A324015 -id:A324015 - cp's OEIS frontend

A124323 Triangle read by rows: T(n,k) is the number of partitions of an n-set having k singleton blocks (0<=k<=n).

1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 4, 4, 6, 0, 1, 11, 20, 10, 10, 0, 1, 41, 66, 60, 20, 15, 0, 1, 162, 287, 231, 140, 35, 21, 0, 1, 715, 1296, 1148, 616, 280, 56, 28, 0, 1, 3425, 6435, 5832, 3444, 1386, 504, 84, 36, 0, 1, 17722, 34250, 32175, 19440, 8610, 2772, 840, 120, 45, 0, 1

Offset: 0

Emeric Deutsch, Oct 28 2006

Row sums are the Bell numbers (A000110). T(n,0)=A000296(n). T(n,k) = binomial(n,k)*T(n-k,0). Sum(k*T(n,k),k=0..n) = A052889(n) = n*B(n-1), where B(q) are the Bell numbers (A000110).
Exponential Riordan array [exp(exp(x)-1-x),x]. - Paul Barry, Apr 23 2009
Sum_{k=0..n} T(n,k)*2^k = A000110(n+1) is the number of binary relations on an n-set that are both symmetric and transitive. - Geoffrey Critzer, Jul 25 2014
Also the number of set partitions of {1, ..., n} with k cyclical adjacencies (successive elements in the same block, where 1 is a successor of n). Unlike A250104, we count {{1}} as having 1 cyclical adjacency. - Gus Wiseman, Feb 13 2019

			T(4,2)=6 because we have 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3 and 1|2|34 (if we take {1,2,3,4} as our 4-set).
Triangle starts:
     1
     0    1
     1    0    1
     1    3    0    1
     4    4    6    0    1
    11   20   10   10    0    1
    41   66   60   20   15    0    1
   162  287  231  140   35   21    0    1
   715 1296 1148  616  280   56   28    0    1
  3425 6435 5832 3444 1386  504   84   36    0    1
From _Gus Wiseman_, Feb 13 2019: (Start)
Row n = 5 counts the following set partitions by number of singletons:
  {{1234}}    {{1}{234}}  {{1}{2}{34}}  {{1}{2}{3}{4}}
  {{12}{34}}  {{123}{4}}  {{1}{23}{4}}
  {{13}{24}}  {{124}{3}}  {{12}{3}{4}}
  {{14}{23}}  {{134}{2}}  {{1}{24}{3}}
                          {{13}{2}{4}}
                          {{14}{2}{3}}
... and the following set partitions by number of cyclical adjacencies:
  {{13}{24}}      {{1}{2}{34}}  {{1}{234}}  {{1234}}
  {{1}{24}{3}}    {{1}{23}{4}}  {{12}{34}}
  {{13}{2}{4}}    {{12}{3}{4}}  {{123}{4}}
  {{1}{2}{3}{4}}  {{14}{2}{3}}  {{124}{3}}
                                {{134}{2}}
                                {{14}{23}}
(End)
From _Paul Barry_, Apr 23 2009: (Start)
Production matrix is
0, 1,
1, 0, 1,
1, 2, 0, 1,
1, 3, 3, 0, 1,
1, 4, 6, 4, 0, 1,
1, 5, 10, 10, 5, 0, 1,
1, 6, 15, 20, 15, 6, 0, 1,
1, 7, 21, 35, 35, 21, 7, 0, 1,
1, 8, 28, 56, 70, 56, 28, 8, 0, 1 (End)

Alois P. Heinz, Rows n = 0..140, flattened
David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.
T. Mansour, A. O. Munagi, Set partitions with circular successions, European Journal of Combinatorics, 42 (2014), 207-216.

Cf. A000110, A052889, A124324.
A250104 is an essentially identical triangle, differing only in row 1.
For columns see A000296, A250105, A250106, A250107.
Cf. A000126, A001610, A032032, A052841, A066982, A080107, A169985, A187784, A324011, A324014, A324015.

G:=exp(exp(z)-1+(t-1)*z): Gser:=simplify(series(G,z=0,14)): for n from 0 to 11 do P[n]:=sort(n!*coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,k),k=0..n) od; # yields sequence in triangular form
# Program from R. J. Mathar, Jan 22 2015:
A124323 := proc(n,k)
    binomial(n,k)*A000296(n-k) ;
end proc:

Flatten[CoefficientList[Range[0,10]! CoefficientList[Series[Exp[x y] Exp[Exp[x] - x - 1], {x, 0,10}], x], y]] (* Geoffrey Critzer, Nov 24 2011 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],Count[#,{}]==k&]],{n,0,9},{k,0,n}] (* _Gus Wiseman, Feb 13 2019 *)

T(n,k) = binomial(n,k)*[(-1)^(n-k)+sum((-1)^(j-1)*B(n-k-j), j=1..n-k)], where B(q) are the Bell numbers (A000110).
E.g.f.: G(t,z) = exp(exp(z)-1+(t-1)*z).
G.f.: 1/(1-xy-x^2/(1-xy-x-2x^2/(1-xy-2x-3x^2/(1-xy-3x-4x^2/(1-... (continued fraction). - Paul Barry, Apr 23 2009

A324011 Number of set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 5, 14, 66, 307, 1554, 8415, 48530, 296582, 1913561, 12988776, 92467629, 688528288, 5349409512, 43270425827, 363680219762, 3170394634443, 28619600156344, 267129951788160, 2574517930001445, 25587989366964056, 261961602231869825

Offset: 0

Gus Wiseman, Feb 12 2019

nonn

These set partitions are fixed points under Callan's bijection phi on set partitions.

			The a(4) = 1, a(6) = 5, and a(7) = 14 set partitions:
  {{13}{24}}  {{135}{246}}    {{13}{246}{57}}
              {{13}{25}{46}}  {{13}{257}{46}}
              {{14}{25}{36}}  {{135}{26}{47}}
              {{14}{26}{35}}  {{135}{27}{46}}
              {{15}{24}{36}}  {{136}{24}{57}}
                              {{136}{25}{47}}
                              {{14}{257}{36}}
                              {{14}{26}{357}}
                              {{146}{25}{37}}
                              {{146}{27}{35}}
                              {{15}{246}{37}}
                              {{15}{247}{36}}
                              {{16}{24}{357}}
                              {{16}{247}{35}}

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

Cf. A000110, A000126, A000296 (singletons allowed, or adjacencies allowed), A001610, A124323, A169985, A261139, A324012, A324014, A324015.

Table[Select[sps[Range[n]],And[Count[#,{_}]==0,Total[If[First[#]==1&&Last[#]==n,1,0]+Count[Subtract@@@Partition[#,2,1],-1]&/@#]==0]&]//Length,{n,0,10}]

a(11)-a(26) from Alois P. Heinz, Feb 12 2019

A324012 Number of self-complementary set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 3, 2, 14, 11, 80, 85, 510

Offset: 0

Gus Wiseman, Feb 12 2019

The complement of a set partition pi of {1, ..., n} is defined as n + 1 - pi (elementwise) on page 3 of Callan. For example, the complement of {{1,5},{2},{3,6},{4}} is {{1,4},{2,6},{3},{5}}. This sequence counts certain self-conjugate set partitions, i.e., fixed points under Callan's conjugation operation.

			The  a(6) = 3 through a(9) = 11 self-complementary set partitions with no singletons or cyclical adjacencies:
  {{135}{246}}    {{13}{246}{57}}  {{1357}{2468}}      {{136}{258}{479}}
  {{13}{25}{46}}  {{15}{246}{37}}  {{135}{27}{468}}    {{147}{258}{369}}
  {{14}{25}{36}}                   {{146}{27}{358}}    {{148}{269}{357}}
                                   {{147}{258}{36}}    {{168}{249}{357}}
                                   {{157}{248}{36}}    {{13}{258}{46}{79}}
                                   {{13}{24}{57}{68}}  {{14}{258}{37}{69}}
                                   {{13}{25}{47}{68}}  {{14}{28}{357}{69}}
                                   {{14}{26}{37}{58}}  {{16}{258}{37}{49}}
                                   {{14}{27}{36}{58}}  {{16}{28}{357}{49}}
                                   {{15}{26}{37}{48}}  {{17}{258}{39}{46}}
                                   {{15}{27}{36}{48}}  {{18}{29}{357}{46}}
                                   {{16}{24}{38}{57}}
                                   {{16}{25}{38}{47}}
                                   {{17}{28}{35}{46}}

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

Cf. A000110, A000126, A000296, A001610, A080107, A169985, A261139, A306417 (all self-conjugate set partitions), A324011 (self-complementarity not required), A324013 (adjacencies allowed), A324014 (singletons allowed), A324015.

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

A324014 Number of self-complementary set partitions of {1, ..., n} with no cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 9, 16, 43, 89, 250, 571, 1639

Offset: 0

Gus Wiseman, Feb 12 2019

The complement of a set partition pi of {1, ..., n} is defined as n + 1 - pi (elementwise) on page 3 of Callan. For example, the complement of {{1,5},{2},{3,6},{4}} is {{1,4},{2,6},{3},{5}}.

			The  a(3) = 1 through a(6) = 9 self-complementary set partitions with no cyclical adjacencies:
  {{1}{2}{3}}  {{13}{24}}      {{14}{25}{3}}      {{135}{246}}
               {{1}{2}{3}{4}}  {{1}{24}{3}{5}}    {{13}{25}{46}}
                               {{1}{2}{3}{4}{5}}  {{14}{25}{36}}
                                                  {{1}{24}{35}{6}}
                                                  {{13}{2}{46}{5}}
                                                  {{14}{2}{36}{5}}
                                                  {{15}{26}{3}{4}}
                                                  {{1}{25}{3}{4}{6}}
                                                  {{1}{2}{3}{4}{5}{6}}

David Callan, On conjugates for set partitions and integer compositions [math.CO].

Cf. A000110, A000296, A001610, A080107 (self-complementary), A169985, A324012 (self-conjugate), A324015.

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

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).

Original entry on oeis.org

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

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A124323 Triangle read by rows: T(n,k) is the number of partitions of an n-set having k singleton blocks (0<=k<=n).

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A324011 Number of 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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A324012 Number of self-complementary 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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A324014 Number of self-complementary set partitions of {1, ..., n} with no cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

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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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