A144880 Partition number array, called M31hat(3).

1, 3, 1, 12, 3, 1, 60, 12, 9, 3, 1, 360, 60, 36, 12, 9, 3, 1, 2520, 360, 180, 144, 60, 36, 27, 12, 9, 3, 1, 20160, 2520, 1080, 720, 360, 180, 144, 108, 60, 36, 27, 12, 9, 3, 1, 181440, 20160, 7560, 4320, 3600, 2520, 1080, 720, 540, 432, 360, 180, 144, 108, 81, 60, 36, 27

Offset: 1

Author

Wolfdieter Lang Oct 09 2008

Keywords

• nonn
• easy
• tabf

Comments

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31hat(3;n,k) with the k-th partition of n in A-St order.

The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].

This is the third (K=3) member of a family of partition number arrays: A107106, A134133,...

Examples

			[1];[3,1];[12,3,1];[60,12,9,3,1];[360,60,36,12,9,3,1];...
a(4,3)= 9 = |S1(3;2,1)|^2. The relevant partition of 4 is (2^2).
		

Links

• W. Lang, First 10 rows of the array and more.
• W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Crossrefs

A144882 (row sums).

A134133 (M31hat(2) array). A144885 (M31hat(4) array).

Formula

a(n,k)= product(|S1(3;j,1)|^e(n,k,j),j=1..n) with |S1(3;n,1)|= A046089(1,n) = [1,3,12,60,...], n>=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.