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.

1, 2, 1, 8, 6, 1, 56, 44, 12, 1, 552, 440, 140, 20, 1, 7202, 5632, 1920, 340, 30, 1, 118456, 89278, 31192, 6160, 700, 42, 1, 2369922, 1708016, 595448, 124432, 16240, 1288, 56, 1, 56230544, 38592786, 13214672, 2830632, 400512, 37296, 2184, 72, 1, 1552048082

Offset

1,2

Links

Table of n, a(n) for n=1..46.

Example

The T(3,2) = 6 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},{2},{3}}
   {{1,3},{2}}  {{1},{2},{3}}
Triangle begins:
     1
     2    1
     8    6    1
    56   44   12    1
   552  440  140   20    1
  7202 5632 1920  340   30    1

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

Crossrefs

Row sums are A059849. First column is A181939.

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

Sequence in context: A114193 A231846 A039683 * A108085 A108084 A305860

Adjacent sequences: A318386 A318387 A318388 * A318390 A318391 A318392

Keyword

nonn,tabl

Author

Gus Wiseman, Aug 25 2018

Status

approved