A000126 -id:A000126 - cp's OEIS frontend

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).

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}]

A306357 Number of nonempty subsets of {1, ..., n} containing no three cyclically successive elements.

Original entry on oeis.org

0, 1, 3, 6, 10, 20, 38, 70, 130, 240, 442, 814, 1498, 2756, 5070, 9326, 17154, 31552, 58034, 106742, 196330, 361108, 664182, 1221622, 2246914, 4132720, 7601258, 13980894, 25714874, 47297028, 86992798, 160004702, 294294530, 541292032, 995591266, 1831177830

Offset: 0

Gus Wiseman, Feb 10 2019

nonn

Cyclically successive means 1 is a successor of n.

Set partitions using these subsets are counted by A323949.

			The a(1) = 1 through a(5) = 20 stable subsets:
  {1}  {1}    {1}    {1}    {1}
       {2}    {2}    {2}    {2}
       {1,2}  {3}    {3}    {3}
              {1,2}  {4}    {4}
              {1,3}  {1,2}  {5}
              {2,3}  {1,3}  {1,2}
                     {1,4}  {1,3}
                     {2,3}  {1,4}
                     {2,4}  {1,5}
                     {3,4}  {2,3}
                            {2,4}
                            {2,5}
                            {3,4}
                            {3,5}
                            {4,5}
                            {1,2,4}
                            {1,3,4}
                            {1,3,5}
                            {2,3,5}
                            {2,4,5}

Cf. A000126, A000296, A001610, A001644, A169985, A306357, A323949, A323952, A323955.

stabsubs[g_]:=Select[Rest[Subsets[Union@@g]],Select[g,Function[ed,UnsameQ@@ed&&Complement[ed,#]=={}]]=={}&];
Table[Length[stabsubs[Partition[Range[n],3,1,1]]],{n,15}]

For n >= 3 we have a(n) = A001644(n) - 1.
From Chai Wah Wu, Jan 06 2020: (Start)
a(n) = 2*a(n-1) - a(n-4) for n > 6.
G.f.: x*(x^5 + x^4 - 2*x^3 + x + 1)/(x^4 - 2*x + 1). (End)

A324015 Number of nonempty subsets of {1, ..., n} containing no two cyclically successive elements.

Original entry on oeis.org

0, 1, 2, 3, 6, 10, 17, 28, 46, 75, 122, 198, 321, 520, 842, 1363, 2206, 3570, 5777, 9348, 15126, 24475, 39602, 64078, 103681, 167760, 271442, 439203, 710646, 1149850, 1860497, 3010348, 4870846, 7881195, 12752042, 20633238, 33385281, 54018520, 87403802

Offset: 0

Gus Wiseman, Feb 12 2019

nonn

Cyclically successive means 1 succeeds n.

After a(1) = 1, same as A001610 shifted once to the right. Also, a(n) = A169985(n) - 1.

			The a(6) = 17 stable subsets:
  {1}, {2}, {3}, {4}, {5}, {6},
  {1,3}, {1,4}, {1,5}, {2,4}, {2,5}, {2,6}, {3,5}, {3,6}, {4,6},
  {1,3,5}, {2,4,6}.

Cf. A000045, A000126, A000296, A001610, A001644, A169985, A306357, A324014.

stabsubs[g_]:=Select[Rest[Subsets[Union@@g]],Select[g,Function[ed,UnsameQ@@ed&&Complement[ed,#]=={}]]=={}&];
Table[Length[stabsubs[Partition[Range[n],2,1,1]]],{n,0,10}]

For n <= 3, a(n) = n. Otherwise, a(n) = a(n - 1) + a(n - 2) + 1.

A111644 Expansion of -(1+x^2)/((x^2+4x+1)(x^2-2*x-1)).

Original entry on oeis.org

1, -6, 29, -124, 501, -1962, 7545, -28696, 108393, -407662, 1528981, -5724500, 21408221, -80003026, 298832369, -1115878064, 4166011601, -15551383382, 58047283725, -216656490156, 808623915973, -3017948390522, 11263433318761, -42036421446600, 156883789264441

Offset: 0

Creighton Dement, Aug 10 2005

In reference to the program code, the sequence of Pell numbers A000126 is given by 1kbaseseq[C*J]. A001353 is 1ibaseiseq[C*J].

Floretion Algebra Multiplication Program, FAMP Code: 1basejseq[C*J] with C = - 'j + 'k - j' + k' - 'ii' - 'ij' - 'ik' - 'ji' - 'ki' and J = + j' + k' + 1.5'ii' + .5'jj' + .5'kk' + .5e

Index entries for linear recurrences with constant coefficients, signature (-6,-8,2,1).

Cf. A111639, A111640, A111641, A111642, A111643, A111645, A000126.

CoefficientList[Series[-(1+x^2)/((x^2+4x+1)(x^2-2x-1)),{x,0,40}],x] (* or *) LinearRecurrence[{-6,-8,2,1},{1,-6,29,-124},40] (* Harvey P. Dale, May 23 2015 *)

a(0)=1, a(1)=-6, a(2)=29, a(3)=-124, a(n)=-6*a(n-1)-8*a(n-2)+ 2*a(n-3)+ a(n-4). - Harvey P. Dale, May 23 2015

A323949 Number of set partitions of {1, ..., n} with no block containing three distinct cyclically successive vertices.

Original entry on oeis.org

1, 1, 2, 4, 10, 36, 145, 631, 3015, 15563, 86144, 508311, 3180930, 21018999, 146111543, 1065040886, 8117566366, 64531949885, 533880211566, 4587373155544, 40865048111424, 376788283806743, 3590485953393739, 35312436594162173, 357995171351223109, 3736806713651177702

Offset: 0

Gus Wiseman, Feb 10 2019

nonn

Cyclically successive means 1 is a successor of n.

			The a(1) = 1 through a(4) = 10 set partitions:
  {{1}}  {{1,2}}    {{1},{2,3}}    {{1,2},{3,4}}
         {{1},{2}}  {{1,2},{3}}    {{1,3},{2,4}}
                    {{1,3},{2}}    {{1,4},{2,3}}
                    {{1},{2},{3}}  {{1},{2},{3,4}}
                                   {{1},{2,3},{4}}
                                   {{1,2},{3},{4}}
                                   {{1},{2,4},{3}}
                                   {{1,3},{2},{4}}
                                   {{1,4},{2},{3}}
                                   {{1},{2},{3},{4}}

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A000085, A000110, A000126, A000296, A000325, A001610, A001644, A078012, A169985.

Cf. A306351, A306357, A323950, A323951, A323953, A323954, A323955, A323956.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],Select[Partition[Range[n],3,1,1],Function[ed,UnsameQ@@ed&&Complement[ed,#]=={}]]=={}&],Range[n]]],{n,8}]

a(12)-a(25) from Alois P. Heinz, Feb 10 2019

A323955 Regular triangle read by rows where T(n, k) is the number of set partitions of {1, ..., n} with no block containing k cyclically successive vertices, n >= 1, 2 <= k <= n + 1.

Original entry on oeis.org

1, 1, 2, 1, 4, 5, 4, 10, 14, 15, 11, 36, 46, 51, 52, 41, 145, 184, 196, 202, 203, 162, 631, 806, 855, 869, 876, 877, 715, 3015, 3847, 4059, 4115, 4131, 4139, 4140, 3425, 15563, 19805, 20813, 21056, 21119, 21137, 21146, 21147, 17722, 86144, 109339, 114469

Offset: 1

Gus Wiseman, Feb 10 2019

Cyclically successive means 1 is a successor of n.

			Triangle begins:
    1
    1    2
    1    4    5
    4   10   14   15
   11   36   46   51   52
   41  145  184  196  202  203
  162  631  806  855  869  876  877
  715 3015 3847 4059 4115 4131 4139 4140
Row 4 counts the following partitions:
  {{13}{24}}      {{12}{34}}      {{1}{234}}      {{1234}}
  {{1}{24}{3}}    {{13}{24}}      {{12}{34}}      {{1}{234}}
  {{13}{2}{4}}    {{14}{23}}      {{123}{4}}      {{12}{34}}
  {{1}{2}{3}{4}}  {{1}{2}{34}}    {{124}{3}}      {{123}{4}}
                  {{1}{23}{4}}    {{13}{24}}      {{124}{3}}
                  {{12}{3}{4}}    {{134}{2}}      {{13}{24}}
                  {{1}{24}{3}}    {{14}{23}}      {{134}{2}}
                  {{13}{2}{4}}    {{1}{2}{34}}    {{14}{23}}
                  {{14}{2}{3}}    {{1}{23}{4}}    {{1}{2}{34}}
                  {{1}{2}{3}{4}}  {{12}{3}{4}}    {{1}{23}{4}}
                                  {{1}{24}{3}}    {{12}{3}{4}}
                                  {{13}{2}{4}}    {{1}{24}{3}}
                                  {{14}{2}{3}}    {{13}{2}{4}}
                                  {{1}{2}{3}{4}}  {{14}{2}{3}}
                                                  {{1}{2}{3}{4}}

First column (k = 2) is A000296. Second column (k = 3) is A323949. Rightmost terms are A000110. Second to rightmost terms are A058692.

Cf. A000126, A000325, A001610, A001644, A169985, A306351, A306357, A323950, A323954.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],Select[Partition[Range[n],k,1,1],Function[ed,UnsameQ@@ed&&Complement[ed,#]=={}]]=={}&],Range[n]]],{n,7},{k,2,n+1}]

A323956 Triangle read by rows: T(n, k) = 1 + n * (n - k) for 1 <= k <= n.

Original entry on oeis.org

1, 3, 1, 7, 4, 1, 13, 9, 5, 1, 21, 16, 11, 6, 1, 31, 25, 19, 13, 7, 1, 43, 36, 29, 22, 15, 8, 1, 57, 49, 41, 33, 25, 17, 9, 1, 73, 64, 55, 46, 37, 28, 19, 10, 1, 91, 81, 71, 61, 51, 41, 31, 21, 11, 1, 111, 100, 89, 78, 67, 56, 45, 34, 23, 12, 1

Offset: 1

Gus Wiseman, Feb 10 2019

			Triangle begins:
  n\k:   1   2   3   4   5   6   7   8   9  10  11  12
  ====================================================
    1:   1
    2:   3   1
    3:   7   4   1
    4:  13   9   5   1
    5:  21  16  11   6   1
    6:  31  25  19  13   7   1
    7:  43  36  29  22  15   8   1
    8:  57  49  41  33  25  17   9   1
    9:  73  64  55  46  37  28  19  10   1
   10:  91  81  71  61  51  41  31  21  11   1
   11: 111 100  89  78  67  56  45  34  23  12   1
   12: 133 121 109  97  85  73  61  49  37  25  13   1
  etc.

G. C. Greubel, Rows n = 1..100 of triangle, flattened

First column is A002061. Second column is A000290. Third column is A028387.

Cf. A000126, A001610, A001644, A006000, A169985, A306351, A323952, A323955.

[[1+n*(n-k): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Apr 22 2019

Mathematica
```
Table[1+n*(n-k),{n,12},{k,n}]//Flatten
```

{T(n,k) = 1+n*(n-k)}; \\ G. C. Greubel, Apr 22 2019

[[1+n*(n-k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Apr 22 2019

From Werner Schulte, Feb 12 2019: (Start)
G.f.: Sum_{n>0,k=1..n} T(n,k)*x^k*t^n = x*t*((1-t+2*t^2)*(1-x*t) + (1-t)*t)/((1-t)^3*(1-x*t)^2).
Row sums: Sum_{k=1..n} T(n,k) = A006000(n-1) for n > 0.
Recurrence: T(n,k) = T(n,k-1) - n for 1 < k <= n with initial values T(n,1) = n^2-n+1 for n > 0.
Recurrence: T(n,k) = T(n-1,k) + 2*n-k-1 for 1 <= k < n with initial values T(n,n) = 1 for n > 0.
(End)

A354909 Number of integer compositions of n that are not the run-sums of any other composition.

Original entry on oeis.org

0, 0, 1, 1, 3, 7, 16, 33, 74, 155, 329, 688, 1439, 2975, 6154, 12654, 25964, 53091, 108369, 220643, 448520

Offset: 0

Gus Wiseman, Jun 19 2022

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

			The a(0) = 0 through a(6) = 16 compositions:
  .  .  (11)  (111)  (112)   (113)    (114)
                     (211)   (311)    (411)
                     (1111)  (1112)   (1113)
                             (1121)   (1122)
                             (1211)   (1131)
                             (2111)   (1221)
                             (11111)  (1311)
                                      (2112)
                                      (2211)
                                      (3111)
                                      (11112)
                                      (11121)
                                      (11211)
                                      (12111)
                                      (21111)
                                      (111111)

The version for binary words is A000918, complement A000126.
These compositions are ranked by A354904 = positions of zeros in A354578.
The complement is counted by A354910, ranked by A354912.
A003242 counts anti-run compositions, ranked by A333489.
A238279 and A333755 count compositions by number of runs.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353853-A353859 pertain to composition run-sum trajectory.
A353932 lists run-sums of standard compositions, rows ranked by A353847.
Cf. A005811, A027336, A066099, A124767, A274174, A351597, A353849, A353850, A353860, A354905, A354907.

Table[Length[Complement[Join@@Permutations/@IntegerPartitions[n], Total/@Split[#]&/@Join@@Permutations/@IntegerPartitions[n]]],{n,0,15}]

A354910 Number of compositions of n that are the run-sums of some other composition.

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 16, 31, 54, 101, 183, 336, 609, 1121, 2038, 3730, 6804, 12445, 22703, 41501, 75768

Offset: 0

Gus Wiseman, Jun 20 2022

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

			The a(0) = 0 through a(6) = 16 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)
                (12)  (13)   (14)   (15)
                (21)  (22)   (23)   (24)
                      (31)   (32)   (33)
                      (121)  (41)   (42)
                             (122)  (51)
                             (131)  (123)
                             (212)  (132)
                             (221)  (141)
                                    (213)
                                    (222)
                                    (231)
                                    (312)
                                    (321)
                                    (1212)
                                    (2121)

The version for binary words is A000126, complement A000918
The complement is counted by A354909, ranked by A354904.
These compositions are ranked by A354912 = nonzeros of A354578.
A003242 counts anti-run compositions, ranked by A333489.
A238279 and A333755 count compositions by number of runs.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353853-A353859 pertain to composition run-sum trajectory.
A353932 lists run-sums of standard compositions, rows ranked by A353847.
Cf. A005811, A027336, A066099, A239312, A274174, A351014, A351597, A353849, A353850, A353864, A354905, A354907.

Table[Length[Union[Total/@Split[#]&/@ Join@@Permutations/@IntegerPartitions[n]]],{n,0,15}]

A000128 A nonlinear binomial sum.

Original entry on oeis.org

1, 2, 4, 8, 16, 31, 58, 105, 185, 319, 541, 906, 1503, 2476, 4058, 6626, 10790, 17537, 28464, 46155, 74791, 121137, 196139, 317508, 513901, 831686, 1345888, 2177900, 3524140, 5702419, 9226966, 14929821, 24157253, 39087571, 63245353, 102333486

Offset: 1

N. J. A. Sloane

Ralph P. Grimaldi, A generalization of the Fibonacci sequence. Proceedings of the seventeenth Southeastern international conference on combinatorics, graph theory, and computing (Boca Raton, Fla., 1986). Congr. Numer. 54 (1986), 123-128. MR0885268 (89f:11030). - N. J. A. Sloane, Apr 08 2012
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..201
D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 292-298.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Differences are A000126.
Second differences are A000071 (Fibonacci -1).
Cf. A000045.

A000128:=(1-2*z+z**2+z**3)/(z**2+z-1)/(z-1)**3; # conjectured by Simon Plouffe in his 1992 dissertation

LinearRecurrence[{4, -5, 1, 2, -1}, {1, 2, 4, 8, 16}, 40] (* Jean-François Alcover, Feb 04 2016 *)

G.f.: (1 - 2 x + x^2 + x^3) / ((1 - x - x^2 )*(1 - x)^3).
a(n) = F(n+4) - n*(n+1)/2 - 3, with F(n) = A000045(n). - Ralf Stephan, Aug 19 2004
a(n) = 2*a(n-1) - a(n-3) + n - 3, n > 3, and a(1) = 1, a(2) = 2, a(3) = 4. - Chunqing Liu, Sep 23 2023

More terms from Michel ten Voorde, Oct 06 2002

cp's OEIS Frontend

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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A306357 Number of nonempty subsets of {1, ..., n} containing no three cyclically successive elements.

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A324015 Number of nonempty subsets of {1, ..., n} containing no two cyclically successive elements.

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A111644 Expansion of -(1+x^2)/((x^2+4*x+1)*(x^2-2*x-1)).

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A323949 Number of set partitions of {1, ..., n} with no block containing three distinct cyclically successive vertices.

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A323955 Regular triangle read by rows where T(n, k) is the number of set partitions of {1, ..., n} with no block containing k cyclically successive vertices, n >= 1, 2 <= k <= n + 1.

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A323956 Triangle read by rows: T(n, k) = 1 + n * (n - k) for 1 <= k <= n.

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A354909 Number of integer compositions of n that are not the run-sums of any other composition.

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A111644 Expansion of -(1+x^2)/((x^2+4x+1)(x^2-2*x-1)).