Spt function - Alchetron, The Free Social Encyclopedia

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.

Example

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

43 + 12 + 22 + 1 + 11 + 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

S ( q ) = ∑ n = 1 ∞ s p t ( n ) q n = 1 ( q ) ∞ ∑ n = 1 ∞ q n ∏ m = 1 n − 1 ( 1 − q m ) 1 − q n

where ( q ) ∞ = ∏ 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 π i z , the function

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 S L 2 ( Z ) with multiplier system χ η − 1 , where χ η is the multiplier system for η ( z ) .

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

s p t ( 5 n + 4 ) ≡ 0 mod ( 5 ) s p t ( 7 n + 5 ) ≡ 0 mod ( 7 ) s p t ( 13 n + 6 ) ≡ 0 mod ( 13 )

References

Spt function Wikipedia

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Contents

Example

Properties

References