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A350609 Triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n) = number of subdigraphs of the transitive tournament on n nodes that have k weak components.

%I A350609 #33 Jan 27 2022 16:19:15
%S A350609 1,1,1,4,2,2,31,15,10,8,474,228,162,96,64,14357,7057,5242,3296,1792,
%T A350609 1024,865024,438662,342394,222720,130048,65536,32768,103931595,
%U A350609 54542867,44669602,30110848,18337792,10027008,4718592,2097152,24935913222,13548525896,11608243634,8093078016,5130403840,2945449984,1518338048,671088640,268435456
%N A350609 Triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n) = number of subdigraphs of the transitive tournament on n nodes that have k weak components.
%C A350609 The sum of row n is 2^(n*(n-1)/2) = A006125(n).
%C A350609 For references and links see A350608.
%H A350609 Don Knuth, <a href="/A350609/b350609.txt">Rows n = 1..16, flattened</a>
%e A350609 For example, the entries for n=3 are {4,2,2}, because the empty subgraph and the subgraphs with a single arc have 1 weak component {123}; 1->2,1->3 and 1->3,2->3 have 2 weak components (namely {1,23} and {12,3}); finally 1->2,2->3 and 1->2,1->3,2->3 have 3 weak components (namely {1,2,3}).
%e A350609 Triangle T(n,k) begins:
%e A350609        1;
%e A350609        1,      1;
%e A350609        4,      2,      2;
%e A350609       31,     15,     10,      8;
%e A350609      474,    228,    162,     96,     64;
%e A350609    14357,   7057,   5242,   3296,   1792,  1024;
%e A350609   865024, 438662, 342394, 222720, 130048, 65536, 32768;
%e A350609   ...
%Y A350609 Column k=1 gives A350608.
%Y A350609 Main diagonal gives A006125(n-1).
%Y A350609 Cf. A350610.
%K A350609 nonn,tabl
%O A350609 1,4
%A A350609 _Don Knuth_, Jan 16 2022