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A318389 Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with meet {{1},...,{n}} and join of length k.

%I A318389 #5 Aug 26 2018 20:11:59
%S A318389 1,2,1,8,6,1,56,44,12,1,552,440,140,20,1,7202,5632,1920,340,30,1,
%T A318389 118456,89278,31192,6160,700,42,1,2369922,1708016,595448,124432,16240,
%U A318389 1288,56,1,56230544,38592786,13214672,2830632,400512,37296,2184,72,1,1552048082
%N A318389 Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with meet {{1},...,{n}} and join of length k.
%e A318389 The T(3,2) = 6 pairs of set partitions:
%e A318389   {{1},{2},{3}}  {{1},{2,3}}
%e A318389   {{1},{2},{3}}  {{1,2},{3}}
%e A318389   {{1},{2},{3}}  {{1,3},{2}}
%e A318389    {{1},{2,3}}  {{1},{2},{3}}
%e A318389    {{1,2},{3}}  {{1},{2},{3}}
%e A318389    {{1,3},{2}}  {{1},{2},{3}}
%e A318389 Triangle begins:
%e A318389      1
%e A318389      2    1
%e A318389      8    6    1
%e A318389     56   44   12    1
%e A318389    552  440  140   20    1
%e A318389   7202 5632 1920  340   30    1
%t A318389 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A318389 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 A318389 spmeet[a_,b_]:=DeleteCases[Union@@Outer[Intersection,a,b,1],{}];spmeet[a_,b_,c__]:=spmeet[spmeet[a,b],c];
%t A318389 Table[Length[Select[Tuples[sps[Range[n]],2],And[Max@@Length/@spmeet@@#==1,Length[csm[Union@@#]]==k]&]],{n,5},{k,n}]
%Y A318389 Row sums are A059849. First column is A181939.
%Y A318389 Cf. A000110, A000258, A001247, A008277, A048994, A060639, A318390, A318391, A318392, A318393.
%K A318389 nonn,tabl
%O A318389 1,2
%A A318389 _Gus Wiseman_, Aug 25 2018