A242882 -id:A242882 - cp's OEIS frontend

A351292 Number of patterns of length n with all distinct run-lengths.

1, 1, 1, 5, 5, 9, 57, 61, 109, 161, 1265, 1317, 2469, 3577, 5785, 43901, 47165, 86337, 127665, 204853, 284197, 2280089, 2398505, 4469373, 6543453, 10570993, 14601745, 22502549, 159506453, 171281529, 314077353, 462623821, 742191037, 1031307185, 1580543969, 2141246229

Offset: 0

Gus Wiseman, Feb 10 2022

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.

			The a(1) = 1 through a(5) = 9 patterns:
  (1)  (1,1)  (1,1,1)  (1,1,1,1)  (1,1,1,1,1)
              (1,1,2)  (1,1,1,2)  (1,1,1,1,2)
              (1,2,2)  (1,2,2,2)  (1,1,1,2,2)
              (2,1,1)  (2,1,1,1)  (1,1,2,2,2)
              (2,2,1)  (2,2,2,1)  (1,2,2,2,2)
                                  (2,1,1,1,1)
                                  (2,2,1,1,1)
                                  (2,2,2,1,1)
                                  (2,2,2,2,1)
The a(6) = 57 patterns grouped by sum:
  111111  111112  111122  112221  111223  111233  112333  122333
          111211  111221  122211  111322  111332  113332  133322
          112111  122111  211122  112222  112223  122233  221333
          211111  221111  221112  211222  113222  133222  223331
                                  221113  122222  211333  333122
                                  222112  211133  222133  333221
                                  222211  221222  222331
                                  223111  222113  233311
                                  311122  222122  331222
                                  322111  222221  332221
                                          222311  333112
                                          233111  333211
                                          311222
                                          322211
                                          331112
                                          332111

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for runs instead of run-lengths is A351200.
A000670 counts patterns, ranked by A333217.
A005649 counts anti-run patterns, complement A069321.
A005811 counts runs in binary expansion.
A032011 counts patterns with distinct multiplicities.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A060223 counts Lyndon patterns, necklaces A019536, aperiodic A296975.
A131689 counts patterns by number of distinct parts.
A238130 and A238279 count compositions by number of runs.
A165413 counts distinct run-lengths in binary expansion, runs A297770.
A345194 counts alternating patterns, up/down A350354.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351202 = permutations of prime factors.
- A351638 = word structures.
Row sums of A350824.
Cf. A003242, A098504, A098859, A106356, A239455, A242882, A325545, A328592, A329740, A351014, A351293.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],UnsameQ@@Length/@Split[#]&]],{n,0,6}]

P(n) = {Vec(-1 + prod(k=1, n, 1 + y*x^k + O(x*x^n)))}
R(u,k) = {k*[subst(serlaplace(p)/y, y, k-1) | p<-u]}
seq(n)={my(u=P(n), c=poldegree(u[#u])); concat([1], sum(k=1, c, R(u, k)*sum(r=k, c, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Feb 11 2022

From Andrew Howroyd, Feb 12 2022: (Start)
a(n) = Sum_{k=1..n} R(n,k)*(Sum_{r=k..n} binomial(r, k)*(-1)^(r-k)), where R(n,k) = Sum_{j=1..floor((sqrt(8*n+1)-1)/2)} k*(k-1)^(j-1) * j! * A008289(n,j).
G.f.: 1 + Sum_{r>=1} Sum_{k=1..r} R(k,x) * binomial(r, k)*(-1)^(r-k), where R(k,x) = Sum_{j>=1} k*(k-1)^(j-1) * j! * [y^j](Product_{k>=1} 1 + y*x^k).
(End)

Terms a(10) and beyond from Andrew Howroyd, Feb 11 2022

A382771 Number of ways to permute the prime indices of n so that the run-lengths are all different.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 2, 1, 2, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 1, 2, 2, 0, 1, 2, 1, 2, 0, 2, 1, 2, 0, 2, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 2, 0, 0, 1, 2, 1, 0, 2, 2, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Apr 07 2025

nonn

The first x with a(x) > 0 but A382857(x) > 1 is a(216) = 4, A382857(216) = 4.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			The a(96) = 4 permutations are:
  (1,1,1,1,1,2)
  (1,1,1,2,1,1)
  (1,1,2,1,1,1)
  (2,1,1,1,1,1)
The a(216) = 4 permutations are:
  (1,1,2,2,2,1)
  (1,2,2,2,1,1)
  (2,1,1,1,2,2)
  (2,2,1,1,1,2)
The a(360) = 6 permutations are:
  (1,1,1,2,2,3)
  (1,1,1,3,2,2)
  (2,2,1,1,1,3)
  (2,2,3,1,1,1)
  (3,1,1,1,2,2)
  (3,2,2,1,1,1)

Positions of 1 are A000961.
Positions of positive terms are A351294, conjugate A381432.
Positions of 0 are A351295, conjugate A381433, equal A382879.
Sorted positions of first appearances are A382772, equal A382878.
For prescribed signature we have A382773, equal A382858.
The restriction to factorials is A382774, equal A335407.
For equal instead of distinct run-lengths we have A382857.
For run-sums instead of run-lengths we have A382876, equal A382877.
Positions of terms > 1 are A383113.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A239455 counts Look-and-Say partitions, complement A351293.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
Cf. A000720, A001221, A001222, A003242, A048767, A051903, A051904, A130091, A238130, A351013, A351202.

Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]],UnsameQ@@Length/@Split[#]&]],{n,30}]

a(A181821(n)) = a(A304660(n)) = A382773(n).

a(n!) = A382774(n).

A384176 Number of subsets of {1..n} without all distinct lengths of maximal runs (increasing by 1).

Original entry on oeis.org

0, 0, 0, 1, 3, 8, 20, 51, 121, 276, 612, 1335, 2881, 6144, 12950, 27029, 55977, 115222, 236058, 481683, 979443

Offset: 0

Gus Wiseman, Jun 16 2025

			The subset {1,3,4,8,9} has maximal runs ((1),(3,4),(8,9)), with lengths (1,2,2), so is counted under a(10).
The a(0) = 0 through a(6) = 20 subsets:
  .  .  .  {1,3}  {1,3}  {1,3}      {1,3}
                  {1,4}  {1,4}      {1,4}
                  {2,4}  {1,5}      {1,5}
                         {2,4}      {1,6}
                         {2,5}      {2,4}
                         {3,5}      {2,5}
                         {1,3,5}    {2,6}
                         {1,2,4,5}  {3,5}
                                    {3,6}
                                    {4,6}
                                    {1,3,5}
                                    {1,3,6}
                                    {1,4,6}
                                    {2,4,6}
                                    {1,2,4,5}
                                    {1,2,4,6}
                                    {1,2,5,6}
                                    {1,3,4,6}
                                    {1,3,5,6}
                                    {2,3,5,6}

For equal instead of distinct lengths the complement is A243815.
These subsets are ranked by the non-members of A328592.
The complement is counted by A384175.
For strict partitions instead of subsets see A384178, A384884, A384886, A384880.
For permutations instead of subsets see A384891, A384892, A010027.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000009, A044813, A242882, A325325, A329739, A351202, A383013, A384177, A384889, A384890.

Table[Length[Select[Subsets[Range[n]],!UnsameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]

A351018 Number of integer compositions of n with all distinct even-indexed parts and all distinct odd-indexed parts.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 18, 27, 46, 77, 122, 191, 326, 497, 786, 1207, 1942, 2905, 4498, 6703, 10574, 15597, 23754, 35043, 52422, 78369, 115522, 169499, 248150, 360521, 532466, 768275, 1116126, 1606669, 2314426, 3301879, 4777078, 6772657, 9677138, 13688079, 19406214

Offset: 0

Gus Wiseman, Feb 09 2022

nonn

Also the number of binary words of length n starting with 1 and having all distinct runs (ranked by A175413, counted by A351016).

			The a(1) = 1 through a(6) = 18 compositions:
  (1)  (2)    (3)    (4)      (5)      (6)
       (1,1)  (1,2)  (1,3)    (1,4)    (1,5)
              (2,1)  (2,2)    (2,3)    (2,4)
                     (3,1)    (3,2)    (3,3)
                     (1,1,2)  (4,1)    (4,2)
                     (2,1,1)  (1,1,3)  (5,1)
                              (1,2,2)  (1,1,4)
                              (2,2,1)  (1,2,3)
                              (3,1,1)  (1,3,2)
                                       (2,1,3)
                                       (2,3,1)
                                       (3,1,2)
                                       (3,2,1)
                                       (4,1,1)
                                       (1,1,2,2)
                                       (1,2,2,1)
                                       (2,1,1,2)
                                       (2,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The case of partitions is A000726.
The version for run-lengths instead of runs is A032020.
These words are ranked by A175413.
A005811 counts runs in binary expansion.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A059966 counts Lyndon compositions, necklaces A008965, aperiodic A000740.
A116608 counts compositions by number of distinct parts.
A238130 and A238279 count compositions by number of runs.
A242882 counts compositions with distinct multiplicities.
A297770 counts distinct runs in binary expansion.
A325545 counts compositions with distinct differences.
A329738 counts compositions with equal run-lengths.
A329744 counts compositions by runs-resistance.
A351014 counts distinct runs in standard compositions.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A003242, A025047, A098504, A098859, A106356, A212322, A328592, A329740, A334028, A349054, A350952, A351205.

Table[Length[Select[Tuples[{0,1},n],#=={}||First[#]==1&&UnsameQ@@Split[#]&]],{n,0,10}]

P(n)=prod(k=1, n, 1 + y*x^k + O(x*x^n));
seq(n)=my(p=P(n)); Vec(sum(k=0, n, polcoef(p,k\2,y)*(k\2)!*polcoef(p,(k+1)\2,y)*((k+1)\2)!)) \\ Andrew Howroyd, Feb 11 2022

a(n>0) = A351016(n)/2.

G.f.: Sum_{k>=0} floor(k/2)! * ceiling(k/2)! * ([y^floor(k/2)] P(x,y)) * ([y^ceiling(k/2)] P(x,y)), where P(x,y) = Product_{k>=1} 1 + y*x^k. - Andrew Howroyd, Feb 11 2022

Terms a(21) and beyond from Andrew Howroyd, Feb 11 2022

A351291 Numbers k such that the k-th composition in standard order does not have all distinct runs.

Original entry on oeis.org

13, 22, 25, 45, 46, 49, 53, 54, 59, 76, 77, 82, 89, 91, 93, 94, 97, 101, 102, 105, 108, 109, 110, 115, 118, 141, 148, 150, 153, 156, 162, 165, 166, 173, 177, 178, 180, 181, 182, 183, 187, 189, 190, 193, 197, 198, 201, 204, 205, 209, 210, 213, 214, 216, 217

Offset: 1

Gus Wiseman, Feb 12 2022

nonn

The n-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of n, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their binary expansions and corresponding compositions begin:
  13:     1101  (1,2,1)
  22:    10110  (2,1,2)
  25:    11001  (1,3,1)
  45:   101101  (2,1,2,1)
  46:   101110  (2,1,1,2)
  49:   110001  (1,4,1)
  53:   110101  (1,2,2,1)
  54:   110110  (1,2,1,2)
  59:   111011  (1,1,2,1,1)
  76:  1001100  (3,1,3)
  77:  1001101  (3,1,2,1)
  82:  1010010  (2,3,2)
  89:  1011001  (2,1,3,1)
  91:  1011011  (2,1,2,1,1)
  93:  1011101  (2,1,1,2,1)
  94:  1011110  (2,1,1,1,2)

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for Heinz numbers of partitions is A130092, complement A130091.
Normal multisets with a permutation of this type appear to be A283353.
Partitions w/o permutations of this type are A351204, complement A351203.
The version using binary expansions is A351205, complement A175413.
The complement is A351290, counted by A351013.
A005811 counts runs in binary expansion, distinct A297770.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has all distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse A085208.
A333489 ranks anti-runs, complement A348612, counted by A003242.
A345167 ranks alternating compositions, counted by A025047.
Counting words with all distinct runs:
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Selected statistics of standard compositions (A066099, reverse A228351):
- Length is A000120.
- Sum is A070939.
- Runs are counted by A124767, distinct A351014.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
Cf. A098859, A106356, A113835, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351015, A351201.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!UnsameQ@@Split[stc[#]]&]

A384886 Number of strict integer partitions of n with all equal lengths of maximal runs (decreasing by 1).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 4, 7, 7, 8, 11, 11, 14, 17, 19, 20, 27, 27, 35, 38, 45, 47, 60, 63, 75, 84, 97, 104, 127, 134, 155, 175, 196, 218, 251, 272, 307, 346, 384, 424, 480, 526, 586, 658, 719, 798, 890, 979, 1078, 1201, 1315, 1451, 1603, 1762, 1934, 2137

Offset: 0

Gus Wiseman, Jun 13 2025

nonn

			The strict partition y = (7,6,5,3,2,1) has maximal runs ((7,6,5),(3,2,1)), with lengths (3,3), so y is counted under a(24).
The a(1) = 1 through a(14) = 14 partitions (A-E = 10-14):
  1  2  3   4   5   6    7   8   9    A     B    C     D    E
        21  31  32  42   43  53  54   64    65   75    76   86
                41  51   52  62  63   73    74   84    85   95
                    321  61  71  72   82    83   93    94   A4
                                 81   91    92   A2    A3   B3
                                 432  631   A1   B1    B2   C2
                                 531  4321  641  543   C1   D1
                                            731  642   742  752
                                                 741   751  842
                                                 831   841  851
                                                 5421  931  941
                                                            A31
                                                            5432
                                                            6521

John Tyler Rascoe, Table of n, a(n) for n = 0..100

For subsets instead of strict partitions we have A243815, distinct lengths A384175.
For distinct instead of equal lengths we have A384178, for anti-runs A384880.
This is the strict case of A384904, distinct lengths A384884.
A000041 counts integer partitions, strict A000009.
A047993 counts partitions with max part = length (A106529).
A098859 counts Wilf partitions (complement A336866), compositions A242882.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
Cf. A000217, A008284, A044813, A047966, A089259, A325324, A325325, A329739, A382857, A383013, A383708, A384176.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&SameQ@@Length/@Split[#,#2==#1-1&]&]],{n,0,15}]

A_q(N) = {Vec(1+sum(k=1,floor(-1/2+sqrt(2+2*N)), sum(i=1,(N/(k*(k+1)/2))+1, q^(k*(k+1)*i^2/2)/prod(j=1,i, 1 - q^(j*k)))) + O('q^(N+1)))} \\ John Tyler Rascoe, Aug 21 2025

G.f.: 1 + Sum_{i,k>0} q^(k*(k+1)*i^2/2)/Product_{j=1..i} (1 - q^(j*k)). - John Tyler Rascoe, Aug 21 2025

A128695 Number of compositions of n with parts in N which avoid the adjacent pattern 111.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 24, 46, 89, 170, 324, 618, 1183, 2260, 4318, 8249, 15765, 30123, 57556, 109973, 210137, 401525, 767216, 1465963, 2801115, 5352275, 10226930, 19541236, 37338699, 71345449, 136324309, 260483548, 497722578, 951030367

Offset: 0

Ralf Stephan, May 08 2007

nonn

			From _Gus Wiseman_, Jul 06 2020: (Start)
The a(0) = 1 through a(5) = 13 compositions:
  ()  (1)  (2)    (3)    (4)      (5)
           (1,1)  (1,2)  (1,3)    (1,4)
                  (2,1)  (2,2)    (2,3)
                         (3,1)    (3,2)
                         (1,1,2)  (4,1)
                         (1,2,1)  (1,1,3)
                         (2,1,1)  (1,2,2)
                                  (1,3,1)
                                  (2,1,2)
                                  (2,2,1)
                                  (3,1,1)
                                  (1,1,2,1)
                                  (1,2,1,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
S. Heubach and T. Mansour, Enumeration of 3-letter patterns in compositions, arXiv:math/0603285 [math.CO], 2006
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Column k=0 of A232435.
Cf. A091616, A232432.
The matching version is A335464.
Contiguously (1,1)-avoiding compositions is A003242.
Contiguously (1,1)-matching compositions are A261983.
Compositions with some part > 2 are A008466
Compositions by number of adjacent equal parts are A106356.
Compositions where each part is adjacent to an equal part are A114901.
Compositions with adjacent parts coprime are A167606.
Compositions with equal parts contiguous are A274174.
Patterns contiguously matched by compositions are A335457.
Patterns contiguously matched by a given partition are A335516.
Cf. A005251, A032020, A131044, A178470, A242882, A333175, A335455.

b:= proc(n, t) option remember; `if`(n=0, 1, add(`if`(abs(t)<>j,
       b(n-j, j), `if`(t=-j, 0, b(n-j, -j))), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Nov 23 2013

nn=33;CoefficientList[Series[1/(1-Sum[(x^i+x^(2i))/(1+x^i+x^(2i)),{i,1,nn}]),{x,0,nn}],x] (* Geoffrey Critzer, Nov 23 2013 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,x_,x_,_}]&]],{n,13}] (* Gus Wiseman, Jul 06 2020 *)

G.f.: 1/(1-Sum(i>=1, x^i*(1+x^i)/(1+x^i*(1+x^i)) ) ).
a(n) ~ c * d^n, where d is the root of the equation Sum_{k>=1} 1/(d^k + 1/(1 + d^k)) = 1, d=1.9107639262818041675000243699745706859615884029961947632387839..., c=0.4993008137128378086219448701860326113802027003939127932922782... - Vaclav Kotesovec, May 01 2014, updated Jul 07 2020
For n>=2, a(n) = A091616(n) + A003242(n). - Vaclav Kotesovec, Jul 07 2020

A329745 Number of compositions of n with runs-resistance 2.

Original entry on oeis.org

0, 0, 2, 3, 6, 15, 22, 41, 72, 129, 213, 395, 660, 1173, 2031, 3582, 6188, 10927, 18977, 33333, 58153, 101954, 178044, 312080, 545475, 955317, 1670990, 2925711, 5118558, 8960938, 15680072, 27447344, 48033498, 84076139, 147142492, 257546234, 450748482, 788937188

Offset: 1

Gus Wiseman, Nov 21 2019

nonn

A composition of n is a finite sequence of positive integers with sum n.
For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.
These are non-constant compositions with equal run-lengths (A329738).

			The a(3) = 2 through a(6) = 15 compositions:
  (1,2)  (1,3)    (1,4)    (1,5)
  (2,1)  (3,1)    (2,3)    (2,4)
         (1,2,1)  (3,2)    (4,2)
                  (4,1)    (5,1)
                  (1,3,1)  (1,2,3)
                  (2,1,2)  (1,3,2)
                           (1,4,1)
                           (2,1,3)
                           (2,3,1)
                           (3,1,2)
                           (3,2,1)
                           (1,1,2,2)
                           (1,2,1,2)
                           (2,1,2,1)
                           (2,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Column k = 2 of A329744.
Column k = n - 2 of A329750.
Cf. A000740, A003242, A008965, A098504, A242882, A318928, A329743, A329746, A329747, A329767.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==2&]],{n,10}]

seq(n)={my(b=Vec(1/(1 - sum(k=1, n, x^k/(1+x^k) + O(x*x^n)))-1)); vector(n, k, sumdiv(k, d, b[d]-1))} \\ Andrew Howroyd, Dec 30 2020

a(n) = A329738(n) - A000005(n).

a(n) = Sum_{d|n} (A003242(d) - 1). - Andrew Howroyd, Dec 30 2020

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

A357184 Numbers k such that the k-th composition in standard order has the same length as its alternating sum.

Original entry on oeis.org

0, 1, 9, 19, 22, 28, 34, 69, 74, 84, 104, 132, 135, 141, 153, 177, 225, 265, 271, 274, 283, 286, 292, 307, 310, 316, 328, 355, 358, 364, 376, 400, 451, 454, 460, 472, 496, 520, 523, 526, 533, 538, 553, 562, 593, 610, 673, 706, 833, 898, 1041, 1047, 1053, 1058

Offset: 1

Gus Wiseman, Sep 28 2022

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The sequence together with the corresponding compositions begins:
    0: ()
    1: (1)
    9: (3,1)
   19: (3,1,1)
   22: (2,1,2)
   28: (1,1,3)
   34: (4,2)
   69: (4,2,1)
   74: (3,2,2)
   84: (2,2,3)
  104: (1,2,4)
  132: (5,3)
  135: (5,1,1,1)
  141: (4,1,2,1)
  153: (3,1,3,1)
  177: (2,1,4,1)
  225: (1,1,5,1)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
For product equal to sum we have A335404, counted by A335405.
For sum equal to twice alternating sum we have A348614, counted by A262977.
These compositions are counted by A357182.
For absolute value we have A357184, counted by A357183.
The case of partitions is counted by A357189.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A025047 counts alternating compositions, ranked by A345167.
A032020 counts strict compositions, ranked by A233564.
A124754 gives alternating sums of standard compositions.
A238279 counts compositions by sum and number of maximal runs.
A357136 counts compositions by alternating sum.
Cf. A000120, A070939, A114220, A114901, A178470, A242882, A262046, A301987.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],Length[stc[#]]==ats[stc[#]]&]

A384884 Number of integer partitions of n with all distinct lengths of maximal gapless runs (decreasing by 0 or 1).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 13, 18, 25, 35, 46, 60, 79, 104, 131, 170, 215, 271, 342, 431, 535, 670, 830, 1019, 1258, 1547, 1881, 2298, 2787, 3359, 4061, 4890, 5849, 7010, 8361, 9942, 11825, 14021, 16558, 19561, 23057, 27084, 31821, 37312, 43627, 50999, 59500, 69267

Offset: 0

Gus Wiseman, Jun 13 2025

nonn

			The partition y = (6,6,4,3,3,2) has maximal gapless runs ((6,6),(4,3,3,2)), with lengths (2,4), so y is counted under a(24).
The a(1) = 1 through a(8) = 18 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (211)   (221)    (222)     (322)      (332)
                    (1111)  (311)    (321)     (331)      (422)
                            (2111)   (411)     (421)      (431)
                            (11111)  (2211)    (511)      (521)
                                     (3111)    (2221)     (611)
                                     (21111)   (3211)     (2222)
                                     (111111)  (4111)     (3221)
                                               (22111)    (4211)
                                               (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

For subsets instead of strict partitions we have A384175.
The strict case is A384178, for anti-runs A384880.
For anti-runs we have A384885.
For equal instead of distinct lengths we have A384887.
A000041 counts integer partitions, strict A000009.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A355394 counts partitions without a neighborless part, singleton case A355393.
A356236 counts partitions with a neighborless part, singleton case A356235.
A356606 counts strict partitions without a neighborless part, complement A356607.
Cf. A008284, A044813, A047993, A242882, A287170, A325324, A325325, A356226, A356230, A356233, A356234, A384176, A384177, A384886.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#,#2>=#1-1&]&]],{n,0,15}]

cp's OEIS Frontend

A351292 Number of patterns of length n with all distinct run-lengths.

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A382771 Number of ways to permute the prime indices of n so that the run-lengths are all different.

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A384176 Number of subsets of {1..n} without all distinct lengths of maximal runs (increasing by 1).

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A351018 Number of integer compositions of n with all distinct even-indexed parts and all distinct odd-indexed parts.

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A351291 Numbers k such that the k-th composition in standard order does not have all distinct runs.

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A384886 Number of strict integer partitions of n with all equal lengths of maximal runs (decreasing by 1).

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A128695 Number of compositions of n with parts in N which avoid the adjacent pattern 111.

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A329745 Number of compositions of n with runs-resistance 2.

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