A179822 Maximally refined partitions into distinct parts (of any natural number) with largest part n.

1, 1, 2, 3, 5, 7, 12, 16, 26, 37, 58, 79, 128, 171, 271, 376, 576, 783, 1239, 1654, 2567, 3505, 5382, 7245, 11247, 15036, 23187, 31370, 47672, 64146, 98887, 131784, 201340, 271350, 412828, 551744, 843285, 1125417, 1715207, 2299452, 3479341, 4654468, 7090529

Offset: 0

Author

Moshe Shmuel Newman, Jan 10 2011

Keywords

• nonn

Comments

For the definition, see sequence A179009. This sequence counts the same objects using a different statistic, the largest part rather than the sum of the parts.

a(n) is the number of subsets of {1..n-1} containing the sum of any two distinct elements whose sum is <= n. This differs from A326080 in that the set may not contain n itself. These sets are the complements of the set of parts in the first definition. - Andrew Howroyd, Apr 13 2021

Examples

			The partitions counted by n=4 are:
  4+1, 4+2+1, 4+3+1, 4+3+2, 4+3+2+1.
The partitions counted by n=5 are:
  5+2+1, 5+3+1, 5+3+2+1, 5+4+2+1, 5+4+3+1, 5+4+3+2, 5+4+3+2+1.
		

Crossrefs

Cf. A179009, A179817, A326080.

Programs

PARI

a(n)={
  my(ok(k,b)=for(i=1, (k-1)\2, if(bittest(b,i) && bittest(b,k-i), return(0))); 1);
  my(recurse(k,b)=if(k==n, ok(k,b), self()(k+1, bitor(b,1<Andrew Howroyd, Apr 13 2021

Extensions

a(19)-a(42) from Andrew Howroyd, Apr 13 2021