Stephen Dunn - cp's OEIS frontend

User: Stephen Dunn

Stephen Dunn has authored 2 sequences.

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

Stephen Dunn, Nov 30 2019

nonn

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.

			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.

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

Cf. A329426, A056542.

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()

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

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

A329426 Number of non-isomorphic directed graphs where every vertex has outdegree 1, and no self-loops.

Original entry on oeis.org

1, 2, 6, 20, 97, 550, 3794, 29826, 266527, 2649156, 29040865, 347548542, 4509961264, 63050417976, 944767674590, 15103712944100, 256594870255076, 4616238126871328, 87670085904641440, 1752759735606185804, 36796608121601906104, 809312755145598475440, 18609995953274373396982

Offset: 2

Stephen Dunn, Nov 30 2019

nonn

			For n = 2, a(2) = 1 + A329427(2) + A056542(1) = 1 + 0 + 0 = 1, which is the graph A <--> B.
For n = 3, a(3) = 1 + A329427(3) + A056542(2) = 1 + 0 + 1 = 2, which are graphs A --> B <--> C and A --> B --> C --> A.
The middle term is nonzero when there are graphs with more than 1 component.

Stephen Dunn, Table of n, a(n) for n = 2..100

Cf. A329427, A056542.

fun A329427(n: Long): Long = (2L..(n/2)).map { a(it) * a(n-it) }.sum()
fun A056542(n: Long): Long = if (n == 1L) 0 else n * A056542(n-1) + 1
fun a(n: Long): Long = 1 + A329427(n) + A056542(n-1)

a(n) = 1 + A329427(n) + A056542(n-1).

a(n) = 1 + A056542(n-1) + Sum_{2..floor(n/2)} a(i)*a(n-i).

cp's OEIS Frontend

User: Stephen Dunn

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

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