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.

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

0,5

Comments

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

Links

Table of n, a(n) for n=0..65.

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

Example

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

Mathematica

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}]

Crossrefs

Row sums are A000110. First column is A324011.

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

Sequence in context: A263138 A274637 A178516 * A323886 A340829 A352551

Adjacent sequences: A306415 A306416 A306417 * A306419 A306420 A306421

Keyword

nonn,tabl

Author

Gus Wiseman, Feb 14 2019

Status

approved