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.

%I A380166 #27 Mar 31 2025 22:58:13
%S A380166 1,0,1,2,0,0,1,22,102,160,80,0,0,0,1,672,45914,973300,9396760,
%T A380166 49410424,155188488,304369008,376231680,284951040,120806400,21964800,
%U A380166 0,0,0,0,1,458324,2493351562,1695612148252,328854102958150,26894789756402464,1153061834890296576,29635726970329429536
%N 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.
%C A380166 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.
%H A380166 M. C. King and N. 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 A380166 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).
%e A380166 Triangle begins:
%e A380166   1;
%e A380166   0, 1, 2;
%e A380166   0, 0, 1, 22, 102,   160,     80;
%e A380166   0, 0, 0,  1, 672, 45914, 973300, 9396760, 49410424, 155188488, 304369008, 376231680, 284951040, 120806400, 21964800;
%e A380166   ...
%e A380166 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.
%Y A380166 Row lengths give A000325(n).
%Y A380166 Right border gives A056972(n).
%Y A380166 Row sums give A379758(n).
%K A380166 nonn,tabf
%O A380166 1,4
%A A380166 _Noah A Rosenberg_, Jan 13 2025