A361210 Number of labeled digraphs on [n] with exactly 1 in-node and exactly 1 out-node.

0, 1, 2, 15, 588, 83295, 40993230, 70413420511, 433343743592312, 9825711749274316671, 840137012096473747415610, 275596225117501271622460109871, 351011149451321734143551287903432452, 1749719217881846572487198585072701742763487, 34317835907818751756576624929762210160396817182918

Offset: 0

Author

Geoffrey Critzer, Apr 09 2023

Keywords

• nonn

Comments

Here, an in-node is a node whose outdegree is zero. An out-node is a node whose in-degree is zero. The in-node is not necessarily distinct from the out-node.

Links

• E. de Panafieu and S. Dovgal, Symbolic method and directed graph enumeration, arXiv:1903.09454 [math.CO], 2019.
• R. W. Robinson, Counting digraphs with restrictions on the strong components, Combinatorics and Graph Theory '95 (T.-H. Ku, ed.), World Scientific, Singapore (1995), 343-354.

Crossrefs

Cf. A086193 (no out-nodes nor in-nodes).

Programs

Mathematica

nn = 14; B[n_] := n! 2^Binomial[n, 2] ; e[z_] := Sum[z^n/B[n], {n, 0, nn}];
g[z_] := Sum[2^(n (n - 1)) z^n/B[n], {n, 0, nn}];egf[ggf_] := Normal[Series[ggf, {z, 0, nn}]] /. Table[z^i -> z^i*2^Binomial[i, 2], {i, 0, nn}];Table[n!, {n, 0, nn}] Map[Coefficient[#, u v] &, CoefficientList[Series[Exp[(u - 1) ( v - 1) z] egf[e[(u - 1) z] g[z] e[(v - 1) z]], {z, 0, nn}], z]]