A125031 -id:A125031 - cp's OEIS frontend

A123903 Total number of "Emperors" in all tournaments on n labeled nodes.

0, 1, 2, 6, 32, 320, 6144, 229376, 16777216, 2415919104, 687194767360, 387028092977152, 432345564227567616, 959230691832896684032, 4231240368651202111471616, 37138201178561408246973726720, 649037107316853453566312041152512, 22596875928343569839364720024765857792

Offset: 0

N. J. A. Sloane, Nov 20 2006

nonn

An "Emperor" is a player who beats everybody else.

a(n) is the number of isolated nodes in all simple labeled graphs on n nodes. - Geoffrey Critzer, Oct 19 2011

Alois P. Heinz, Table of n, a(n) for n = 0..50
S. B. Maurer, The king chicken theorems, Math. Mag., 53 (1980), 67-80.

Cf. A123553, A125031, A006125.

List([0..20], n-> n*2^Binomial(n-1,2)); # G. C. Greubel, Aug 06 2019

[n*2^Binomial(n-1,2): n in [0..20]]; // G. C. Greubel, Aug 06 2019

a:= n-> n*2^((n-1)*(n-2)/2):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 26 2013

a=Sum[2^Binomial[n,2]x^n/n!,{n,0,20}];
Range[0,20]!CoefficientList[Series[x a,{x,0,20}],x]
Table[n*2^Binomial[n-1,2], {n,0,20}] (* G. C. Greubel, Aug 06 2019 *)

A123903(n):=n*2^((n-1)*(n-2)/2)$ makelist(A123903(n),n,0,17); /* Martin Ettl, Nov 13 2012 */

vector(20, n, n--; n*2^binomial(n-1,2)) \\ G. C. Greubel, Aug 06 2019

[n*2^binomial(n-1,2) for n in (0..20)] # G. C. Greubel, Aug 06 2019

a(n) = n*2^((n-1)*(n-2)/2).
E.g.f.: x * Sum_{n>=0} 2^C(n,2) x^n/n!. - Geoffrey Critzer, Oct 19 2011
a(n) = n * A006125(n-1). - Anton Zakharov, Dec 21 2016

A013976 Number of tournaments on n nodes with a unique winner.

Original entry on oeis.org

1, 2, 6, 32, 600, 20544, 1218224, 160241152, 42129744768, 21293228876800, 22220602090444032, 45959959305969143808, 188702851949391611599872, 1566370829898884830446395392, 25801614282517307546368494213120, 851300764257208779000509484701188096

Offset: 1

Michael Stob (stob(AT)udu.calvin.edu)

Andrew Howroyd, Table of n, a(n) for n = 1..20
Yaakov Malinovsky and John W. Moon, On Round-Robin Tournaments with a Unique Maximum Score and Some Related Results, arXiv:2208.14932 [math.CO], 2022.
Index entries for sequences related to tournaments

Cf. A125031.

Winners(n)={
  local(M=Map(Mat([x^0, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(merge(r, p, v)=acc(p + sum(i=1, poldegree(p)-r-1, polcoef(p,i)*(1-x^i)), v));
  my(recurse(r, p, i, q, v, e)=if(i<0, merge(r, x^e+q, v), my(t=polcoef(p, i)); for(k=0, t, self()(r, p, i-1, (k+x*(t-k))*x^i+q, binomial(t, k)*v, e+k))));
  for(k=2, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], my(p=src[i, 1]); recurse(n-k, p, poldegree(p), 0, src[i, 2], 0)));
  Mat(M);
}
a(n)={my(M=Winners(n)); sum(i=1, matsize(M)[1], if(pollead(M[i,1])==1, M[i,2]))} \\ Andrew Howroyd, Feb 29 2020

a(1)=1 inserted and a(12) and beyond from Andrew Howroyd, Feb 28 2020

A123553 A "king chicken" in a tournament graph (a directed labeled graph on n nodes with a single arc between every pair of nodes) is a player A who for any other player B either beats B directly or beats someone who beats B. Sequence gives total number of king chickens in all 2^(n(n-1)/2) tournaments.

Original entry on oeis.org

1, 2, 12, 128, 2680, 109824, 8791552, 1376518144, 422360211456, 254460936847360, 301592785058791424, 704473043710859280384, 3248469673423387574140928, 29616255381502146777580568576, 534589619577015738514639410954240, 19128875195554152154920492396852543488

Offset: 1

N. J. A. Sloane, Nov 14 2006

nonn

H. G. Landau showed in 1951 that there may be several king chickens in a tournament and that a player is a king chicken if he has the highest score. The converse is not true and there can be more king chickens than highest scorers. The smallest example has 4 players: A beats B and C, B beats C and D, C beats D, D beats A; D is a king chicken despite having fewer points than A and B. Maurer showed in 1980 that there is one king chicken if one player beats all others and otherwise there are at least three.

			For n = 3 there are 8 tournaments: six of the form A beats B and C and B beats C, with one king chicken (A) and two of the form A beats B beats C beats A, with three king chickens each (A or B or C), for a total of 6*1 + 2*3 = 12.

Christian Sievers, Table of n, a(n) for n = 1..81
S. B. Maurer, The king chicken theorems, Math. Mag., 53 (1980), 67-80.
Index entries for sequences related to tournaments

Cf. A006125, A013976, A125032, A125031 (highest scorers), A123903 (Emperors).

a(n)=n*sum(k=0,n-1,binomial(n-1,k)*2^(binomial(k,2)+binomial(n-1-k,2))*(2^k-1)^(n-1-k)) \\ Christian Sievers, Nov 01 2023

a(n) >= A006125(n)*3 - A006125(n-1)*n*2 with equality for n<=4.

a(n) = n * Sum_{k=0..n-1} C(n-1,k) * 2^(C(k,2)+C(n-1-k,2)) * (2^k-1)^(n-1-k) where C(n,k) is the binomial coefficient. - Christian Sievers, Nov 01 2023

Corrected and edited by Martin Fuller, Nov 16 2006

a(7) and beyond from Christian Sievers, Nov 01 2023

A125032 Triangle read by rows: T(n,k) = number of tournaments with n players which have the k-th score sequence. The score sequences are in the same order as A068029 and start with the empty score sequence.

Original entry on oeis.org

1, 1, 2, 6, 2, 24, 8, 8, 24, 120, 40, 40, 120, 40, 120, 240, 280, 24, 720, 240, 240, 720, 240, 720, 1440, 1680, 144, 240, 80, 720, 1440, 2880, 1680, 1680, 1680, 8640, 2400, 144, 2400, 2640, 5040, 1680, 1680, 5040, 1680, 5040, 10080, 11760, 1008, 1680, 560

Offset: 1

Martin Fuller, Nov 16 2006

The score sequences are sorted by number of players and then lexicographically.

There are A000571(m) score sequences for m players. The sum of all the a(n) for m players is A006125(m)=2^(m(m-1)/2).

			There are two score sequences with 3 players: [0,1,2] from 6 tournaments and [1,1,1] from 2 tournaments. These score sequences come 4th and 5th respectively, so a(4)=6 and a(5)=2.

Martin Fuller, Table of n, a(n) for n = 1..2242
Eric Weisstein's World of Mathematics, Score Sequence
Index entries for sequences related to tournaments

Cf. A000571, A006125, A068029, A125031 (number of highest scorers), A123553.

Other sequences that can be calculated using this one: A013976, A125031.

cp's OEIS Frontend

A123903 Total number of "Emperors" in all tournaments on n labeled nodes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

Maxima

PARI

Sage

Formula

A013976 Number of tournaments on n nodes with a unique winner.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A125032 Triangle read by rows: T(n,k) = number of tournaments with n players which have the k-th score sequence. The score sequences are in the same order as A068029 and start with the empty score sequence.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs