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A361718 Triangular array read by rows. T(n,k) is the number of labeled directed acyclic graphs on [n] with exactly k nodes of indegree 0.

%I A361718 #28 Jan 04 2024 18:11:12
%S A361718 1,0,1,0,2,1,0,15,9,1,0,316,198,28,1,0,16885,10710,1610,75,1,0,
%T A361718 2174586,1384335,211820,10575,186,1,0,654313415,416990763,64144675,
%U A361718 3268125,61845,441,1,0,450179768312,286992935964,44218682312,2266772550,43832264,336924,1016,1
%N A361718 Triangular array read by rows. T(n,k) is the number of labeled directed acyclic graphs on [n] with exactly k nodes of indegree 0.
%C A361718 Also the number of sets of n nonempty subsets of {1..n}, k of which are singletons, such that there is only one way to choose a different element from each. For example, row n = 3 counts the following set-systems:
%C A361718   {{1},{1,2},{1,3}}    {{1},{2},{1,3}}    {{1},{2},{3}}
%C A361718   {{1},{1,2},{2,3}}    {{1},{2},{2,3}}
%C A361718   {{1},{1,3},{2,3}}    {{1},{3},{1,2}}
%C A361718   {{2},{1,2},{1,3}}    {{1},{3},{2,3}}
%C A361718   {{2},{1,2},{2,3}}    {{2},{3},{1,2}}
%C A361718   {{2},{1,3},{2,3}}    {{2},{3},{1,3}}
%C A361718   {{3},{1,2},{1,3}}    {{1},{2},{1,2,3}}
%C A361718   {{3},{1,2},{2,3}}    {{1},{3},{1,2,3}}
%C A361718   {{3},{1,3},{2,3}}    {{2},{3},{1,2,3}}
%C A361718   {{1},{1,2},{1,2,3}}
%C A361718   {{1},{1,3},{1,2,3}}
%C A361718   {{2},{1,2},{1,2,3}}
%C A361718   {{2},{2,3},{1,2,3}}
%C A361718   {{3},{1,3},{1,2,3}}
%C A361718   {{3},{2,3},{1,2,3}}
%H A361718 E. de Panafieu and S. Dovgal, <a href="https://arxiv.org/abs/1903.09454">Symbolic method and directed graph enumeration</a>, arXiv:1903.09454 [math.CO], 2019.
%H A361718 R. W. Robinson, <a href="http://cobweb.cs.uga.edu/~rwr/publications/components.pdf">Counting digraphs with restrictions on the strong components</a>, Combinatorics and Graph Theory '95 (T.-H. Ku, ed.), World Scientific, Singapore (1995), 343-354.
%F A361718 T(n,k) = A368602(n,k) * binomial(n,k). - _Gus Wiseman_, Jan 03 2024
%e A361718 Triangle begins:
%e A361718   1;
%e A361718   0,     1;
%e A361718   0,     2,     1;
%e A361718   0,    15,     9,    1;
%e A361718   0,   316,   198,   28,  1;
%e A361718   0, 16885, 10710, 1610, 75, 1;
%e A361718   ...
%t A361718 nn = 8; B[n_] := n! 2^Binomial[n, 2] ;ggf[egf_] := Normal[Series[egf, {z, 0, nn}]] /. Table[z^i -> z^i/2^Binomial[i, 2], {i, 0, nn}];Table[Take[(Table[B[n], {n, 0, nn}] CoefficientList[ Series[ggf[Exp[(u - 1) z]]/ggf[Exp[-z]], {z, 0, nn}], {z, u}])[[i]], i], {i, 1, nn + 1}] // Grid
%t A361718 nv=4;Table[Length[Select[Subsets[Subsets[Range[n]],{n}], Count[#,{_}]==k&&Length[Select[Tuples[#], UnsameQ@@#&]]==1&]],{n,0,nv},{k,0,n}]
%Y A361718 Cf. A058876 (mirror), A361579, A224069.
%Y A361718 Row-sums are A003024, unlabeled A003087.
%Y A361718 Column k = 1 is A003025(n) = |n*A134531(n)|.
%Y A361718 Column k = n-1 is A058877.
%Y A361718 For fixed sinks we get A368602.
%Y A361718 A058891 counts set-systems, unlabeled A000612.
%Y A361718 A323818 counts covering connected set-systems, unlabeled A323819.
%Y A361718 Cf. A000169, A059201, A082402, A088957, A133686, A334282, A350415, A367904, A367908, A368600, A368601.
%K A361718 nonn,tabl
%O A361718 0,5
%A A361718 _Geoffrey Critzer_, Apr 02 2023