cp's OEIS Frontend

A380767 Number of sequences in which the games of a single-elimination tournament with n teams can be played if arbitrarily many arenas are available and the tournament bracket is chosen to be the bracket with the largest such number of sequences.

%I A380767 #12 Feb 23 2025 21:50:21
%S A380767 1,1,3,5,19,63,365,1199,7177,36209,295355,1652085,15193115,114570449,
%T A380767 1323338487,8732267521,93577466255,822198823101,10952623368043
%N A380767 Number of sequences in which the games of a single-elimination tournament with n teams can be played if arbitrarily many arenas are available and the tournament bracket is chosen to be the bracket with the largest such number of sequences.
%C A380767 a(n) is also the number of tie-permitting labeled histories for the labeled topology with n leaves that possesses the largest number of tie-permitting labeled histories.
%C A380767 Terms for n=2 to 8 appear in Tables 2 and 3 of King & Rosenberg (2023); terms for n=9 to 20 are supplied by Emily H. Dickey.
%H A380767 Matthew C. King and Noah A. Rosenberg, <a href="https://doi.org/10.1080/0025570X.2023.2266389">A mathematical connection between single-elimination sports tournaments and evolutionary trees</a>, Math. Mag. 96 (2023), 484-497.
%F A380767 a(n) is computed as the maximum over unlabeled binary rooted trees T with n leaves (trees in the set enumerated by A001190) of the quantity computed for tree T in eq. 3 of King & Rosenberg (2023) (by summing terms in Theorem 3).
%e A380767 For 5 teams A, B, C, D, E, the maximizing tournament structure is ((A,B),((C,D),E)). The 5 game sequences enumerated are: (1) Game (A,B), then game (C,D), then game ((C,D),E), then game ((A,B),((C,D),E)); (2) Game (C,D), then game (A,B), then game ((C,D),E), then game ((A,B),((C,D),E)); (3) Game (C,D), then game ((C,D),E), then game (A,B), then game ((A,B),((C,D),E)); (4) Game (A,B) and game (C,D) simultaneously, then game ((C,D),E), then game ((A,B),((C,D),E)); (5) Game (C,D), then game (A,B) and game ((C,D),E) simultaneously, then game ((A,B),((C,D),E)).
%Y A380767 Cf. A379758 and A380166 for game sequences with fully symmetric tournaments.
%Y A380767 Cf. A001190.
%K A380767 nonn,more
%O A380767 2,3
%A A380767 _Noah A Rosenberg_, Feb 02 2025