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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.
20
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
1,3
EXAMPLE
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
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}]
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Aug 25 2018
STATUS
approved