A342113 - OEIS

%I #6 Mar 07 2021 17:59:59

%S 1,1,7,145,7999,1103041,365051647,281898887425,497570152386559,

%T 1976049386530790401,17439288184770966867967,

%U 338596445913833207323643905,14343481992486219718322674565119

%N 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 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 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 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 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 Cf. A339934.

%K nonn

%O 0,3

%A _Geoffrey Critzer_, Feb 28 2021