A000595 -id:A000595 - cp's OEIS frontend

A000666 Number of symmetric relations on n nodes.

1, 2, 6, 20, 90, 544, 5096, 79264, 2208612, 113743760, 10926227136, 1956363435360, 652335084592096, 405402273420996800, 470568642161119963904, 1023063423471189431054720, 4178849203082023236058229792, 32168008290073542372004082199424

Offset: 0

N. J. A. Sloane

Each node may or may not be related to itself.
Also the number of rooted graphs on n+1 nodes.
The 1-to-1 correspondence is as follows: Given a rooted graph on n+1 nodes, replace each edge joining the root node to another node with a self-loop at that node and erase the root node. The result is an undirected graph on n nodes which is the graph of the symmetric relation.
Also the number of the graphs with n nodes whereby each node is colored or not colored. A loop can be interpreted as a colored node. - Juergen Will, Oct 31 2011

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, pp. 101, 241.
M. D. McIlroy, Calculation of numbers of structures of relations on finite sets, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sept. 15, 1955, pp. 14-22.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David Applegate, Table of n, a(n) for n = 0..80 [Shortened file because terms grow rapidly: see Applegate link below for additional terms]
David Applegate, Table of n, a(n) for n = 0..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
R. L. Davis, The number of structures of finite relations, Proc. Amer. Math. Soc. 4 (1953), 486-495.
R. L. Davis, Structure of dominance relations, Bull. Math. Biophys., 16 (1954), 131-140. [Annotated scanned copy]
F. Harary, The number of linear, directed, rooted, and connected graphs, Trans. Am. Math. Soc. 78 (1955) 445-463, eq. (24).
Frank Harary, Edgar M. Palmer, Robert W. Robinson and Allen J. Schwenk, Enumeration of graphs with signed points and lines, J. Graph Theory 1 (1977), no. 4, 295-308.
Adib Hassan, Po-Chien Chung and Wayne B. Hayes, Graphettes: Constant-time determination of graphlet and orbit identity including (possibly disconnected) graphlets up to size 8, arXiv:1708.04341 [cs.DS], 2017.
Mathematics Stack Exchange, Enumerating Graphs with Self-Loops, Jan 23 2014.
M. D. McIlroy, Calculation of numbers of structures of relations on finite sets, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sep. 15, 1955, pp. 14-22. [Annotated scanned copy]
W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. Ann., 174 (1967), 53-78.
W. Oberschelp, The number of non-isomorphic m-graphs, Presented at Mathematical Institute Oberwolfach, July 3 1967 [Scanned copy of manuscript]
W. Oberschelp, Strukturzahlen in endlichen Relationssystemen, in Contributions to Mathematical Logic (Proceedings 1966 Hanover Colloquium), pp. 199-213, North-Holland Publ., Amsterdam, 1968. [Annotated scanned copy]
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
Marko Riedel, Maple code for sequence.
Marko Riedel et al., Enumerating Graphs with Self-Loops
R. W. Robinson, Notes - "A Present for Neil Sloane"
R. W. Robinson, Notes - computer printout
Sage, Common Graphs (Graph Generators)
Lorenzo Sauras-Altuzarra, Graphs
J. M. Tangen and N. J. A. Sloane, Correspondence, 1976-1976
Eric Weisstein's World of Mathematics, Rooted Graph

Cf. A000595, A001172, A001174, A006905, A000250, A054921 (connected relations).

Maple
```
# see Riedel link above
```

Join[{1,2}, Table[CycleIndex[Join[PairGroup[SymmetricGroup[n]], Permutations[Range[n*(n-1)/2+1, n*(n+1)/2]], 2],s] /. Table[s[i]->2, {i,1,n^2-n}], {n,2,8}]] (* Geoffrey Critzer, Nov 04 2011 *)
Table[Module[{eds,pms,leq},
eds=Select[Tuples[Range[n],2],OrderedQ];
pms=Map[Sort,eds/.Table[i->Part[#,i],{i,n}]]&/@Permutations[Range[n]];
leq=Function[seq,PermutationCycles[Ordering[seq],Length]]/@pms;
Total[Thread[Power[2,leq]]]/n!
],{n,0,8}] (* This is after Geoffrey Critzer's program but does not use the (deprecated) Combinatorica package. - Gus Wiseman, Jul 21 2016 *)
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i-1}], {i, 2, Length[v]}] + Sum[Quotient[v[[i]], 2] + 1, {i, 1, Length[v]}];
a[n_] := a[n] = (s = 0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 17}] (* Jean-François Alcover, Nov 13 2017, after Andrew Howroyd *)

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, v[i]\2 + 1)}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)); s/n!} \\ Andrew Howroyd, Oct 22 2017

from itertools import combinations
from math import prod, factorial, gcd
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A000666(n): return int(sum(Fraction(1<>1)+1)*r+(q*r*(r-1)>>1) for q, r in p.items()),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # Chai Wah Wu, Jul 02 2024

Euler transform of A054921. - N. J. A. Sloane, Oct 25 2018
Let G_{n+1,k} be the number of rooted graphs on n+1 nodes with k edges and let G_{n+1}(x) = Sum_{k=0..n(n+1)/2} G_{n+1,k} x^k. Thus a(n) = G_{n+1}(1). Let S_n(x_1, ..., x_n) denote the cycle index for Sym_n. (cf. the link in A000142).
Compute x_1*S_n and regard it as the cycle index of a set of permutations on n+1 points and find the corresponding cycle index for the action on the n(n+1)/2 edges joining those points (the corresponding "pair group"). Finally, by replacing each x_i by 1+x^i gives G_{n+1}(x). [Harary]
Example, n=2. S_2 = (1/2)*(x_1^2+x_2), x_1*S_2 = (1/2)*(x_1^3+x_1*x_2). The pair group is (1/2)*(x_1^2+x_1*x_2) and so G_3(x) = (1/2)*((1+x)^3+(1+x)*(1+x^2)) = 1+2*x+2*x^2+x^3; set x=1 to get a(2) = 6.
a(n) ~ 2^(n*(n+1)/2)/n! [McIlroy, 1955]. - Vaclav Kotesovec, Dec 19 2016

Description corrected by Christian G. Bower
More terms from Vladeta Jovovic, Apr 18 2000
Entry revised by N. J. A. Sloane, Mar 06 2007

A002724 Number of inequivalent n X n binary matrices, where equivalence means permutations of rows or columns.

Original entry on oeis.org

1, 2, 7, 36, 317, 5624, 251610, 33642660, 14685630688, 21467043671008, 105735224248507784, 1764356230257807614296, 100455994644460412263071692, 19674097197480928600253198363072, 13363679231028322645152300040033513414, 31735555932041230032311939400670284689732948

Offset: 0

N. J. A. Sloane

A diagonal of the array A(m,n) described in A028657. - N. J. A. Sloane, Sep 01 2013

Also, number of bipartite graphs with both partite sets of size n, one of which is marked. For connected bipartite graphs, see A363846. - Max Alekseyev, Jun 24 2023

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..26 from Alois P. Heinz)
Manuel Kauers and Jakob Moosbauer, Good pivots for small sparse matrices, arXiv:2006.01623 [cs.SC], 2020.
A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]
Mathematics Stack Exchange, How many n-by-m binary matrices are there up to row and column permutations
B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218.
Marko Riedel, Maple code with two different algorithms
M. Zivkovic, Classification of small (0,1) matrices, arXiv:math/0511636 [math.CO], 2005.
Index entries for sequences related to binary matrices
Index to number of inequivalent matrices modulo permutation of rows and columns

Cf. A002623, A002727, A006148, A002728, A002725, A052269, A052271, A052272, A091059, A363845, A363846, A000595.
Cf. A028657 (this sequence is the diagonal). - N. J. A. Sloane, Sep 01 2013
Column k=2 of A246106.

Maple
```
# See Marko Riedel link.
```

b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Union[Flatten[Table[ Function[{p}, p + j*x^i] /@ b[n - i*j, i - 1], {j, 0, n/i}]]]]];
g[n_, k_] := g[n, k] = Sum[Sum[2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]* Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/ Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n + k, n + k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n - k]];
Table[A[n, n], {n, 0, 15}] (* Jean-François Alcover, Aug 10 2018, after Alois P. Heinz *)

a(n) = A(n,n) \\ A defined in A028657. - Andrew Howroyd, Mar 01 2023

a(n) = Sum_{1*s_1+2*s_2+...=n, 1*t_1+2*t_2+...=n} (fixA[s_1, s_2, ...;t_1, t_2, ...]/(1^s_1*s_1!*2^s_2*s_2!*...*1^t_1*t_1!*2^t_2*t_2!*...)) where fixA[...] = 2^Sum_{i, j>=1} (gcd(i, j)*s_i*t_j). - Christian G. Bower, Dec 18 2003

a(n) = A028657(2*n, n). - Max Alekseyev, Jun 24 2023

More terms from Vladeta Jovovic, Feb 04 2000

a(15) from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 24 2008

A006905 Number of transitive relations on n labeled nodes.

Original entry on oeis.org

1, 2, 13, 171, 3994, 154303, 9415189, 878222530, 122207703623, 24890747921947, 7307450299510288, 3053521546333103057, 1797003559223770324237, 1476062693867019126073312, 1679239558149570229156802997, 2628225174143857306623695576671, 5626175867513779058707006016592954, 16388270713364863943791979866838296851, 64662720846908542794678859718227127212465

Offset: 0

Simon Plouffe and N. J. A. Sloane

D. Ford and J. McKay, personal communication, 1991.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

S. R. Finch, Transitive relations, topologies and partial orders.
S. R. Finch, Transitive relations, topologies and partial orders, June 5, 2003. [Cached copy, with permission of the author]
Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
J. Klaska, Transitivity and Partial Order, Mathematica Bohemica, 122 (1), 75-82 (1997). Based on a correspondence between transitive relations and partial orders, the author obtains a formula and calculates the first 14 terms. - Jeff McSweeney (jmcsween(AT)mtsu.edu), May 13 2000
Firdous Ahmad Mala, Three Open Problems in Enumerative Combinatorics, J. Appl. Math. Computation (2023) Vol. 7, No. 1, 24-27.
Firdous Ahmad Mala, Why the number of transitive relations is not an integer polynomial, BOHR Int'l J. Eng. (2023) Vol. 2, No. 1, pp. 30-31.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

Cf. A000595, A001173, A340264. See A091073 for unlabeled case.

P = Cases[Import["https://oeis.org/A001035/b001035.txt", "Table"], {, }][[All, 2]];
a[n_] := Sum[P[[k+1]] Sum[Binomial[n, s] StirlingS2[n-s, k-s], {s, 0, k}], {k, 0, n}];
a /@ Range[0, 18] (* Jean-François Alcover, Dec 29 2019, after Charles R Greathouse IV *)
transitive[r_]:=Catch[Do[If[a[[2]]==b[[1]]&&FreeQ[r,{a[[1]],b[[2]]}],Throw[False]],{a,r},{b,r}];True];
a[n_]:=Count[Subsets[Tuples[Range[n],2]],?transitive]; (* _Juan José Alba González, Jul 04 2022 *)

\\ P = [1, 1, 3, 19, ...] is A001035, starting from 0.
a(n)=sum(k=0,n,P[k+1]*sum(s=0,k,binomial(n,s)*stirling(n-s,k-s,2)))
\\ Charles R Greathouse IV, Sep 05 2011

E.g.f.: A(x + exp(x) - 1) where A(x) is the e.g.f. for A001035. - Geoffrey Critzer, Jul 28 2014

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 28 2003
a(15)-a(16) from Charles R Greathouse IV, Aug 30 2006
a(17)-a(18) from Charles R Greathouse IV, Sep 05 2011

A001174 Number of oriented graphs (i.e., digraphs with no bidirected edges) on n unlabeled nodes. Also number of complete digraphs on n unlabeled nodes. Number of antisymmetric relations (i.e., oriented graphs with loops) on n unlabeled nodes is A083670.

Original entry on oeis.org

1, 2, 7, 42, 582, 21480, 2142288, 575016219, 415939243032, 816007449011040, 4374406209970747314, 64539836938720749739356, 2637796735571225009053373136, 300365896158980530053498490893399

Offset: 1

N. J. A. Sloane

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 133, c_p.
M. D. McIlroy, Calculation of numbers of structures of relations on finite sets, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sept. 15, 1955, pp. 14-22.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..50
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
R. L. Davis, The number of structures of finite relations, Proc. Amer. Math. Soc. 4 (1953), 486-495.
Musa Demirci, Ugur Ana, and Ismail Naci Cangul, Properties of Characteristic Polynomials of Oriented Graphs, Proc. Int'l Conf. Adv. Math. Comp. (ICAMC 2020) Springer, see p. 60.
F. Harary and E. M. Palmer, Enumeration of mixed graphs, Proc. Amer. Math. Soc., 17 (1966), 682-687.
T. R. Hoffman and J. P. Solazzo, Complex Two-Graphs via Equiangular Tight Frames, arXiv preprint arXiv:1408.0334 [math.CO], 2014-2017.
M. D. McIlroy, Calculation of numbers of structures of relations on finite sets, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sep. 15, 1955, pp. 14-22. [Annotated scanned copy]
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
Eric Weisstein's World of Mathematics, Oriented Graph

Cf. A000595, A001173, A281446.

Cf. A047656 (labeled case), A054941 (connected labeled case), A086345 (connected unlabeled case).

permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total @ Quotient[v - 1, 2];
a[n_] := Module[{s = 0}, Do[s += permcount[p]*3^edges[p], {p, IntegerPartitions[n]}]; s/n!];
Array[a, 15] (* Jean-François Alcover, Jul 06 2018, after Andrew Howroyd *)

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, (v[i]-1)\2)}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)); s/n!} \\ Andrew Howroyd, Oct 23 2017

from itertools import combinations
from math import prod, gcd, factorial
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A001174(n): return int(sum(Fraction(3**(sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))+sum((q-1>>1)*r+(q*r*(r-1)>>1) for q, r in p.items())),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # Chai Wah Wu, Jul 15 2024

There's an explicit formula - see for example Harary and Palmer (book), Eq. (5.4.14).

a(n) ~ 3^(n*(n-1)/2)/n! [McIlroy, 1955]. - Vaclav Kotesovec, Dec 19 2016

More terms from Vladeta Jovovic

Revised description from Vladeta Jovovic, Jan 20 2005

A326204 Number of Hamiltonian labeled n-vertex digraphs (with loops).

Original entry on oeis.org

0, 2, 4, 120, 19104

Offset: 0

Gus Wiseman, Jun 14 2019

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

			The a(2) = 4 digraph edge-sets:
  {12,21}
  {11,12,21}
  {12,21,22}
  {11,12,21,22}

Wikipedia, Hamiltonian path

The unlabeled case is A326226.
The case without loops is A326219.
The undirected case (without loops) is A326208.
Non-Hamiltonian digraphs are A326220.
Digraphs containing a Hamiltonian path are A326214.
Cf. A000595, A002416, A003024, A003216, A057864.

Table[Length[Select[Subsets[Tuples[Range[n],2]],FindHamiltonianCycle[Graph[Range[n],DirectedEdge@@@#]]!={}&]],{n,0,4}] (* Mathematica 8.0+. Warning: Using HamiltonianGraphQ instead of FindHamiltonianCycle returns a(4) = 19200 which is incorrect *)

A326225 Number of Hamiltonian unlabeled n-vertex digraphs (without loops).

Original entry on oeis.org

0, 1, 1, 4, 61, 3725, 844141, 626078904

Offset: 0

Gus Wiseman, Jun 15 2019

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

			Non-isomorphic representatives of the a(3) = 4 digraph edge-sets:
  {12,23,31}
  {12,13,21,32}
  {12,13,21,23,31}
  {12,13,21,23,31,32}

The labeled case is A326219.
The case with loops is A326226.
The undirected case is A003216.
Non-Hamiltonian unlabeled digraphs (without loops) are A326222.
Cf. A000595, A002416, A003087, A003216, A053763, A246446, A326204, A326218, A326223.

a(5)-a(7) from Sean A. Irvine, Jun 16 2019

A229865 Number of n X n 0..1 arrays with corresponding row and column sums equal.

Original entry on oeis.org

1, 2, 8, 80, 2432, 247552, 88060928, 112371410944, 523858015518720, 9041009511609073664, 583447777113052431515648, 141885584718620229407228821504, 130832005909904417592540055577034752, 459749137931232137234615429529864283095040, 6182706200522446492946534924719926752508110700544

Offset: 0

R. H. Hardin, Oct 01 2013

nonn

Also known as labeled Eulerian digraphs allowing loops. - Brendan McKay, May 12 2019

			Some solutions for n=4:
  0 0 0 1     0 0 1 0     0 0 0 1     0 0 1 0     0 0 1 1
  0 1 0 0     1 0 0 0     1 0 1 0     0 0 1 1     1 0 0 1
  0 0 0 1     0 1 0 0     0 1 0 1     0 1 1 1     1 1 1 0
  1 0 1 0     0 0 0 1     0 1 1 0     1 1 0 0     0 1 1 1
From _Gus Wiseman_, Jun 22 2019: (Start)
The a(3) = 8 Eulerian digraph edge-sets:
  {}
  {11}
  {22}
  {11,22}
  {12,21}
  {11,12,21}
  {12,21,22}
  {11,12,21,22}
(End)

Mohammad Behzad Kang and Andrew Salch, The mod p cohomology of the Morava stabilizer group at large primes, arXiv:2410.24171 [math.AT], 2024. See p. 46.

Column 1 of A229870.
The unlabeled version is A308111.
Cf. A000595, A002416, A002720, A007080.
Cf. A326209, A326237, A326251, A326252, A326253.

Table[Length[Select[Subsets[Tuples[Range[n],2]],Sort[First/@#]==Sort[Last/@#]&]],{n,4}] (* Gus Wiseman, Jun 22 2019 *)

a(n) = 2^n * A007080(n). - Andrew Howroyd, Sep 11 2019

a(0)=1 prepended by Alois P. Heinz, May 14 2019

Terms a(11) and beyond from Andrew Howroyd, Sep 11 2019

A326226 Number of unlabeled n-vertex Hamiltonian digraphs (with loops).

Original entry on oeis.org

0, 2, 3, 24, 858

Offset: 0

Gus Wiseman, Jun 14 2019

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

			Non-isomorphic representatives of the a(2) = 3 digraph edge-sets:
  {12,21}
  {11,12,21}
  {11,12,21,22}

Wikipedia, Hamiltonian path
Gus Wiseman, Non-isomorphic representatives of the a(3) = 24 Hamiltonian digraphs

The labeled case is A326204.
The case without loops is A326225.
The undirected case is A003216 (without loops) or A326215 (with loops).
Unlabeled non-Hamiltonian digraphs are A326223.
Unlabeled digraphs with a Hamiltonian path are A326221.
Cf. A000595, A002416, A003087, A057864, A246446, A326208.

dinorm[m_]:=If[m=={},{},If[Union@@m!=Range[Max@@Flatten[m]],dinorm[m/. Apply[Rule,Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}],{1}]],First[Sort[dinorm[m,1]]]]];
dinorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#1>=aft&]}]},Union@@(dinorm[#1,aft+1]&)/@Union[Table[Map[Sort,m/. {par+aft-1->aft,aft->par+aft-1},{0}],{par,First/@Position[mx,Max[mx]]}]]]];
Table[Length[Select[Union[dinorm/@Subsets[Tuples[Range[n],2]]],FindHamiltonianCycle[Graph[Range[n],DirectedEdge@@@#]]!={}&]],{n,0,4}] (* Mathematica 8.0+. Warning: Using HamiltonianGraphQ instead of FindHamiltonianCycle returns a(4) = 867 which is incorrect *)

cp's OEIS Frontend

A000666 Number of symmetric relations on n nodes.

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A002724 Number of inequivalent n X n binary matrices, where equivalence means permutations of rows or columns.

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Programs

Maple

Mathematica

PARI

Formula

Extensions

A006905 Number of transitive relations on n labeled nodes.

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Programs

Mathematica

PARI

Formula

Extensions

A001174 Number of oriented graphs (i.e., digraphs with no bidirected edges) on n unlabeled nodes. Also number of complete digraphs on n unlabeled nodes. Number of antisymmetric relations (i.e., oriented graphs with loops) on n unlabeled nodes is A083670.

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Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

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Extensions

A326204 Number of Hamiltonian labeled n-vertex digraphs (with loops).

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Mathematica

A326225 Number of Hamiltonian unlabeled n-vertex digraphs (without loops).

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A229865 Number of n X n 0..1 arrays with corresponding row and column sums equal.

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