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.^[1]

The first few values of spt(n) are:

1, 3, 5, 10, 14, 26, 35, 57, 80, 119, 161, 238, 315, 440, 589 ... (sequence A092269 in the OEIS)

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 +たす 1 +たす 2 +たす 2 +たす 4 =わ 10.

Like the partition function, spt(n) has a generating function. It is given by

${\displaystyle S(q)=\sum _{n=1}^{\infty }\mathrm {spt} (n)q^{n}={\frac {1}{(q)_{\infty }}}\sum _{n=1}^{\infty }{\frac {q^{n}\prod _{m=1}^{n-1}(1-q^{m})}{1-q^{n}}}}${\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 ${\displaystyle (q)_{\infty }=\prod _{n=1}^{\infty }(1-q^{n})}${\displaystyle (q)_{\infty }=\prod _{n=1}^{\infty }(1-q^{n})}.

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

${\displaystyle {\tilde {S}}(z):=q^{-1/24}S(q)-{\frac {1}{12}}{\frac {E_{2}(z)}{\eta (z)}}}${\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 ${\displaystyle SL_{2}(\mathbb {Z} )}${\displaystyle SL_{2}(\mathbb {Z} )} with multiplier system ${\displaystyle \chi _{\eta }^{-1}}${\displaystyle \chi _{\eta }^{-1}}, where ${\displaystyle \chi _{\eta }}${\displaystyle \chi _{\eta }} is the multiplier system for ${\displaystyle \eta (z)}${\displaystyle \eta (z)}.

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

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

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

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

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