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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.
8
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
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.
Sequence in context: A350009 A350010 A193734 * A319511 A110608 A318397
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Aug 25 2018
STATUS
approved