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.

1, 1, 3, 5, 19, 63, 365, 1199, 7177, 36209, 295355, 1652085, 15193115, 114570449, 1323338487, 8732267521, 93577466255, 822198823101, 10952623368043

Offset: 2

Author

Noah A Rosenberg, Feb 02 2025

Keywords

• nonn
• more

Comments

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.

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.

Examples

			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)).
		

Links

• Matthew C. King and Noah A. Rosenberg, A mathematical connection between single-elimination sports tournaments and evolutionary trees, Math. Mag. 96 (2023), 484-497.

Crossrefs

Cf. A379758 and A380166 for game sequences with fully symmetric tournaments.

Cf. A001190.

Formula

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).