Spt function

Last updated March 04, 2024

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]

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

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 $q=e^{2\pi iz}$ , 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 $SL_{2}(\mathbb {Z} )$ 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).

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References

↑ Andrews, George E. (2008-11-01). "The number of smallest parts in the partitions of n". Journal für die Reine und Angewandte Mathematik (Crelles Journal). 2008 (624): 133–142. doi:10.1515/CRELLE.2008.083. ISSN 1435-5345. S2CID 123142859.

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[1] Andrews, George E. (2008-11-01). "The number of smallest parts in the partitions of n". Journal für die Reine und Angewandte Mathematik (Crelles Journal). 2008 (624): 133–142. doi:10.1515/CRELLE.2008.083. ISSN 1435-5345. S2CID 123142859.

[1]

Spt function

Contents

Example

Properties

Related Research Articles

References