A124777 Number of naturally labeled partially ordered sets associated with compositions in standard order.

1, 1, 1, 1, 1, 4, 1, 1, 1, 11, 13, 8, 1, 4, 1, 1

Offset: 0

Author

Franklin T. Adams-Watters, Nov 06 2006

Keywords

• more
• nonn
• tabf

Comments

The standard order of compositions is given by A066099.

The k-th term of the composition is the number of objects with rank k. The rank of an object is one more than the maximum rank of any smaller object in the ordering (1 for a minimal element), or equivalently the size of the largest chain of which the object is the maximal element.

Examples

			Composition number 11 is 2,1,1; there are 3 partial orders
associated with this (shown below); these can be naturally labeled
respectively in 1, 4 and 3 ways, so a(11) = 1+4+3 = 8.
..O..*O..*..O
..|..*|..*./|
..O..*O..*O.|
./.\.*|..*|.|
O...O*O.O*O.O
The table starts:
1
1
1 1
1 4 1 1
		

Crossrefs

Cf. A066099, A124775, A124776, A011782 (row lengths), A006455 (row sums).