A002086 -id:A002086 - cp's OEIS frontend

A049288 Number of nonisomorphic circulant tournaments, i.e., Cayley tournaments for cyclic group of order 2n-1.

1, 1, 1, 2, 3, 4, 6, 16, 16, 30, 88, 94, 205, 457, 586, 1096, 3280, 5472, 7286, 21856, 26216, 49940, 174848, 182362, 399472, 1048576, 1290556, 3355456, 7456600, 9256396, 17895736, 59654816, 89478656, 130150588, 390451576, 490853416, 954437292

Offset: 1

Valery A. Liskovets

Further values for prime-squared orders can be found in A038789.

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

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

Cf. A002086, A002087, A038789, A049297, A049287, A049289, A060966.

a(n) <= A002086(n). - Andrew Howroyd, Apr 28 2017

a(n) = A002086(n) for squarefree 2n-1. - Andrew Howroyd, Apr 28 2017

a(14)-a(37) from Andrew Howroyd, Apr 28 2017

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

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

A000198 Largest order of automorphism group of a tournament with n nodes.

Original entry on oeis.org

1, 1, 3, 3, 5, 9, 21, 21, 81, 81, 81, 243, 243, 441, 1215, 1701, 1701, 6561, 6561, 6561, 45927, 45927, 45927, 137781, 137781, 229635, 1594323, 1594323, 1594323, 4782969, 4782969, 7971615, 14348907, 33480783, 33480783, 129140163, 129140163, 129140163

Offset: 1

N. J. A. Sloane

J. W. Moon, Topics on Tournaments. Holt, NY, 1968, p. 81.
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).

Joseph Myers, Table of n, a(n) for n = 1..1000
B. Alspach, A combinatorial proof of a conjecture of Goldberg and Moon, Canad. Math. Bull. 11 (1968), 655-661.
B. Alspach, On point-symmetric tournaments, Canad. Math. Bull., 13 (1970), 317-323. [Annotated copy] See g(n) as defined on page 317 (NOT on page 322).
B. Alspach and J. L. Berggren, On the determination of the maximum order of the group of a tournament, Canad. Math. Bull 16 (1973), 11-14.
J. D. Dixon, The maximum order of the group of a tournament, Canad. Math. Bull. 10 (1967), 503-505.
Index entries for sequences related to tournaments

Cf. A000568, A374164.

a:= proc(n) local t, r; t:= irem(n, 9);
   `if`(3^ilog[3](n)=n, 3^((3^ilog[3](n)-1)/2),
   `if`(irem(n, 5, 'r')=0 and 3^ilog[3](r)=r, 5*3^((5*3^ilog[3](r)-5)/2),
   `if`(irem(n, 7, 'r')=0 and 3^ilog[3](r)=r, 7*3^((7*3^ilog[3](r)-5)/2),
   `if`(irem(n, 3, 'r')=0, 3^r*a(r),
   `if`(t in {1, 2, 4}, a(n-1),
   `if`(t = 8, max(a(n-1), a(5)*a(n-5)),
   `if`(t = 5, max(a(2)*a(n-2), a(5)*a(n-5), a(7)*a(n-7)),
        a(7)*a(n-7) )))))))
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Jun 29 2012

a[n_] := a[n] = With[{t = Mod[n, 9]}, Which[ IntegerQ[Log[3, n]], 3^((1/2)*(n-1)),{q, r} = QuotientRemainder[n, 5]; r == 0 && IntegerQ[Log[3, q]], 5*3^((1/2)*(n-5)),{q, r} = QuotientRemainder[n, 7];r == 0 && IntegerQ[Log[3, q]], 7*3^((1/2)*(n-5)), {q, r} = QuotientRemainder[n, 3]; r == 0, 3^q*a[q],MemberQ[{1, 2, 4}, t], a[n-1],t == 8, Max[a[n-1], a[5]*a[n-5]], t == 5, Max[a[2]*a[n-2],a[5]*a[n-5], a[7]*a[n-7]],True, a[7]*a[n-7]]]; Table[a[n], {n, 1, 38}] (* Jean-François Alcover, Nov 12 2012, after Alois P. Heinz *)

from functools import lru_cache
@lru_cache(maxsize=None)
def A000198(n):
    if n <= 7: return (1, 1, 3, 3, 5, 9, 21)[n-1]
    if (r:=n%9) in {0,3,6}:
        return 3**(m:=n//3)*A000198(m)
    elif r in {1,2,4}:
        return A000198(n-1)
    elif r == 5:
        return max(A000198(n-2),5*A000198(n-5),21*A000198(n-7))
    elif r == 7:
        return 21*A000198(n-7)
    elif r == 8:
        return max(A000198(n-1),5*A000198(n-5)) # Chai Wah Wu, Jul 01 2024

a(3^k) = 3^((3^k - 1)/2), a(5*3^k) = 5*3^((5*3^k - 5)/2), a(7*3^k) = 7*3^((7*3^k - 5)/2), and, for all other n, a(n) = max(a(i)a(n-i)) where the maximum is taken over 1 <= i <= n-1 (from Alspach and Berggren (1973) Theorem 4).

a(3r) = (3^r)a(r), a(n) = a(n-1) for n = 1, 2 or 4 mod 9, a(9k+8) = max(a(9k+7), a(5)a(9k+3)), a(9k+5) = max(a(2)a(9k+3), a(5)a(9k), a(7)a(9k-2)), a(9k+7) = a(7)a(9k) (from Alspach and Berggren (1973) Theorem 5).

Edited and extended by Joseph Myers, Jun 28 2012

cp's OEIS Frontend

A049288 Number of nonisomorphic circulant tournaments, i.e., Cayley tournaments for cyclic group of order 2n-1.

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

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A000198 Largest order of automorphism group of a tournament with n nodes.

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