A131449 Number of organic (also called increasing) vertex labelings of rooted ordered trees with n non-root vertices.

1, 1, 2, 1, 6, 3, 3, 2, 1, 24, 12, 12, 12, 8, 8, 6, 6, 4, 4, 3, 3, 2, 1, 120, 60, 60, 60, 60, 40, 40, 40, 30, 30, 30, 30, 30, 24, 20, 20, 20, 20, 20, 15, 15, 15, 15, 12, 12, 12, 10, 10, 10, 10, 8, 8, 6, 6, 5, 5, 4, 4, 3, 3, 2, 1, 720

Offset: 0

Author

Wolfdieter Lang, Aug 07 2007

Keywords

• nonn
• more
• tabf

Comments

Organic vertex labeling with numbers 1,2,...,n means that the sequence of vertex labels along the (unique) path from the root with label 0 to any leaf (non-root vertex of degree 1) is increasing.

Row lengths sequence, i.e. the number of rooted ordered trees, C(n):=A000108(n) (Catalan numbers): [1,1,2,5,14,42,...].

Number of rooted trees with n non-root vertices [1,1,2,4,9,20,...]=A000081(n+1).

Row sums give [1,1,3,155,105,945,...]= A001147(n), n>=0. A035342(n,1), n>=1, first column of triangle S2(3).

Examples

			[0! ]; [1! ]; [2!,1]; [3!,3,3,2,1], [4!,12,12,12,8,8,6,6,4,4,3,3,2,1];...
n=3: 3 labelings (0,1,2)(0,3), (0,1,3) (0,2) and (0,2,3) (0,1) for the rooted tree o-o-x-o.
n=3: 3 labelings (0,3)(0,1,2), (0,2)(0,1,3) and (0,1)(0,2,3) for the rooted tree o-x-o-o.
		

Links

• W. Lang, First 6 rows.
• W. Lang, Rooted ordered trees with n=5 non-root vertices and number of labelings.