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A368602 Triangle read by rows where T(n,k) is the number of labeled acyclic digraphs on {1..n} with sinks {1..k}.

%I A368602 #11 Jan 04 2024 15:33:53
%S A368602 1,0,1,0,1,1,0,5,3,1,0,79,33,7,1,0,3377,1071,161,15,1,0,362431,92289,
%T A368602 10591,705,31,1,0,93473345,19856703,1832705,93375,2945,63,1,0,
%U A368602 56272471039,10249747713,789619327,32382465,782719,12033,127,1
%N A368602 Triangle read by rows where T(n,k) is the number of labeled acyclic digraphs on {1..n} with sinks {1..k}.
%C A368602 Also the number of set-systems with n vertices and n edges such that {i} is a singleton edge iff i <= k, and such that there is only one way to choose a different vertex from each edge.
%F A368602 T(n,k) = A361718(n,k)/binomial(n,k).
%e A368602 Triangle begins:
%e A368602     1
%e A368602     0    1
%e A368602     0    1    1
%e A368602     0    5    3    1
%e A368602     0   79   33    7    1
%e A368602     0 3377 1071  161   15    1
%e A368602     ...
%e A368602 Row n = 3 counts the following set-systems:
%e A368602   {{1},{1,2},{1,3}}    {{1},{2},{1,3}}    {{1},{2},{3}}
%e A368602   {{1},{1,2},{2,3}}    {{1},{2},{2,3}}
%e A368602   {{1},{1,3},{2,3}}    {{1},{2},{1,2,3}}
%e A368602   {{1},{1,2},{1,2,3}}
%e A368602   {{1},{1,3},{1,2,3}}
%t A368602 Table[Length[Select[Subsets[Subsets[Range[n]],{n}], Union@@Cases[#,{_}]==Range[k] && Length[Select[Tuples[#],UnsameQ@@#&]]==1&]], {n,0,3},{k,0,n}]
%Y A368602 Column k = n-1 is A000225 = A058877(n)/n.
%Y A368602 Column k = 1 is A134531 (up to sign) or A003025(n)/n, non-fixed A350415.
%Y A368602 For any choice of k sinks we get A361718.
%Y A368602 A058891 counts set-systems, unlabeled A000612.
%Y A368602 A059201 counts covering T_0 set-systems.
%Y A368602 A323818 counts covering connected set-systems, unlabeled A323819.
%Y A368602 Cf. A000169, A003024, A003087, A082402, A088957, A334282, A367862, A367904, A367908, A368600, A368601.
%K A368602 nonn,tabl
%O A368602 0,8
%A A368602 _Gus Wiseman_, Jan 02 2024
%E A368602 More terms from _Alois P. Heinz_, Jan 04 2024