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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.

%I A383656 #18 May 08 2025 02:21:19
%S A383656 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,
%T A383656 0,6824,5166,2422,805,175,21,1,0,127166,76712,30072,9177,2100,322,28,
%U A383656 1,0,3205924,1526910,486800,129360,28497,4788,546,36,1,0,108975934,40603534,10292970,2285240,455805,76629,9870,870,45,1
%N 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.
%C A383656 For precise definition see the links: David Bevan and others (2023) or D.E. Knuth (1997).
%H A383656 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 A383656 D. E. Knuth, <a href="/A323841/a323841.pdf">Letter to Daniel Ullman and others</a>, Apr 29 1997 [Annotated scanned copy, with permission].
%F A383656 E.g.f.: f(x)^y where f(x) is the e.g.f. for A135922.
%e A383656 Triangle begins:
%e A383656  1;
%e A383656  0, 1;
%e A383656  0, 1, 1;
%e A383656  0, 2, 3, 1;
%e A383656  0, 8, 11, 6, 1;
%e A383656  0, 52, 60, 35, 10, 1;
%e A383656  0, 502, 472, 255, 85, 15, 1;
%e A383656  ...
%t A383656 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
%Y A383656 Cf. A323843 (column k=1), A135922 (row sums).
%K A383656 nonn,tabl
%O A383656 0,8
%A A383656 _Geoffrey Critzer_, May 04 2025