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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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS 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: A243320 A319897 A193734 * A319511 A110608 A318397 Adjacent sequences:  A318387 A318388 A318389 * A318391 A318392 A318393 KEYWORD nonn,tabl AUTHOR Gus Wiseman, Aug 25 2018 STATUS approved

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Last modified January 19 21:47 EST 2020. Contains 331066 sequences. (Running on oeis4.)