7n+5)\equiv 0mod(7)$

$\mathrm{s}\mathrm{p}\mathrm{t}(13n+6)\equiv 0mod(13).$

References

Template:Numtheory-stub Template:Combin-stub