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: (1,1,1,1), (1,1,2), (1,3), (2,2) and (4). These partitions have 4, 2, 1, 2 and 1 smallest parts respectively. So spt(4) = 4 + 2 + 1 + 2 + 1 = 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). There are connections to Maass forms, and under certain conditions the generating function is an eigenform for some Hecke operators.[1]

While a closed formula is not known for spt(n), there are Ramanajuan-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) [2]

References[edit]

  1. ^ Frank Garvan. Congruences for Andrews’ spt-function modulo 32760 and extension of Atkin’s Hecke-type partition congruences. 
  2. ^ George Andrews. The number of smallest parts in the partitions of n.