cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Martin Fuller, Nov 16 2006

Keywords

Comments

All highest scorers are also king chickens, A123553.

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    \\ 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

Extensions

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

Views

Author

N. J. A. Sloane, Nov 14 2006

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    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

Formula

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

Extensions

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

Views

Author

Martin Fuller, Nov 16 2006

Keywords

Comments

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).

Examples

			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.

Links

Crossrefs

Cf. A000571, A006125, A068029, A125031 (number of highest scorers), A123553.
Other sequences that can be calculated using this one: A013976, A125031.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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