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, 98672511931269

Offset

2,3

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 21 appear in Tables 3 and 4 of Dickey & Rosenberg (2026).

Links

Table of n, a(n) for n=2..21.

Emily H. Dickey and Noah A. Rosenberg, Labeled histories and maximally probable labeled topologies with multifurcation, arXiv:2511.16799 [q-bio.PE], 2025. See pp. 12-13.

Emily H. Dickey and Noah A. Rosenberg, Labeled histories and maximally probable labeled topologies with multifurcation, Discr. Appl. Math. 391 (2026), 192-203. See Tables 3 and 4.

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

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).

Example

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)).

Crossrefs

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

Cf. A001190.

Sequence in context: A106918 A182357 A181519 * A265778 A257866 A251617

Adjacent sequences: A380764 A380765 A380766 * A380768 A380769 A380770

Keyword

nonn,more

Author

Noah A Rosenberg, Feb 02 2025

Status

approved