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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.

%I A383655 #15 May 07 2025 11:35:22
%S A383655 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,
%T A383655 8686,4739,1512,385,70,21,0,1,152191,69488,18956,4032,770,112,28,0,1,
%U A383655 3632916,1369719,312696,56868,9072,1386,168,36,0,1,118317913,36329160,6848595,1042320,142170,18144,2310,240,45,0,1
%N 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.
%C A383655 For precise definition see the links: David Bevan and others (2023)  or D.E. Knuth (1997).
%H A383655 David Bevan, Gi-Sang Cheon and Sergey Kitaev, <a href="https://arxiv.org/abs/2311.08023">On naturally labelled posets and permutations avoiding 12-34</a>, arXiv:2311.08023 [math.CO], 2023.
%H A383655 D. E. Knuth, <a href="/A323841/a323841.pdf">Letter to Daniel Ullman and others</a>, Apr 29 1997 [Annotated scanned copy, with permission].
%F A383655 E.g.f.: exp((y-1)*x)*f(x) where f(x) is the e.g.f. for A135922.
%e A383655 Triangle T(n,k) begins:
%e A383655    1;
%e A383655    0,  1;
%e A383655    1,  0,  1;
%e A383655    2,  3,  0,  1;
%e A383655   11,  8,  6,  0, 1;
%e A383655   72, 55, 20, 10, 0, 1;
%e A383655   ...
%t A383655 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
%Y A383655 Cf. A323842 (column k=0), A135922 (row sums).
%K A383655 nonn,tabl
%O A383655 0,7
%A A383655 _Geoffrey Critzer_, May 04 2025