A165652 Number of disconnected 2-regular graphs on n vertices.

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 8, 9, 12, 16, 20, 24, 32, 38, 48, 59, 72, 87, 109, 129, 157, 190, 229, 272, 330, 390, 467, 555, 659, 778, 926, 1086, 1283, 1509, 1774, 2074, 2437, 2841, 3322, 3871, 4509, 5236, 6094, 7055, 8181, 9464, 10944, 12624, 14577, 16778, 19322, 22209

Offset

0,9

Comments

a(n) is also the number of partitions of n such that each part i satisfies 2<i<n.

For n>=2, it appears that a(n+1) is the number of (1,0)-separable partitions of n, as defined at A239482. For example, the four (1,0)-separable partitions of 9 are 621, 531, 441, 31212, corresponding to a(10) = 4. - Clark Kimberling, Mar 21 2014.

Links

Andrew van den Hoeven, Table of n, a(n) for n = 0..10000

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

a = A008483 - A179184 = Euler_tranformation(A179184) - A179184.

For n > 2, since there is exactly one connected 2-regular graph on n vertices (the n cycle C_n) then a(n) = A008483(n) - 1.

(A008483(n) is also the number of not necessarily connected 2-regular graphs on n vertices.)

Column D(n, 2) in the triangle A068933.

Example

The a(6)=1 graph is C_3+C_3. The a(7)=1 graph is C_3+C_4. The a(8)=2 graphs are C_3+C_5, C_4+C_4. The a(9)=3 graphs are 3C_3, C_3+C_6, C_4+C_5.

Prog

(Magma) p := NumberOfPartitions; a := func< n | n lt 3 select 0 else p(n) - p(n-1) - p(n-2) + p(n-3) - 1 >;

Crossrefs

2-regular simple graphs: A179184 (connected), this sequence (disconnected), A008483 (not necessarily connected).

Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A157928 (k=0), A157928 (k=1), this sequence (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8).

Disconnected 2-regular simple graphs with girth at least g: this sequence (g=3), A185224 (g=4), A185225 (g=5), A185226 (g=6), A185227 (g=7), A185228 (g=8), A185229 (g=9).

Cf. A239482.

Sequence in context: A042962 A027584 A161240 * A191851 A063678 A005424

Adjacent sequences: A165649 A165650 A165651 * A165653 A165654 A165655

Keyword

easy,nonn

Author

Jason Kimberley, Sep 28 2009

Status

approved