# Spt function

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The '''spt function''' (smallest parts function) is a function in [number theory](/source/number_theory) that counts the sum of the number of smallest parts in each [integer partition](/source/integer_partition) of a positive integer. It is related to the [partition function](/source/Partition_function_(number_theory)).<ref>{{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|url-access=subscription }}</ref>

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](/source/generating_function). It is given by
:<math>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>(q)_{\infty}=\prod_{n=1}^{\infty} (1-q^n)</math>.

The function <math>S(q)</math> is related to a [mock modular form](/source/mock_modular_form). Let <math>E_2(z)</math> denote the weight 2 quasi-modular [Eisenstein series](/source/Eisenstein_series) and let <math>\eta(z)</math> denote the [Dedekind eta function](/source/Dedekind_eta_function). Then for <math>q=e^{2\pi i z}</math>, the function
:<math>\tilde{S}(z):=q^{-1/24}S(q)-\frac{1}{12}\frac{E_2(z)}{\eta(z)}</math>
is a [mock modular form](/source/mock_modular_form) of weight 3/2 on the full [modular group](/source/modular_group) <math>SL_2(\mathbb{Z})</math> with multiplier system <math>\chi_{\eta}^{-1}</math>, where <math>\chi_{\eta}</math> is the multiplier system for <math>\eta(z)</math>.
<!--
under certain conditions the generating function is an [eigenform](/source/eigenform) for some [Hecke operator](/source/Hecke_operator)s.<ref>{{cite journal | title = Congruences for Andrews’ spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences | author = Frank Garvan}}</ref>
*** this is only true modulo weakly holomorphic modular forms
-->

While a closed formula is not known for spt(''n''), there are Ramanujan-like [congruences](/source/Ramanujan's_congruences) including
:<math>\mathrm{spt}(5n+4) \equiv 0 \mod(5) </math>
:<math>\mathrm{spt}(7n+5) \equiv 0 \mod(7) </math>
:<math>\mathrm{spt}(13n+6) \equiv 0 \mod(13).</math>

==References==
{{Reflist}}

Category:Combinatorics
Category:Integer sequences

{{Numtheory-stub}}
{{combin-stub}}

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Adapted from the Wikipedia article [Spt function](https://en.wikipedia.org/wiki/Spt_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Spt_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.