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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.
10
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
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.
Sequence in context: A114193 A231846 A039683 * A108085 A108084 A305860
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Aug 25 2018
STATUS
approved