A176724 Triangle for number of partitions which define multiset repetition classes.

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 2, 1, 1, 1, 0, 0, 0, 1, 0, 1, 2, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 2, 1, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 2, 2, 1, 1, 1

Offset: 1

Author

Wolfdieter Lang, Jul 14 2010

Keywords

• nonn
• easy
• tabl

Comments

For definitions, references, links and examples see the corresponding partition array A176723.

Row sums coincide with those of array A176723 for n>=1, and they are given by A007294.

If for n=0 a 1 is added (the empty partition defines the empty multiset class) the tabl structure will be lost.

Examples

			1;
0,1;
0,1,1;
0,0,1,1;
0,0,0,1,1;
0,0,1,1,1,1;
0,0,0,1,1,1,1;
...
		

Links

• W. Lang: First 15 rows and row sums.

Crossrefs

a(7,5)=1 because there is only one 5 part partition of 7 which is 5-multiset repetition class defining, namely (1^3,2^2) (see row n=7 of the partition array A176723). This defines the 5-multiset class representative {1,1,1,2,2}.

Formula

a(n,m) is the number of m part partitions of n which define m-multiset repetition classes. Multiset repetition class defining is equivalent to the following constraint on the exponents of a partition (1^e[1],2^e[2],...,M^e[M]):

e[1] >= e[2]>=...>=e[M]>=1, i.e., positive nonincreasing with largest part M. This will satisfy T(M) <= n where T(M) = A000217(M) are the triangular numbers; for each n every sufficiently small positive M does occur.

Extensions

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