A383656 Triangular array read by rows: T(n,k) is the number of n-node Stanley graphs containing exactly k connected components, n>=0, 0<=k<=n.

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 8, 11, 6, 1, 0, 52, 60, 35, 10, 1, 0, 502, 472, 255, 85, 15, 1, 0, 6824, 5166, 2422, 805, 175, 21, 1, 0, 127166, 76712, 30072, 9177, 2100, 322, 28, 1, 0, 3205924, 1526910, 486800, 129360, 28497, 4788, 546, 36, 1, 0, 108975934, 40603534, 10292970, 2285240, 455805, 76629, 9870, 870, 45, 1

Offset: 0

Author

Geoffrey Critzer, May 04 2025

Keywords

• nonn
• tabl

Comments

For precise definition see the links: David Bevan and others (2023) or D.E. Knuth (1997).

Examples

			Triangle begins:
 1;
 0, 1;
 0, 1, 1;
 0, 2, 3, 1;
 0, 8, 11, 6, 1;
 0, 52, 60, 35, 10, 1;
 0, 502, 472, 255, 85, 15, 1;
 ...
		

Links

• David Bevan, Gi-Sang Cheon, and Sergey Kitaev, On naturally labelled posets and permutations avoiding 12-34, arXiv:2311.08023 [math.CO], 2023.
• D. E. Knuth, Letter to Daniel Ullman and others, Apr 29 1997 [Annotated scanned copy, with permission].

Crossrefs

Cf. A323843 (column k=1), A135922 (row sums).

Programs

Mathematica

nn = 8; Prepend[Table[(Range[0, nn]! CoefficientList[Series[(Exp[-x] g[x])^y, {x, 0, nn}], {x, y}])[[i,1 ;; i]], {i, 2, nn + 1}], {1}] // Grid

Formula

E.g.f.: f(x)^y where f(x) is the e.g.f. for A135922.