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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 ) = ∑ n = 1 ∞ s p t ( n ) q n = 1 ( q ) ∞ ∑ n = 1 ∞ q n ∏ m = 1 n − 1 ( 1 − q m ) 1 − q n {\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}}}}

where ( q ) ∞ = ∏ n = 1 ∞ ( 1 − q n ) {\displaystyle (q)_{\infty }=\prod _{n=1}^{\infty }(1-q^{n})} .

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

S ~ ( z ) := q − 1 / 24 S ( q ) − 1 12 E 2 ( z ) η ( z ) {\displaystyle {\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 S L 2 ( Z ) {\displaystyle SL_{2}(\mathbb {Z} )} with multiplier system χ η − 1 {\displaystyle \chi _{\eta }^{-1}} , where χ η {\displaystyle \chi _{\eta }} is the multiplier system for η ( z ) {\displaystyle \eta (z)} .

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

s p t ( 5 n + 4 ) ≡ 0 mod ( 5 ) {\displaystyle \mathrm {spt} (5n+4)\equiv 0\mod (5)} s p t ( 7 n + 5 ) ≡ 0 mod ( 7 ) {\displaystyle \mathrm {spt} (7n+5)\equiv 0\mod (7)} s p t ( 13 n + 6 ) ≡ 0 mod ( 13 ) . {\displaystyle \mathrm {spt} (13n+6)\equiv 0\mod (13).}

References

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