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.

1, 1, 2, 1, 6, 8, 1, 14, 48, 56, 1, 30, 200, 560, 552, 1, 62, 720, 3640, 8280, 7202, 1, 126, 2408, 19600, 77280, 151242, 118456, 1, 254, 7728, 95256, 579600, 1915732, 3316768, 2369922, 1, 510, 24200, 435120, 3836952, 19056492, 54726672, 85317192, 56230544, 1

Offset: 1

Author

Gus Wiseman, Aug 25 2018

Keywords

• nonn
• tabl

Examples

			The T(3,3) = 8 pairs of set partitions:
  {{1},{2},{3}}  {{1,2,3}}
   {{1},{2,3}}  {{1,2},{3}}
   {{1},{2,3}}  {{1,3},{2}}
   {{1,2},{3}}  {{1},{2,3}}
   {{1,2},{3}}  {{1,3},{2}}
   {{1,3},{2}}  {{1},{2,3}}
   {{1,3},{2}}  {{1,2},{3}}
    {{1,2,3}}  {{1},{2},{3}}
Triangle begins:
    1
    1    2
    1    6    8
    1   14   48   56
    1   30  200  560  552
    1   62  720 3640 8280 7202
		

Crossrefs

Row sums are A060639. Last column is A181939.

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

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
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]]]]]]]]];
spmeet[a_,b_]:=DeleteCases[Union@@Outer[Intersection,a,b,1],{}];spmeet[a_,b_,c__]:=spmeet[spmeet[a,b],c];
Table[Length[Select[Tuples[sps[Range[n]],2],And[Length[spmeet@@#]==k,Length[csm[Union@@#]]==1]&]],{n,6},{k,n}]

Formula

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