A318393 Regular tetrangle where T(n,k,i) is the number of pairs of set partitions of {1,...,n} with meet of length k and join of length i.

1, 1, 2, 1, 1, 6, 3, 8, 6, 1, 1, 14, 7, 48, 36, 6, 56, 44, 12, 1, 1, 30, 15, 200, 150, 25, 560, 440, 120, 10, 552, 440, 140, 20, 1, 1, 62, 31, 720, 540, 90, 3640, 2860, 780, 65, 8280, 6600, 2100, 300, 15, 7202, 5632, 1920, 340, 30, 1, 1, 126, 63, 2408, 1806

Offset: 1

Author

Gus Wiseman, Aug 25 2018

Keywords

• nonn
• tabf

Examples

			The T(3,3,1) = 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}}
Tetrangle begins:
   1   1     1       1            1
       2 1   6 3     14 7         30  15
             8 6 1   48 36 6      200 150 25
                     56 44 12 1   560 440 120 10
                                  552 440 140 20  1
		

Crossrefs

Cf. A000110, A000258, A001247, A008277, A048994, A059849, A060639, A181939, A318389, A318390, A318391, A318392.

Programs

Mathematica

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]]]]]]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
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@@#]]==j]&]],{n,6},{k,n},{j,k}]