A219889 -id:A219889 - cp's OEIS frontend

A007107 Number of labeled 2-regular digraphs with n nodes.

1, 0, 0, 1, 9, 216, 7570, 357435, 22040361, 1721632024, 166261966956, 19459238879565, 2714812050902545, 445202898702992496, 84798391618743138414, 18567039007438379656471, 4631381194792101913679985, 1305719477625154539392776080, 413153055417968797025496881656

Offset: 0

N. J. A. Sloane

Or number of n X n matrices with exactly two 1's in each row and column which are not in the main diagonal, other entries 0 (cf. A001499). - Vladimir Shevelev, Mar 22 2010

Number of 2-factors of the n-crown graph. - Andrew Howroyd, Feb 28 2016

R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1982.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..254 (first 49 terms from R. W. Robinson)
O. Gonzalez, C. Beltran and I. Santamaria, On the Number of Interference Alignment Solutions for the K-User MIMO Channel with Constant Coefficients, arXiv preprint arXiv:1301.6196 [cs.IT], 2013. - From _N. J. A. Sloane_, Feb 19 2013
R. J. Mathar, OEIS A007107, Mar 15 2019

Cf. column t=0 of A284989.

Cf. A007108 (log transform), A197458 (row and column sum <=2), A219889 (unlabeled), A001499 (loops allowed).

a:= proc(n) option remember; `if`(n<5, ((n-1)*(n-2)/2)^2,
      (n-1)*(2*(n^3-2*n^2+n+1)*a(n-1)/(n-2)+((n^2-2*n+2)*
      (n+1)*a(n-2) +(2*n^2-6*n+1)*n*a(n-3)+(n-3)*(a(n-4)*
      (n^3-5*n^2+3)-(n-4)*(n-1)*(n+1)*a(n-5))))/(2*n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 10 2017

Table[Sum[Sum[Sum[(-1)^(k+j-s)*n!*(n-k)!*(2n-k-2j-s)!/(s!*(k-s)!*(n-k-j)!^2*j!*2^(2n-2k-j)),{j,0,n-k}],{s,0,k}],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 09 2014 after Shanzhen Gao *)

a(n)=sum(k=0,n, sum(s=0,k, sum(j=0,n-k, (-1)^(k+j-s)*n!*(n-k)!*(2*n-k-2*j-s)!/(s!*(k-s)!*(n-k-j)!^2*j!*2^(2*n-2*k-j))))) \\ Charles R Greathouse IV, Feb 08 2017

a(n) = Sum_{k=0..n} Sum_{s=0..k} Sum_{j=0..n-k} (-1)^(k+j-s)*n!*(n-k)!*(2n-k-2j-s)!/(s!*(k-s)!*(n-k-j)!^2*j!*2^(2n-2k-j)). - Shanzhen Gao, Nov 05 2007

a(n) ~ 2*sqrt(Pi) * n^(2*n+1/2) / exp(2*n+5/2). - Vaclav Kotesovec, May 09 2014

A219890 Number of directed 3-regular graphs on n nodes.

Original entry on oeis.org

1, 2, 23, 415, 19041, 1104045, 79818336, 6958257941, 720814138673, 87579097910309

Offset: 4

N. J. A. Sloane, Nov 30 2012

Gunnar Brinkmann, Generating regular directed graphs, Discrete Math., 313 (2012), 1-7.

Column k=3 of A350910.

Cf. A096368, A219889, A219890, A219891, A219892, A219893.

A219891 Number of directed 4-regular graphs on n nodes.

Original entry on oeis.org

1, 4, 92, 19041, 6510087, 2983458766, 1775668832509, 1349509467853512

Offset: 5

N. J. A. Sloane, Nov 30 2012

Gunnar Brinkmann, Generating regular directed graphs, Discrete Math., 313 (2012), 1-7.

Column k=4 of A350910.

Cf. A096368, A219889, A219890, A219892, A219893.

A219892 Number of directed 5-regular graphs on n nodes.

Original entry on oeis.org

1, 4, 624, 1104045, 2983458766, 10922621656743

Offset: 6

N. J. A. Sloane, Nov 30 2012

Gunnar Brinkmann, Generating regular directed graphs, Discrete Math., 313 (2012), 1-7.

Column k=5 of A350910.

Cf. A096368, A219889, A219890, A219891, A219893.

A219893 Number of directed 6-regular graphs on n nodes.

Original entry on oeis.org

1, 7, 5021, 79818336, 1775668832509

Offset: 7

N. J. A. Sloane, Nov 30 2012

Gunnar Brinkmann, Generating regular directed graphs, Discrete Math., 313 (2012), 1-7.

Column k=6 of A350910.

Cf. A096368, A219889, A219890, A219891, A219892.

A350910 Triangle read by rows: T(n,k) is the number of k-regular digraphs on n unlabeled nodes, k = 0..n-1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 5, 2, 1, 1, 4, 23, 23, 4, 1, 1, 4, 92, 415, 92, 4, 1, 1, 7, 624, 19041, 19041, 624, 7, 1, 1, 8, 5021, 1104045, 6510087, 1104045, 5021, 8, 1, 1, 12, 47034, 79818336, 2983458766, 2983458766, 79818336, 47034, 12, 1

Offset: 1

Andrew Howroyd, Jan 29 2022

			Triangle begins:
  1;
  1, 1;
  1, 1,   1;
  1, 2,   2,     1;
  1, 2,   5,     2,     1;
  1, 4,  23,    23,     4,   1;
  1, 4,  92,   415,    92,   4, 1;
  1, 7, 624, 19041, 19041, 624, 7, 1;
  ...

Columns k=0..6 are A000012, A002865, A219889, A219890, A219891, A219892, A219893.
Row sums are A350911.
Cf. A051031 (graphs), A329228 (semi-regular), A350912.

A306827 Isomorphism classes of connected 2-in 2-out digraphs, allowing parallel edges but not loops.

Original entry on oeis.org

1, 2, 5, 13, 59, 285, 1987, 16057, 149430, 1551863, 17747299, 221015026

Offset: 2

Brendan McKay, Mar 12 2019

From R. J. Mathar, Mar 15 2019: (Start)
The multiset transform (graphs with n>=2 nodes and k>=1 components) is
1
2
5    1
13    2
59    8    1
285   23    2
1987  100    8    1
16057  468   27    2
149430 2943  115    8    1
The row sums (Euler transform) are A307180. (End)

			For n=3, the a(3)=2 solutions consist of two superimposed directed triangles, one with the triangles oriented consistently and the other with the triangles oriented oppositely.

J. Hu, A. H. MacDonald and B. D. McKay, Correlations in two-dimensional vortex liquids, Physical Review B, 49, 21 (1994) 15263-15270.
R. J. Mathar, OEIS A306827

Cf. A219889 (no multiedges, not necessarily connected), A306892 (loops permitted), A307180 (Euler transf.)

A005641 Number of 2-diregular digraphs with n nodes.

Original entry on oeis.org

1, 3, 8, 27, 131, 711, 5055, 41607, 389759, 4065605, 46612528, 581713045, 7846380548, 113718755478, 1762208816647, 29073392136390, 508777045979418

Offset: 2

N. J. A. Sloane

nonn

These graphs may have loops, whereas A219889 does not admit them. Both sequences do not admit multiedges. - R. J. Mathar, Mar 27 2019

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Richard J. Mathar, 2-regular Digraphs of the Lovelock Lagrangian, arXiv:1903.12477 [math.GM], 2019.
M. V. S. Ramanath and T. R. Walsh, Enumeration and generation of a class of regular digraphs, J. Graph Theory 11 (1987), no. 4, 471-479. (Annotated scanned copy)

Cf. A005642 (inverse Euler transform, connected graphs).

A219894 Number of directed 2-regular graphs on n nodes without directed 2-cycles.

Original entry on oeis.org

1, 4, 9, 55, 453, 4357, 47598, 569769, 7371560, 102525991, 1525164223

Offset: 5

N. J. A. Sloane, Nov 30 2012

Gunnar Brinkmann, Generating regular directed graphs, Discrete Math., 313 (2012), 1-7.

Cf. A096368, A219889, A219890, A219891, A219892, A219893, A219895.

a(14) and a(15) from Brendan McKay, Mar 12 2019

A219895 Number of directed 3-regular graphs on n nodes without directed 2-cycles.

Original entry on oeis.org

3, 26, 1547, 146487, 16018987, 1985766270, 278601715904

Offset: 7

N. J. A. Sloane, Nov 30 2012

Gunnar Brinkmann, Generating regular directed graphs, Discrete Math., 313 (2012), 1-7.

Cf. A096368, A219889, A219890, A219891, A219892, A219893, A219894.

cp's OEIS Frontend

A007107 Number of labeled 2-regular digraphs with n nodes.

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A219890 Number of directed 3-regular graphs on n nodes.

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A219891 Number of directed 4-regular graphs on n nodes.

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A219892 Number of directed 5-regular graphs on n nodes.

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A219893 Number of directed 6-regular graphs on n nodes.

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A350910 Triangle read by rows: T(n,k) is the number of k-regular digraphs on n unlabeled nodes, k = 0..n-1.

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A306827 Isomorphism classes of connected 2-in 2-out digraphs, allowing parallel edges but not loops.

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A005641 Number of 2-diregular digraphs with n nodes.

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A219894 Number of directed 2-regular graphs on n nodes without directed 2-cycles.

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A219895 Number of directed 3-regular graphs on n nodes without directed 2-cycles.

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