%I #6 Aug 26 2018 20:11:46

%S 1,1,2,1,6,8,1,14,48,56,1,30,200,560,552,1,62,720,3640,8280,7202,1,

%T 126,2408,19600,77280,151242,118456,1,254,7728,95256,579600,1915732,

%U 3316768,2369922,1,510,24200,435120,3836952,19056492,54726672,85317192,56230544,1

%N 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 T(n,k) = S(n,k) * A181939(k) where S = A008277.

%e The T(3,3) = 8 pairs of set partitions:

%e {{1},{2},{3}} {{1,2,3}}

%e {{1},{2,3}} {{1,2},{3}}

%e {{1},{2,3}} {{1,3},{2}}

%e {{1,2},{3}} {{1},{2,3}}

%e {{1,2},{3}} {{1,3},{2}}

%e {{1,3},{2}} {{1},{2,3}}

%e {{1,3},{2}} {{1,2},{3}}

%e {{1,2,3}} {{1},{2},{3}}

%e Triangle begins:

%e 1

%e 1 2

%e 1 6 8

%e 1 14 48 56

%e 1 30 200 560 552

%e 1 62 720 3640 8280 7202

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t 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 spmeet[a_,b_]:=DeleteCases[Union@@Outer[Intersection,a,b,1],{}];spmeet[a_,b_,c__]:=spmeet[spmeet[a,b],c];

%t Table[Length[Select[Tuples[sps[Range[n]],2],And[Length[spmeet@@#]==k,Length[csm[Union@@#]]==1]&]],{n,6},{k,n}]

%Y Row sums are A060639. Last column is A181939.

%Y Cf. A000110, A000258, A001247, A008277, A048994, A059849, A318389, A318391, A318392, A318393.

%K nonn,tabl

%O 1,3

%A _Gus Wiseman_, Aug 25 2018