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A342113 Number of surjective compatible pairs (C,O), where O is an acyclic orientation of simple labeled graph G on n nodes and C:V(G) -> {1,2,...}.

%I A342113 #6 Mar 07 2021 17:59:59
%S A342113 1,1,7,145,7999,1103041,365051647,281898887425,497570152386559,
%T A342113 1976049386530790401,17439288184770966867967,
%U A342113 338596445913833207323643905,14343481992486219718322674565119
%N A342113 Number of surjective compatible pairs (C,O), where O is an acyclic orientation of simple labeled graph G on n nodes and C:V(G) -> {1,2,...}.
%C A342113 A pair (C,O) is a surjective compatible pair if O is an acyclic orientation of the edges of a simple labeled graph G on n nodes, and C is a surjective function from V(G)->{1,2,...k} for some positive integer k such that for all u,v in V(G) if u->v under the orientation then C(u)>= C(v).
%H A342113 R. P. Stanley, <a href="https://doi.org/10.1016/j.disc.2006.03.010">Acyclic orientation of graphs</a>, Discrete Math. 5 (1973), 171-178.
%F A342113 Let E(x) = Sum_{n>=0}x^n/(n! 2^Binomial(n,2)).  Then Sum_{n>=0}a_n x^n/(n! 2^Binomial(n,2)) = 1/(2 - E(-x)^-1).
%t A342113 nn = 12; b[n_] := q^Binomial[n, 2] n! /. q -> 2; e[z_] := Sum[z^n/b[n], {n, 0, nn}];Table[b[n], {n, 0, nn}] CoefficientList[ Series[1/(1 - (1/e[-z] - 1)), {z, 0, nn}], z]
%Y A342113 Cf. A339934.
%K A342113 nonn
%O A342113 0,3
%A A342113 _Geoffrey Critzer_, Feb 28 2021