1,2

A partition is indecomposable if it is not [1] and cannot be represented as the product of two smaller partitions, where the product of two partitions is the multiset of all products of parts from the two multiplicands. Another way to define the product of partitions is to regard the partition as a finite sequence b(k) being the number of parts of size k; then the Dirichlet g.f. of b * c is the product of the Dirichlet g.f.s of b and c.

The (formal) Dirichlet generating function for A000041 is Product_{n>1} 1/(1-n^{-s})^a(n). (Formal because this g.f. does not converge for any value of s.)

The product of [2,2,1] * [2,1,1] is the partition with parts:

4 4 2

2 2 1

2 2 1

which is [4^2,2^5,1^2]. In terms of Dirichlet g.f.s, this is (2*2^s + 1^s) * (2^s + 2*1^s) = (2*4^s + 5*2^s + 2*1^s).

Of the partitions of 6, [6] = [3] * [2], [4,2] = [2] * [2,1], [3^2] = [3] * [1^2], [2^3] = [2] * [1^3], [2^2,1^2] = [2,1] * [1^2] and [1^6] = [1^3] * [1^2]. This leaves [5,1], [4,1^2], [3,2,1], [3,1^3] and [2,1^4] as the 5 indecomposable partitions of 6.

Cf. A000041, A090751.

Sequence in context: A051886 A118007 A158747 this_sequence A129022 A122076 A014784

Adjacent sequences: A122694 A122695 A122696 this_sequence A122698 A122699 A122700

nonn

Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Sep 22 2006