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,...}.

1, 1, 7, 145, 7999, 1103041, 365051647, 281898887425, 497570152386559, 1976049386530790401, 17439288184770966867967, 338596445913833207323643905, 14343481992486219718322674565119

Offset: 0

Author

Geoffrey Critzer, Feb 28 2021

Keywords

• nonn

Comments

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).

Links

• R. P. Stanley, Acyclic orientation of graphs, Discrete Math. 5 (1973), 171-178.

Crossrefs

Cf. A339934.

Programs

Mathematica

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]

Formula

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).