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A306418 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.

%I A306418 #7 Feb 14 2019 20:22:25
%S A306418 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,
%T A306418 282,343,231,7,0,0,0,66,1406,1476,1088,104,0,0,0,0,307,7592,7383,4929,
%U A306418 909,27,0,0,0,0,1554,44227,40514,22950,6240,470,20,0,0,0,0
%N A306418 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.
%C A306418 See Callan's article for details on this transformation (SeparateIS).
%H A306418 David Callan, <a href="https://arxiv.org/abs/math/0508052">On conjugates for set partitions and integer compositions</a>, arXiv:math/0508052 [math.CO], 2005.
%e A306418 Triangle begins:
%e A306418     1
%e A306418     0    1
%e A306418     0    2    0
%e A306418     0    2    3    0
%e A306418     1    2   12    0    0
%e A306418     0   12   35    5    0    0
%e A306418     5   56  100   42    0    0    0
%e A306418    14  282  343  231    7    0    0    0
%e A306418    66 1406 1476 1088  104    0    0    0    0
%e A306418   307 7592 7383 4929  909   27    0    0    0    0
%t A306418 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A306418 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}],{}]];
%t A306418 Table[Length[Select[sps[Range[n]],Length[FixedPointList[qbj,#]]-2==k&]],{n,0,8},{k,0,n}]
%Y A306418 Row sums are A000110. First column is A324011.
%Y A306418 Cf. A000126, A000296, A001610, A306416, A306417, A324012.
%K A306418 nonn,tabl
%O A306418 0,5
%A A306418 _Gus Wiseman_, Feb 14 2019