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 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)

Example

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.

Properties

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

[math]\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} }[/math]

where [math]\displaystyle{ (q)_{\infty}=\prod_{n=1}^{\infty} (1-q^n) }[/math].

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

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

is a mock modular form of weight 3/2 on the full modular group [math]\displaystyle{ SL_2(\mathbb{Z}) }[/math] with multiplier system [math]\displaystyle{ \chi_{\eta}^{-1} }[/math], where [math]\displaystyle{ \chi_{\eta} }[/math] is the multiplier system for [math]\displaystyle{ \eta(z) }[/math].

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

[math]\displaystyle{ \mathrm{spt}(5n+4) \equiv 0 \mod(5) }[/math]

[math]\displaystyle{ \mathrm{spt}(7n+5) \equiv 0 \mod(7) }[/math]

[math]\displaystyle{ \mathrm{spt}(13n+6) \equiv 0 \mod(13). }[/math]

References

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