A000568 -id:A000568 - cp's OEIS frontend

A121272 Number of outcomes of unlabeled n-team round-robin tournaments that are not uniquely defined by their score vectors.

0, 0, 0, 0, 5, 45, 438, 6849, 191483, 9732967, 903753099, 154108310917, 48542114686488, 28401423719121589, 31021002160355165644, 63530415842308265098260, 244912778438520759443242406

Offset: 1

Tanya Khovanova, Aug 23 2006

nonn

This sequence is the difference between A000568 (Number of outcomes of unlabeled n-team round-robin tournaments) and A000570 (Number of tournaments on n nodes determined by their score vectors).

			All tournaments with 4 or fewer teams are uniquely defined by their score vectors. Hence a(1) = a(2) = a(3) = a(4) = 0.
For five-team tournaments only two score sequences do not define the tournament uniquely: {1,1,2,3,3} and {1,2,2,2,3}. The first sequence corresponds to two different tournaments and the second sequence to three different tournaments. Thus a(5) = 5.

Eric Weisstein's World of Mathematics, Score Sequence.

Cf. A000568, A000570.

a(n) = A000568(n) - A000570(n). - Michel Marcus, Nov 01 2019

A255599 Number of (semi-)regular tournaments of order n.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 3, 85, 15, 13333, 1223, 19434757, 1495297, 276013571133, 18400989629

Offset: 1

Danny Rorabaugh, Feb 27 2015

A tournament of odd order n is regular if the out-degree of each vertex is (n-1)/2. A tournament of even order n is semi-regular if the out-degree of each vertex is n/2-1 or n/2.

B. McKay, Digraphs

Cf. A000568, A096368.

a(1)-a(2) prepended and a(14)-a(15) added by Brendan McKay, Mar 21 2019

A256373 The decimal values of binary sequences representing the "bits" form of adjacency matrices of non-isomorphic tournament graphs.

Original entry on oeis.org

0, 5, 34, 36, 100, 165, 520, 528, 544, 549, 565, 814, 1552, 1589, 2568, 2577, 2592, 2593, 2597, 4629, 4663, 8245, 8328, 8469, 8748, 8757, 8765, 16448, 16484, 16512, 16549, 16640, 16645, 16896, 16901, 16904, 16932

Offset: 1

Dan Parrish, Mar 26 2015

nonn

For the tournaments on n vertices, the matrices consist of the first A000568(n)terms of the sequence.

			The 4x4 non-isomorphic tournament matrices are as follows:
The first has "bits" value 0, then second 5, the third 34, the fourth 36, where we read the bit pattern only from the elements above the diagonal, starting at the first row, left to right, then top to bottom.
There are twelve 5x5 non-isomorphic tournament matrices;  their bit patterns are  0, 5, 34, 36, 100, 165, 520, 528, 544, 549, 565, 814.

Brendan McKay, Digraphs, Australian National University.

A354607 Triangular array read by rows: T(n,k) is the number of labeled tournaments on [n] that have exactly k irreducible (strongly connected) components, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 2, 0, 6, 0, 24, 16, 0, 24, 0, 544, 240, 120, 0, 120, 0, 22320, 6608, 2160, 960, 0, 720, 0, 1677488, 315840, 70224, 20160, 8400, 0, 5040, 0, 236522496, 27001984, 3830400, 758016, 201600, 80640, 0, 40320, 0, 64026088576, 4268194560, 366729600, 46448640, 8628480, 2177280, 846720, 0, 362880

Offset: 0

Geoffrey Critzer, Jul 08 2022

			Triangle T(n,k) begins:
  1;
  0,     1;
  0,     0,    2;
  0,     2,    0,    6;
  0,    24,   16,    0,  24;
  0,   544,  240,  120,   0, 120;
  0, 22320, 6608, 2160, 960,   0, 720;
  ...

N. J. A. Sloane, Illustration of first 5 terms
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 11 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)

Cf. A006125 (row sums), A054946 (column k=1), A000142 (main diagonal).

nn = 10; G[x_] := Sum[2^Binomial[n, 2] x^n/n!, {n, 0, nn}]; Table[
Take[(Range[0, nn]! CoefficientList[Series[1/(1 - y (1 - 1/ G[x])), {x, 0, nn}], {x, y}])[[i]], i], {i, 1, nn}]

E.g.f.: 1/(1-y*(1-1/A(x))) where A(x) is the e.g.f. for A006125.

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A121272 Number of outcomes of unlabeled n-team round-robin tournaments that are not uniquely defined by their score vectors.

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A255599 Number of (semi-)regular tournaments of order n.

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A256373 The decimal values of binary sequences representing the "bits" form of adjacency matrices of non-isomorphic tournament graphs.

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A354607 Triangular array read by rows: T(n,k) is the number of labeled tournaments on [n] that have exactly k irreducible (strongly connected) components, n >= 0, 0 <= k <= n.

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