A359686 Triangle read by rows: T(n,k) is the minimum number of connected endofunctions that are spanning subgraphs of a semi-regular loopless digraph on n vertices each with out-degree k.

1, 1, 8, 0, 14, 78, 0, 22, 213, 944, 0, 0, 529, 3400, 13800, 0, 0

Offset: 2

Author

Yali Harrary, Jan 11 2023

Keywords

• nonn
• tabl
• more

Comments

An endofunction represented as a digraph is one in which every vertex has out-degree 1. Loopless means that there are no fixed points in the function. The digraph of a connected endofunction is unicyclic (contains exactly one cycle).

In the case k = 1, the graphs considered have vertices with out-degree 1 and the entire graph is the only spanning subgraph that is an endofunction. Hence T(n,1) = 0. (n > 3 because when n = 2, 3 it still will be unicyclic.)

In the case k = n-1, the graphs considered are the complete digraphs and every connected endofunction on the vertex set is a subgraph. Hence T(n, n-1) = A000435(n), which gives the total number of connected endofunctions without fixed points.

Examples

			Triangle begins:
  2 | 1;
  3 | 1,  8;
  4 | 0, 14,  78;
  5 | 0, 22, 213,  944;
  6 | 0,  0, 529, 3400, 13800;
  ...
In the following examples, the notation 1->{2,3} is shorthand for the set of arcs {(1,2), (1,3)}.
T(5,2) = 22 is attained with the digraph described by: 1->{4,5}, 2->{3,5}, 3->{2,4}, 4->{1,3}, 5->{1,2}.
		

Crossrefs

Cf. A000435, A359628.

Formula

T(n,1) = 0 for n > 3, otherwise 1.

T(n,n-1) = A000435(n).

T(n,k) = 0 for 2*k + 2 < n.