A123553 -id:A123553 - 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

A125031 Total number of highest scorers in all 2^(n(n-1)/2) tournaments with n players.

Original entry on oeis.org

1, 2, 12, 104, 1560, 53184, 3422384, 430790144, 111823251840, 56741417927680, 57729973360342272, 118195918779085344768, 479770203506298422135808, 3914602958361039682677710848, 63809077054456699374663196416000, 2076906726499655025703507210668998656

Offset: 1

Martin Fuller, Nov 16 2006

nonn

All highest scorers are also king chickens, A123553.

			With 4 players there are 32 tournaments with 1 highest scorer, 24 tournaments with 2 highest scorers and 8 tournaments with 3 highest scorers. Therefore a(4)=32*1+24*2+8*3=104.

Andrew Howroyd, Table of n, a(n) for n = 1..20
Index entries for sequences related to tournaments

Cf. A006125, A013976, A123553, A125032, A123903.

\\ Requires Winners from A013976.
a(n)={my(M=Winners(n)); sum(i=1, matsize(M)[1], pollead(M[i, 1])*M[i, 2])} \\ Andrew Howroyd, Feb 29 2020

a(5)-a(10) also computed by Gordon Royle, Nov 16 2006

Terms a(12) and beyond from Andrew Howroyd, Feb 28 2020

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.

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

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A125031 Total number of highest scorers in all 2^(n(n-1)/2) tournaments with n players.

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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.

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