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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. 10

%I

%S 1,2,1,8,6,1,56,44,12,1,552,440,140,20,1,7202,5632,1920,340,30,1,

%T 118456,89278,31192,6160,700,42,1,2369922,1708016,595448,124432,16240,

%U 1288,56,1,56230544,38592786,13214672,2830632,400512,37296,2184,72,1,1552048082

%N 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 The T(3,2) = 6 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},{2},{3}}

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

%e Triangle begins:

%e 1

%e 2 1

%e 8 6 1

%e 56 44 12 1

%e 552 440 140 20 1

%e 7202 5632 1920 340 30 1

%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[Max@@Length/@spmeet@@#==1,Length[csm[Union@@#]]==k]&]],{n,5},{k,n}]

%Y Row sums are A059849. First column is A181939.

%Y Cf. A000110, A000258, A001247, A008277, A048994, A060639, A318390, A318391, A318392, A318393.

%K nonn,tabl

%O 1,2

%A _Gus Wiseman_, Aug 25 2018

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Last modified May 26 17:16 EDT 2020. Contains 334630 sequences. (Running on oeis4.)