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 A324323 Regular triangle read by rows where T(n,k) is the number of topologically connected set partitions of {1,...,n} with k blocks, 0 <= k <= n. 5
 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 5, 0, 0, 0, 0, 1, 16, 4, 0, 0, 0, 0, 1, 42, 42, 0, 0, 0, 0, 0, 1, 99, 258, 27, 0, 0, 0, 0, 0, 1, 219, 1222, 465, 0, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,18 COMMENTS A set partition of {1,...,n} is topologically connected if the graph whose vertices are the blocks and whose edges are crossing pairs of blocks is connected, where two blocks cross each other if they are of the form {{...x...y...},{...z...t...}} for some x < z < y < t or z < x < t < y. LINKS EXAMPLE Triangle begins:     1     0    1     0    1    0     0    1    0    0     0    1    1    0    0     0    1    5    0    0    0     0    1   16    4    0    0    0     0    1   42   42    0    0    0    0     0    1   99  258   27    0    0    0    0     0    1  219 1222  465    0    0    0    0    0 Row n = 6 counts the following set partitions:   {{123456}}  {{1235}{46}}  {{13}{25}{46}}               {{124}{356}}  {{14}{25}{36}}               {{1245}{36}}  {{14}{26}{35}}               {{1246}{35}}  {{15}{24}{36}}               {{125}{346}}               {{13}{2456}}               {{134}{256}}               {{1345}{26}}               {{1346}{25}}               {{135}{246}}               {{1356}{24}}               {{136}{245}}               {{14}{2356}}               {{145}{236}}               {{146}{235}}               {{15}{2346}} MATHEMATICA croXQ[stn_]:=MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x0]&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]]; crosscmpts[stn_]:=csm[Union[Subsets[stn, {1}], Select[Subsets[stn, {2}], croXQ]]]; sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}]; Table[Length[Select[sps[Range[n]], Length[crosscmpts[#]]<=1&&Length[#]==k&]], {n, 0, 6}, {k, 0, n}] CROSSREFS Row sums are A099947. Row k = 2 is A002662. Cf. A000108, A000110, A001263, A016098, A136653, A268814, A268815, A306438, A324011. Cf. A324166, A324172, A324173, A324327, A324328. Sequence in context: A088194 A048894 A047766 * A005078 A284446 A270030 Adjacent sequences:  A324320 A324321 A324322 * A324324 A324325 A324326 KEYWORD nonn,tabl,more AUTHOR Gus Wiseman, Feb 22 2019 STATUS approved

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Last modified February 18 04:48 EST 2020. Contains 332011 sequences. (Running on oeis4.)