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 partition of a positive integer. It is related to the partition function.
The first few values of spt(n) are:
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
where .
The function is related to a mock modular form. Let denote the weight 2 quasi-modular Eisenstein series and let denote the Dedekind eta function. Then for , the function
is a mock modular form of weight 3/2 on the full modular group with multiplier system , where is the multiplier system for .
While a closed formula is not known for spt(n), there are Ramanujan-like congruences including
- ^{[1]}
References
- â†‘ George Andrews. "The number of smallest parts in the partitions of n".