A379758 -id:A379758 - cp's OEIS frontend

A380166 Triangle read by rows: T(n,k) is the number of sequences in which the games of a fully symmetric single-elimination tournament with 2^n teams can be played if arbitrarily many arenas are available and the number of distinct times at which games are played is k, 1 <= k <= 2^n-1.

1, 0, 1, 2, 0, 0, 1, 22, 102, 160, 80, 0, 0, 0, 1, 672, 45914, 973300, 9396760, 49410424, 155188488, 304369008, 376231680, 284951040, 120806400, 21964800, 0, 0, 0, 0, 1, 458324, 2493351562, 1695612148252, 328854102958150, 26894789756402464, 1153061834890296576, 29635726970329429536

Offset: 1

Noah A Rosenberg, Jan 13 2025

T(n,k) is also the number of tie-permitting labeled histories for a fully symmetric labeled topology with 2^n leaves and exactly k times at which events take place.

			Triangle begins:
  1;
  0, 1, 2;
  0, 0, 1, 22, 102,   160,     80;
  0, 0, 0,  1, 672, 45914, 973300, 9396760, 49410424, 155188488, 304369008, 376231680, 284951040, 120806400, 21964800;
  ...
For n = 2 and a tournament with 2^2 = 4 teams and structure ((A,B),(C,D)), game (A,B) can be played before or after game (C,D); the championship game occurs later, producing T(2,3) = 2. Alternatively, game (A,B) can be played simultaneously with (C,D), with the championship game occurring later, producing T(2,2) = 1.

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

Row lengths give A000325(n).
Right border gives A056972(n).
Row sums give A379758(n).

T(1,1) = 1 and for n > 1 and n <= k <= 2^n - 1, T(n,k) = Sum_{c=max(n-1,k-2^(n-1))..min(2^(n-1)-1,k-1)} Sum_{d=max(n-1,k-c-1)..min(2^(n-1)-1,k-1)} ((k-1)! / ((k-1-d)! * (k-1-c)! * (c+d-(k-1))!)) * T(n-1,c) * T(n-1,d).

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.

Original entry on oeis.org

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

Offset: 2

Noah A Rosenberg, Feb 02 2025

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.

			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)).

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

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

Cf. A001190.

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).

A381865 Number of sequences in which the matches of a fully symmetric single-elimination tournament with 3^n players can be played if arbitrarily many matches can occur simultaneously and each match involves 3 players.

Original entry on oeis.org

1, 1, 13, 308682013, 20447648974223714249697186722386536049691073

Offset: 0

Noah A Rosenberg, Mar 08 2025

a(n) is also the number of tie-permitting labeled histories for a fully symmetric strictly trifurcating labeled topology with 3^n leaves.

			Two of the 13 cases with n=2 and 3^2=9 players are: (1) (A,B,C) play, then (D,E,F) play, then (G,H,I) play, then the winners of the three matches play; (2) (A,B,C) play simultaneously with (D,E,F), then the winners of these two matches play against G, then the winner plays against H and I.

Emily H. Dickey and Noah A. Rosenberg, Labelled histories with multifurcation and simultaneity, Phil. Trans. R. Soc. B 380 (2025), 20230307. (see Theorem 15 with r=3)

Cf. A273723 (if matches must be non-simultaneous), A379758 (if matches involve only two players at a time).

A381948 Number of sequences in which the matches of a fully symmetric single-elimination tournament with 4^n players can be played if arbitrarily many matches can occur simultaneously and each match involves 4 players.

Original entry on oeis.org

1, 1, 75, 3016718788056802445, 940214577272785072764883853635996915471902343186386048409875362373502134253520788722829230121857323681047351543536731036815

Offset: 0

Noah A Rosenberg, Mar 10 2025

nonn

a(n) is also the number of tie-permitting labeled histories for a fully symmetric strictly quadfurcating labeled topology with 4^n leaves.

			Two of the 75 cases with n=4 and 4^2=16 players are: (1) (A,B,C,D) play, then (E,F,G,H) play, then (I,J,K,L) play, then (M,N,O,P) play, then the winners of the four matches play; (2) (A,B,C,D) play simultaneously with (E,F,G,H) and (I,J,K,L), then the winners of these three matches play against M, then the winner plays against N, O, and P.

Emily H. Dickey and Noah A. Rosenberg, Labelled histories with multifurcation and simultaneity, Phil. Trans. R. Soc. B 380 (2025), 20230307. (see Theorem 15 with r=4)

Cf. A273725 (if matches must be non-simultaneous), A379758 (if matches involve only two players at a time), A381865 (if matches involve only three players at a time).

cp's OEIS Frontend

A380166 Triangle read by rows: T(n,k) is the number of sequences in which the games of a fully symmetric single-elimination tournament with 2^n teams can be played if arbitrarily many arenas are available and the number of distinct times at which games are played is k, 1 <= k <= 2^n-1.

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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.

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A381865 Number of sequences in which the matches of a fully symmetric single-elimination tournament with 3^n players can be played if arbitrarily many matches can occur simultaneously and each match involves 3 players.

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A381948 Number of sequences in which the matches of a fully symmetric single-elimination tournament with 4^n players can be played if arbitrarily many matches can occur simultaneously and each match involves 4 players.

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