A324011 -id:A324011 - cp's OEIS frontend

A306416 Number of ordered set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

1, 0, 0, 0, 2, 0, 26, 84, 950, 6000, 62522, 556116, 6259598, 69319848, 874356338, 11384093196, 161462123894, 2397736692144, 37994808171962, 631767062124564, 11088109048500158, 203828700127054008, 3928762035148317314, 79079452776283889820, 1661265965479375937030, 36332908076071038467520, 826376466514358722894154

Offset: 0

Gus Wiseman, Feb 14 2019

nonn

			The a(4) = 2 ordered set partitions are: {{1,3},{2,4}}, {{2,4},{1,3}}.

David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.

Cf. A000110, A000126, A000296, A000670, A001610, A032032 (adjacencies allowed), A052841 (singletons allowed), A124323, A169985, A306417, A324011 (orderless case), A324012, A324015.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Length[stn]!,{stn,Select[sps[Range[n]],And[Count[#,{_}]==0,Total[If[First[#]==1&&Last[#]==n,1,0]+Count[Subtract@@@Partition[#,2,1],-1]&/@#]==0]&]}],{n,0,10}]

a(12)-a(26) from Alois P. Heinz, Feb 14 2019

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.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Feb 14 2019

See Callan's article for details on this transformation (SeparateIS).

			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

David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.

Row sums are A000110. First column is A324011.

Cf. A000126, A000296, A001610, A306416, A306417, A324012.

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}]

cp's OEIS Frontend

A306416 Number of ordered set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

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