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A318390 Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with join {{1,...,n}} and meet of length k.

%I A318390 #6 Aug 26 2018 20:11:46
%S A318390 1,1,2,1,6,8,1,14,48,56,1,30,200,560,552,1,62,720,3640,8280,7202,1,
%T A318390 126,2408,19600,77280,151242,118456,1,254,7728,95256,579600,1915732,
%U A318390 3316768,2369922,1,510,24200,435120,3836952,19056492,54726672,85317192,56230544,1
%N A318390 Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with join {{1,...,n}} and meet of length k.
%F A318390 T(n,k) = S(n,k) * A181939(k) where S = A008277.
%e A318390 The T(3,3) = 8 pairs of set partitions:
%e A318390   {{1},{2},{3}}  {{1,2,3}}
%e A318390    {{1},{2,3}}  {{1,2},{3}}
%e A318390    {{1},{2,3}}  {{1,3},{2}}
%e A318390    {{1,2},{3}}  {{1},{2,3}}
%e A318390    {{1,2},{3}}  {{1,3},{2}}
%e A318390    {{1,3},{2}}  {{1},{2,3}}
%e A318390    {{1,3},{2}}  {{1,2},{3}}
%e A318390     {{1,2,3}}  {{1},{2},{3}}
%e A318390 Triangle begins:
%e A318390     1
%e A318390     1    2
%e A318390     1    6    8
%e A318390     1   14   48   56
%e A318390     1   30  200  560  552
%e A318390     1   62  720 3640 8280 7202
%t A318390 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A318390 csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A318390 spmeet[a_,b_]:=DeleteCases[Union@@Outer[Intersection,a,b,1],{}];spmeet[a_,b_,c__]:=spmeet[spmeet[a,b],c];
%t A318390 Table[Length[Select[Tuples[sps[Range[n]],2],And[Length[spmeet@@#]==k,Length[csm[Union@@#]]==1]&]],{n,6},{k,n}]
%Y A318390 Row sums are A060639. Last column is A181939.
%Y A318390 Cf. A000110, A000258, A001247, A008277, A048994, A059849, A318389, A318391, A318392, A318393.
%K A318390 nonn,tabl
%O A318390 1,3
%A A318390 _Gus Wiseman_, Aug 25 2018