A049288 -id:A049288 - cp's OEIS frontend

A049287 Number of nonisomorphic circulant graphs, i.e., undirected Cayley graphs for the cyclic group of order n.

1, 2, 2, 4, 3, 8, 4, 12, 8, 20, 8, 48, 14, 48, 44, 84, 36, 192, 60, 336, 200, 416, 188, 1312, 423, 1400, 928, 3104, 1182, 8768, 2192, 8364, 6768, 16460, 11144, 46784, 14602, 58288, 44424, 136128, 52488, 355200, 99880, 432576, 351424, 762608, 364724, 2122944, 798952, 3356408

Offset: 1

Valery A. Liskovets

Further values for (twice) squarefree and (twice) prime-squared orders can be found in the Liskovets reference.

Terms may be computed by filtering potentially isomorphic graphs of A285620 through nauty. - Andrew Howroyd, Apr 29 2017

Andrew Howroyd, Table of n, a(n) for n = 1..70
V. Gatt, On the Enumeration of Circulant Graphs of Prime-Power Order: the case of p^3, arXiv:1703.06038 [math.CO], 2017.
V. A. Liskovets, Some identities for enumerators of circulant graphs, arXiv:math/0104131 [math.CO], 2001; J. Alg. Comb. 18 (2003) 189.
V. A. Liskovets and R. Poeschel, On the enumeration of circulant graphs of prime-power and squarefree orders.
Brendan McKay, Nauty home page.
R. Poeschel, Publications.
Eric Weisstein's World of Mathematics, Circulant Graph.
Eric Weisstein's World of Mathematics, Circulant Matrix.

Cf. A049297, A049288, A049289, A060966, A285620, A038781, A075545.

CountDistinct /@ Table[CanonicalGraph[CirculantGraph[n, #]] & /@ Subsets[Range[Floor[n/2]]], {n, 25}] (* Eric W. Weisstein, May 13 2017 *)

There is an easy formula for prime orders. Formulae are also known for squarefree and prime-squared orders.
From Andrew Howroyd, Apr 24 2017: (Start)
a(n) <= A285620(n).
a(n) = A285620(n) for n squarefree or twice square free.
a(A000040(n)^2) = A038781(n).
a(n) = Sum_{d|n} A075545(d).
(End)

a(48)-a(50) from Andrew Howroyd, Apr 29 2017

A049289 Number of nonisomorphic self-complementary circulant graphs (Cayley graphs for the cyclic group) of order 4n+1.

Original entry on oeis.org

1, 1, 0, 2, 4, 0, 7, 10, 0, 30, 56, 0, 0, 316, 0, 1096

Offset: 0

Valery A. Liskovets

Further values for (twice) squarefree and (twice) prime-squared orders can be found in the Liskovets reference.

V. A. Liskovets, Some identities for enumerators of circulant graphs, arXiv:math/0104131 [math.CO], 2001.
V. A. Liskovets and R. Poeschel, On the enumeration of circulant graphs of prime-power and squarefree orders
R. Poeschel, Publications

Cf. A049297, A049287, A049288.

A002086 Number of circulant tournaments on 2n+1 nodes up to Cayley isomorphism.

Original entry on oeis.org

1, 1, 2, 4, 4, 6, 16, 16, 30, 88, 94, 208, 472, 586, 1096, 3280, 5472, 7286, 21856, 26216, 49940, 175104, 182362, 399480, 1048576, 1290556, 3355456, 7456600, 9256396, 17895736, 59660288, 89478656, 130150588, 390451576, 490853416, 954437292, 3435974656

Offset: 1

N. J. A. Sloane

nonn

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..200
B. Alspach, On point-symmetric tournaments, Canad. Math. Bull., 13 (1970), 317-323. See g(n) as defined on page 322 (NOT on page 317).
B. Alspach, On point-symmetric tournaments, Canad. Math. Bull., 13 (1970), 317-323. [Annotated copy] See g(n) as defined on page 322 (NOT on page 317).
Index entries for sequences related to tournaments

Cf. A002087, A049288.

IsLeastPoint[s_, f_] := Module[{t = f[s]}, While[t > s, t = f[t]]; s == t];
C0[n_, k_] := Sum[Boole @ IsLeastPoint[u, Mod[#*k, n]&], {u, 1, n-1}]/2;
IsBidrected[s_, r_, f_] := Module[{t = f[s]}, While[t != s && t != r, t = f[t]]; t == r];
IsC[n_, k_] := Sum[Boole @ IsBidrected[u, n-u, Mod[#*k, n]&], {u, 1, n-1}] == 0;
a[n_] := Module[{m = 2*n + 1}, Sum[If [GCD[m, k] == 1 && IsC[m, k], 2^C0[m, k], 0], {k, 1, m}]/EulerPhi[m]];
Array[a, 40] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)

IsLeastPoint(s,f)={my(t=f(s));while(t>s,t=f(t));s==t}
C(n,k)=sum(u=1,n-1,IsLeastPoint(u,v->v*k%n))/2;
IsBidrected(s,r,f)={my(t=f(s));while(t<>s&&t<>r,t=f(t));t==r}
IsC(n,k)=sum(u=1,n-1,IsBidrected(u,n-u,v->v*k%n))==0;
a(n)=my(m=2*n+1);sum(k=1, m, if (gcd(m,k)==1 && IsC(m,k), 2^C(m,k),0))/eulerphi(m); \\ Andrew Howroyd, Sep 30 2017

More terms from Roderick J. Fletcher, Oct 15 1996 (yylee(AT)mail.ncku.edu.tw)
Definition corrected by Andrew Howroyd, Apr 28 2017
Terms a(32) and beyond from Andrew Howroyd, Sep 30 2017

A038789 Number of nonisomorphic circulant p^2-tournaments, indexed by odd primes p.

Original entry on oeis.org

3, 205, 399472, 10481104587335128, 123992391755346585462636, 81988033818127290961528376002383682007296, 4480981113642949878240780781141254929604041319893664

Offset: 1

N. J. A. Sloane, May 04 2000

M. Klin, V. A. Liskovets and R. Poeschel, Analytical enumeration of circulant graphs with prime-squared vertices, Sem. Lotharingien de Combin., B36d, 1996, 36 pages.

Cf. A049288, A038785.

More terms from Valery A. Liskovets, May 09 2001

Offset corrected by Sean A. Irvine, Feb 14 2021

A002087 Number of point-symmetric tournaments with 2n+1 nodes.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 16, 16, 30

Offset: 1

N. J. A. Sloane

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).

B. Alspach, On point-symmetric tournaments, Canad. Math. Bull., 13 (1970), 317-323. See t(n).
B. Alspach, On point-symmetric tournaments, Canad. Math. Bull., 13 (1970), 317-323. [Annotated copy] See t(n).
Index entries for sequences related to tournaments

Cf. A002086, A049288 (note that A002086, A002087, A049288 are distinct sequences).

Reference to Alspach (1970) corrected by Andrew Howroyd, Apr 28 2017

A049309 Number of nonisomorphic self-complementary circulant digraphs (Cayley digraphs for the cyclic group) of order 2n-1.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 8, 20, 20, 30, 88, 94, 214, 457, 596, 1096, 3280, 5560, 7316, 21944, 26272, 49940

Offset: 1

Valery A. Liskovets

There is an easy formula for prime orders. Formulae are also known for squarefree and prime-squared orders.

Further values for squarefree and prime-squared orders can be found in the Liskovets reference.

V. A. Liskovets, Some identities for enumerators of circulant graphs, arXiv:math/0104131 [math.CO], 2001.
V. A. Liskovets and R. Poeschel, On the enumeration of circulant graphs of prime-power and squarefree orders
R. Poeschel, Publications

Cf. A049288, A049289, A049297, A038785.

a(14)-a(22) from Andrew Howroyd, May 06 2017

A054246 Non-Cayley-isomorphic circulant p^2-tournaments, indexed by odd primes p.

Original entry on oeis.org

1, 1, 4, 16, 36, 256, 900, 8836, 343396, 1201216, 53085796

Offset: 3

N. J. A. Sloane, May 04 2000

V. A. Liskovets and R. Poeschel, Non-Cayley-isomorphic self-complementary circulant graphs, J. Graph Th., 34, 2000, 128-141.

M. Klin, V. A. Liskovets and R. Poeschel, Analytical enumeration of circulant graphs with prime-squared vertices, Sem. Lotharingien de Combin., B36d, 1996, 36 pages.

Cf. A038788, A038791.

a(p^2)=A049288(p)^2

More terms from Valery A. Liskovets, May 09 2001

A346179 Number of nonisomorphic vertex-transitive tournaments of order 2n-1.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 16, 16, 30, 110, 94, 214, 694, 586, 1096, 3280, 5472, 7286, 25206, 26216, 49940, 196624, 182362, 407856, 907116

Offset: 1

Brendan McKay, Jul 09 2021

The circulant tournaments A049288 are included.

Up to 49 vertices, non-circulant vertex-transitive tournaments occur on 21, 25, 27, 39, 45 and 49 vertices.

Brendan McKay, vertex-transitive tournaments

If the automorphism group contains a full-length cycle, the tournament is circulant and is counted by A049288.

Cf. A060747.

cp's OEIS Frontend

A049287 Number of nonisomorphic circulant graphs, i.e., undirected Cayley graphs for the cyclic group of order n.

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A049289 Number of nonisomorphic self-complementary circulant graphs (Cayley graphs for the cyclic group) of order 4n+1.

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A002086 Number of circulant tournaments on 2n+1 nodes up to Cayley isomorphism.

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A038789 Number of nonisomorphic circulant p^2-tournaments, indexed by odd primes p.

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A002087 Number of point-symmetric tournaments with 2n+1 nodes.

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A049309 Number of nonisomorphic self-complementary circulant digraphs (Cayley digraphs for the cyclic group) of order 2n-1.

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A054246 Non-Cayley-isomorphic circulant p^2-tournaments, indexed by odd primes p.

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Formula

Extensions

A346179 Number of nonisomorphic vertex-transitive tournaments of order 2n-1.

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