A329427 Number of directed graphs of n vertices with more than 1 component and outdegree 1.

1, 2, 10, 32, 173, 864, 5876, 42654, 369352, 3490396, 37205377, 431835570, 5488938513, 75253166882, 1111054042385, 17529435042906, 294620759901439, 5250432711385802, 98912760811106081, 1963457208200874954, 40962100714228585825, 895889161265034629994, 20497593840242211891900

Offset: 4

Author

Stephen Dunn, Nov 30 2019

Keywords

• nonn

Comments

a(n) gives the number of unique ways a directed graph of n vertices with outdegree 1 can be broken into smaller components of size >= 2. It can be generalized to higher degree by replacing A329426 in the formula with a suitable counting function.

Examples

			a(4) = A329426(2)*A329426(2) = 1*1 = 1, which represents the graph
  V <--> V
  V <--> V.
a(5) = A329426(2)*A329426(3) = 1*2 = 2, which represents the two possible graphs of size 3 (V --> V <--> V, etc.) paired with V <--> V.
a(6) = A329426(2)*A329426(4) + A329426(3)*A329426(3) = 1*6 + 2*2 = 10.
		

Links

• Stephen Dunn, Table of n, a(n) for n = 4..100

Crossrefs

Cf. A329426, A056542.

Programs

Kotlin

fun A056542(n: Long): Long = if (n == 1L) 0 else n * A056542(n-1) + 1
fun A329426(n: Long): Long = 1 + a(n) + A056542(n-1)
fun a(n: Long): Long = (2L..(n/2)).map { A329426(it) * A329426(n-it) }.sum()

Formula

a(n) = Sum_{i=2..floor(n/2)} A329426(i) * A329426(n-i).

Extensions

Term a(26) corrected by Sidney Cadot, Jan 06 2023.