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.

The first few values of spt(''n'') are:

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

Example

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

:
:3 +
: +
:2 + +
: + + +

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
:$S(q)=\sum_^ \mathrm(n) q^n=\frac\sum_^ \frac$
where $(q)_=\prod_^ (1-q^n)$.

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 $q=e^$, the function
:$\tilde(z):=q^S(q)-\frac\frac$
is a mock modular form of weight 3/2 on the full modular group $SL_2(\mathbb)$ with multiplier system $\chi_^$, where $\chi_$ is the multiplier system for $\eta(z)$.

While a closed formula is not known for spt(''n''), there are Ramanujan-like congruences including
:$\mathrm(5n+4) \equiv 0 \mod(5)$
:$\mathrm(7n+5) \equiv 0 \mod(7)$
:$\mathrm(13n+6) \equiv 0 \mod(13)$

References

Combinatorics
Integer sequences

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