Spt function explained

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 ...

Example

For example, there are five partitions of 4 (with smallest parts underlined):

3 +

+

2 + +

+ + +

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) = \sum ^{inf ty} _{n =1} spt (n) q ^{n = (1)/((q) _{infty})} \sum ^{inf ty} _{n =1} (q ^{n} \prod ^{n -1} _{m =1} ( 1- q ^{m)})/(1- q ^{n}) where (q) _{infty} = \prod ^{inf ty} _{n =1} ( 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 η (z) denote the Dedekind eta function. Then for q = e ^{2\pi} , the function\tilde{S}(z): = q ^{-1/24} S(q) - (1)/(12) (E _{2(z)})/(η (z)) is a mock modular form of weight 3/2 on the full modular group SL _{2(Z)} with multiplier system \chi ^{-1} _{η} , where \chi _{η} is the multiplier system for η (z) .

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

spt ( 5 n +4 ) \equiv 0 \mod( 5 ) spt ( 7 n +5 ) \equiv 0 \mod( 7 ) spt ( 13 n +6 ) \equiv 0 \mod( 13 ).

Notes and References

1. Andrews . George E. . George Andrews (mathematician) . 2008-11-01 . The number of smallest parts in the partitions of n . Journal für die Reine und Angewandte Mathematik . en . 2008 . 624 . 133–142 . 10.1515/CRELLE.2008.083 . 123142859 . 1435-5345.