A176725 Number of ways to choose one element from the multiset corresponding to the k-th multiset repetition class defining partition of n in canonical Abramowitz-Stegun order.

0, 1, 1, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 3, 2, 2, 2, 1, 4, 3, 3, 2, 2, 2, 1, 4, 3, 3, 2, 2, 2, 1, 4, 3, 3, 3, 2, 3, 2, 2, 2, 1, 4, 4, 3, 3, 3, 2, 3, 2, 2, 2, 1, 4, 4, 3, 3, 3, 2, 3, 2, 2, 2, 1, 5, 4, 3, 4, 3, 3, 3, 2, 3, 2, 3, 2, 2, 2, 1, 5, 4, 4, 4, 3, 4, 3, 3, 3, 2, 3, 2, 3, 2, 2, 2, 1

Offset: 0

Author

Wolfdieter Lang, Jul 14 2010

Keywords

• nonn
• tabf
• easy

Comments

This array a(n,k), called (Multiset choose 1), is also denoted by MS(1;n,k).

MS(1;n,k) gives the number of ways to choose one element from the multiset encoded by the k-th multiset repetition class defining partition of n. a(n,k)=MS(1;n,k) is the largest part in the k-th multiset defining partition of n, i.e., the largest element in the multiset. The A-St order for partitions is used. For n=0 the empty multiset appears.

The row lengths of this array are A007294(n).

Multisets are sets in which the elements are allowed to appear more than once. Representatives of repetition classes can be characterized as certain partitions of the nonnegative integers n. For the characteristic array of these partitions of n in A-St order see A176723.

This is the first member of an l-family of arrays for multiset choose l, called MS(l;n,k). This investigation was stimulated by the quoted Griffiths and Mező paper.

Examples

			[0],
[1],
[1],
[2,1],
[2,1],
[2,1],
[3,2,2,1],
[3,2,2,1],
[3,2,2,1],
[3,3,2,2,2,1],
...
a(6,2)=MS(1;6,2)=2 because the second multiset repetition class defining partition of n=6 is [1^2,2^2] (from the characteristic array A176723) encoding the 4-multiset representative {1,1,2,2}, and there are 2 ways to choose 1 element from this set.
		

Links

• M. Griffiths and I. Mező, A Generalization of Stirling Numbers of the Second Kind via a Special Multiset, Journal of Integer Sequences 13 (2010) 10.2.5.
• W. Lang, First 15 rows and corresponding multisets.

Formula

a(n,k)= largest part in the k-th multiset repetition class defining partition of n>=1. a(0,1):=0. See the characteristic array A176723 in order to find this partition.

a(n,k) = largest element of the k-th multiset repetition class representative in row n>=1, a(0,1):=0.

Extensions

Changed in response to comments by Franklin T. Adams-Watters - Wolfdieter Lang, Apr 02 2011