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.{{Cite journal |last=Andrews |first=George E. |author-link=George Andrews (mathematician) |date=2008-11-01 |title=The number of smallest parts in the partitions of n |url=https://www.degruyter.com/document/doi/10.1515/CRELLE.2008.083/html |journal= Journal für die Reine und Angewandte Mathematik |language=en |volume=2008 |issue=624 |pages=133–142 |doi=10.1515/CRELLE.2008.083 |s2cid=123142859 |issn=1435-5345}}

The first few values of spt(n) are:

:1, 3, 5, 10, 14, 26, 35, 57, 80, 119, 161, 238, 315, 440, 589 ... {{OEIS|id=A092269}}

Example

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

:{{underline|4}}

:3 + {{underline|1}}

:{{underline|2}} + {{underline|2}}

:2 + {{underline|1}} + {{underline|1}}

:{{underline|1}} + {{underline|1}} + {{underline|1}} + {{underline|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

: $S(q)=\sum_{n=1}^{\infty} \mathrm{spt}(n) q^n=\frac{1}{(q)_{\infty}}\sum_{n=1}^{\infty} \frac{q^n \prod_{m=1}^{n-1}(1-q^m)}{1-q^n}$

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

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 $q=e^{2\pi i z}$ , 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 $SL_2(\mathbb{Z})$ 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

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

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

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

References

Category:Combinatorics

Category:Integer sequences