A383655 Triangle read by rows: T(n,k) is the number of n-node Stanley graphs containing exactly k isolated points, n>=0, 0<=k<=n.

1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 11, 8, 6, 0, 1, 72, 55, 20, 10, 0, 1, 677, 432, 165, 40, 15, 0, 1, 8686, 4739, 1512, 385, 70, 21, 0, 1, 152191, 69488, 18956, 4032, 770, 112, 28, 0, 1, 3632916, 1369719, 312696, 56868, 9072, 1386, 168, 36, 0, 1, 118317913, 36329160, 6848595, 1042320, 142170, 18144, 2310, 240, 45, 0, 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 T(n,k) begins:
   1;
   0,  1;
   1,  0,  1;
   2,  3,  0,  1;
  11,  8,  6,  0, 1;
  72, 55, 20, 10, 0, 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. A323842 (column k=0), A135922 (row sums).

Programs

Mathematica

nn = 10; g[x_] :=Total[Table[Sum[QBinomial[n, k, 2] x^n/n!, {k, 0, n}], {n, 0, nn}]]; Table[(Range[0, nn]! CoefficientList[Series[Exp[y x] Exp[-x] g[x] Exp[-x], {x, 0, nn}], {x, y}])[[i, 1 ;; i]], {i, 1, nn + 1}] // Grid

Formula

E.g.f.: exp((y-1)*x)*f(x) where f(x) is the e.g.f. for A135922.