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

1,3

Links

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

Formula

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

Example

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

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}]

Crossrefs

Row sums are A060639. Last column is A181939.

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

Sequence in context: A350009 A350010 A193734 * A319511 A110608 A318397

Adjacent sequences: A318387 A318388 A318389 * A318391 A318392 A318393

Keyword

nonn,tabl

Author

Gus Wiseman, Aug 25 2018

Status

approved