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A306418
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Regular triangle read by rows where T(n, k) is the number of set partitions of {1, ..., n} requiring k steps of removing singletons and cyclical adjacency initiators until reaching a fixed point, n >= 0, 0 <= k <= n.
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2
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1, 0, 1, 0, 2, 0, 0, 2, 3, 0, 1, 2, 12, 0, 0, 0, 12, 35, 5, 0, 0, 5, 56, 100, 42, 0, 0, 0, 14, 282, 343, 231, 7, 0, 0, 0, 66, 1406, 1476, 1088, 104, 0, 0, 0, 0, 307, 7592, 7383, 4929, 909, 27, 0, 0, 0, 0, 1554, 44227, 40514, 22950, 6240, 470, 20, 0, 0, 0, 0
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OFFSET
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0,5
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COMMENTS
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See Callan's article for details on this transformation (SeparateIS).
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LINKS
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EXAMPLE
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Triangle begins:
1
0 1
0 2 0
0 2 3 0
1 2 12 0 0
0 12 35 5 0 0
5 56 100 42 0 0 0
14 282 343 231 7 0 0 0
66 1406 1476 1088 104 0 0 0 0
307 7592 7383 4929 909 27 0 0 0 0
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MATHEMATICA
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sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
qbj[stn_]:=With[{ini=Join@@Table[Select[s, If[#==Max@@Max@@@stn, MemberQ[s, First[Union@@stn]], MemberQ[s, (Union@@stn)[[Position[Union@@stn, #][[1, 1]]+1]]]]&], {s, stn}], sng=Join@@Select[stn, Length[#]==1&]}, DeleteCases[Table[Complement[s, Union[sng, ini]], {s, stn}], {}]];
Table[Length[Select[sps[Range[n]], Length[FixedPointList[qbj, #]]-2==k&]], {n, 0, 8}, {k, 0, n}]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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