A068933 Triangular array D(n, r) = number of disconnected r-regular graphs with n nodes, 0 <= r < n.

0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 2, 1, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 1, 1, 4, 2, 1, 0, 0, 0, 0, 0, 1, 0, 5, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 8, 9, 3, 1, 0, 0, 0, 0, 0, 0, 1, 0, 9, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 12, 31, 25, 3, 1, 0, 0, 0, 0, 0

Offset

1,31

Comments

A graph is called r-regular if every node has exactly r edges. Row sums give A068932.

Links

Jason Kimberley, Rows 1..23 of A068933 triangle, flattened.

Jason Kimberley, Disconnected regular graphs (with girth at least 3)

Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g

Formula

D(n, r) = A051031(n, r) - A068934(n, r).

Example

This sequence can be computed using the information in A068934. We'll abbreviate A068934(n, r) as C(n, r). To compute D(13, 4), note that the connected components of a 4-regular graph must have at least 5 elements. So a disconnected 13-node 4-regular graph must have two components and their sizes are either 8 and 5, or 7 and 6. So D(13, 4) = C(8, 4)*C(5, 4) + C(7, 4)*C(6, 4) = 6*1 + 2*1 = 8.
0;
1, 0;
1, 0, 0;
1, 1, 0, 0;
1, 0, 0, 0, 0;
1, 1, 1, 0, 0, 0;
1, 0, 1, 0, 0, 0, 0;
1, 1, 2, 1, 0, 0, 0, 0;
1, 0, 3, 0, 0, 0, 0, 0, 0;
1, 1, 4, 2, 1, 0, 0, 0, 0, 0;
1, 0, 5, 0, 1, 0, 0, 0, 0, 0, 0;
1, 1, 8, 9, 3, 1, 0, 0, 0, 0, 0, 0;
1, 0, 9, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0;
1, 1, 12, 31, 25, 3...

Crossrefs

Cf. A051031, A068932, A068934.

Sequence in context: A321921 A236419 A114448 * A015472 A049816 A328958

Adjacent sequences: A068930 A068931 A068932 * A068934 A068935 A068936

Keyword

nonn,tabl

Author

David Wasserman, Mar 08 2002

Status

approved