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

1, 2, 6, 24, 280, 8640, 233520, 23157120, 5329376640, 1314169920000, 1016970317932800, 1772428331094220800, 3441650619022551936000, 22088285526822118789785600, 291368298787833283829100288000

Offset: 1

Author

N. J. A. Sloane, Jun 11 2016

Keywords

• nonn

Comments

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).

Examples

			With 4 players there 6 = 4*3/2 games played with 2^(4*3/2) = 64 possible outcomes.
The possible score sequences and the number of ways each can happen are as follows:
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)
3111 8
2220 8
2211 24
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.)
The score sequences with 4 players are partitions of 6 into 4 parts.
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.
		

References

• 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).

Links

• Shalosh B. Ekhad, On the Most Commonly-Occurring Score Vectors for American Tournaments of n-players, and their Corresponding Records, published in the personal journal of Shalosh B. Ekhad and Doron Zeilberger, Jun 13 2016; Local copy, pdf file only, no active links
• 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). [Annotated scanned copy, scanned at 300 dpi. Do not replace with a smaller file as the print is very tiny and hard to read.]

Crossrefs

Cf. A000571.

Extensions

a(1)-a(9) confirmed by N. J. A. Sloane, Jun 11 2016

a(10)-a(15) from Shalosh B. Ekhad, Jun 13 2016