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A365638 Triangular array read by rows: T(n, k) is the number of ways that a k-element transitive tournament can occur as a subtournament of a larger tournament on n labeled vertices.

%I A365638 #27 Dec 31 2024 15:37:01
%S A365638 1,1,1,2,4,2,8,24,24,6,64,256,384,192,24,1024,5120,10240,7680,1920,
%T A365638 120,32768,196608,491520,491520,184320,23040,720,2097152,14680064,
%U A365638 44040192,55050240,27525120,5160960,322560,5040,268435456,2147483648,7516192768,11274289152,7046430720,1761607680,165150720,5160960,40320
%N A365638 Triangular array read by rows: T(n, k) is the number of ways that a k-element transitive tournament can occur as a subtournament of a larger tournament on n labeled vertices.
%C A365638 A tournament is a directed digraph obtained by assigning a direction for each edge in an undirected complete graph. In a transitive tournament all nodes can be strictly ordered by their reachability.
%H A365638 Paul Erdős, <a href="https://www.renyi.hu/~p_erdos/1964-22.pdf">On a problem in graph theory</a>, The Mathematical Gazette, 47: 220-223 (1963).
%F A365638 T(n, k) = binomial(n, k)*k!*2^(binomial(n, 2) - binomial(k, 2)).
%F A365638 T(n, 0) = A006125(n).
%F A365638 T(n, 1) = A095340(n).
%F A365638 T(n, 2) = A103904(n).
%F A365638 T(n, n) = n!.
%F A365638 T(n, n-1) = A002866(n-1).
%F A365638 T(n, n-2) = A052670(n).
%F A365638 T(n, k) = A008279(n, k) * A117260(n, k). - _Peter Luschny_, Dec 31 2024
%e A365638 Triangle begins:
%e A365638      1
%e A365638      1,     1
%e A365638      2,     4,     2
%e A365638      8,    24,    24,     6
%e A365638     64,   256,   384,   192,    24
%e A365638   1024,  5120, 10240,  7680,  1920,  120
%p A365638 T := (n, k) -> 2^(((n-1)*n - (k-1)*k)/2) * n! / (n-k)!:
%p A365638 seq(seq(T(n, k), k = 0..n), n = 0..8);  # _Peter Luschny_, Nov 02 2023
%o A365638 (PARI) T(n, k) = binomial(n, k)*k!*2^(binomial(n, 2) - binomial(k, 2))
%Y A365638 Cf. A002866, A006125, A052670, A095340, A103904, A008279, A117260, A379614 (row sums).
%Y A365638 Cf. A122027, A224886, A259105, A350608, A350609, A350610.
%K A365638 nonn,easy,tabl
%O A365638 0,4
%A A365638 _Thomas Scheuerle_, Sep 14 2023