cp's OEIS Frontend

A274098 Number of ways the most probable score sequence happens in an n-person round-robin tournament.

%I A274098 #46 Mar 30 2021 14:43:59
%S A274098 1,2,6,24,280,8640,233520,23157120,5329376640,1314169920000,
%T A274098 1016970317932800,1772428331094220800,3441650619022551936000,
%U A274098 22088285526822118789785600,291368298787833283829100288000
%N A274098 Number of ways the most probable score sequence happens in an n-person round-robin tournament.
%C A274098 There are n players, each player plays all the others, so there are n(n-1)/2 games and 2^(n(n-1)/2) possible outcomes (there are no ties). At the end of the tournament we record the score sequence, which is the partition of n(n-1)/2 into n parts specified by the numbers of victories of the players. Then a(n) is the number of ways the most probable score sequence can occur. The number of different score sequences is given by A000571(n).
%D A274098 P. A. MacMahon, An American tournament treated by the calculus of symmetric functions, Quart. J. Pure Appl. Math., 49 (1920), 1-36. Gives a(1) to a(9).
%H A274098 Shalosh B. Ekhad, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/Percy.html">On the Most Commonly-Occurring Score Vectors for American Tournaments of n-players, and their Corresponding Records</a>, published in the personal journal of Shalosh B. Ekhad and Doron Zeilberger, Jun 13 2016; <a href="/A274098/a274098_1.pdf">Local copy, pdf file only, no active links</a>
%H A274098 P. A. MacMahon, <a href="/A274098/a274098.pdf">An American tournament treated by the calculus of symmetric functions</a>, Quart. J. Pure Appl. Math., 49 (1920), 1-36. Gives a(1) to a(9). [Annotated scanned copy, scanned at 300 dpi. Do not replace with a smaller file as the print is very tiny and hard to read.]
%e A274098 With 4 players there 6 = 4*3/2 games played with 2^(4*3/2) = 64 possible outcomes.
%e A274098 The possible score sequences and the number of ways each can happen are as follows:
%e A274098 3210 24 (meaning one player won 3 times, one player won twice, one player won once, and one player had no wins, and this can happen in 24 ways)
%e A274098 3111 8
%e A274098 2220 8
%e A274098 2211 24
%e A274098 The most probable score sequence is either 3210 or 2211, and either can happen in 24 ways, so a(4)=24. (Usually there is a unique most probable score sequence.)
%e A274098 The score sequences with 4 players are partitions of 6 into 4 parts.
%e A274098 For 6 players the most probable score sequence is 4,3,3,2,2,1. It is unique, and happens in 8640 of the 2^15 possible outcomes, so a(6) = 8640.
%Y A274098 Cf. A000571.
%K A274098 nonn
%O A274098 1,2
%A A274098 _N. J. A. Sloane_, Jun 11 2016
%E A274098 a(1)-a(9) confirmed by _N. J. A. Sloane_, Jun 11 2016
%E A274098 a(10)-a(15) from _Shalosh B. Ekhad_, Jun 13 2016