Spt function

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The spt function (smallest parts function) is a function in number theory that counts the sum of the number of smallest parts in each partition of a positive integer. It is related to the partition function.

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 OEIS)

Example[edit]

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[edit]

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

S(q)=\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) [1]

References[edit]

  1. ^ George Andrews. "The number of smallest parts in the partitions of n".