A106356 -id:A106356 - cp's OEIS frontend

A351290 Numbers k such that the k-th composition in standard order has all distinct runs.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 50, 51, 52, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 78

Offset: 1

Gus Wiseman, Feb 10 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:
   0:      0  ()
   1:      1  (1)
   2:     10  (2)
   3:     11  (1,1)
   4:    100  (3)
   5:    101  (2,1)
   6:    110  (1,2)
   7:    111  (1,1,1)
   8:   1000  (4)
   9:   1001  (3,1)
  10:   1010  (2,2)
  11:   1011  (2,1,1)
  12:   1100  (1,3)
  14:   1110  (1,1,2)
  15:   1111  (1,1,1,1)

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

The version for Heinz numbers and prime multiplicities is A130091.
The version using binary expansions is A175413, complement A351205.
The version for run-lengths instead of runs is A329739.
These compositions are counted by A351013.
The complement is A351291.
A005811 counts runs in binary expansion, distinct A297770.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse A085208.
A333489 ranks anti-runs, complement A348612.
A345167 ranks alternating compositions, counted by A025047.
A351204 counts partitions where every permutation has all distinct runs.
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:
- Length is A000120.
- Parts are A066099, reverse A228351.
- 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[#]]&]

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

Original entry on oeis.org

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

A374698 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are distinct.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 12, 16, 17, 18, 20, 22, 24, 26, 32, 33, 34, 37, 38, 40, 41, 44, 48, 50, 52, 64, 65, 66, 68, 69, 70, 72, 76, 80, 81, 88, 96, 98, 100, 104, 128, 129, 130, 132, 133, 134, 137, 140, 144, 145, 148, 150, 152, 154, 160, 161, 164, 166, 176, 180

Offset: 1

Gus Wiseman, Jul 27 2024

nonn

The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.

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 maximal strictly increasing subsequences of the 212th composition in standard order are ((1,2),(2,3)), with leaders (1,2), so 212 is in the sequence.
The terms together with corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   4: (3)
   5: (2,1)
   6: (1,2)
   8: (4)
   9: (3,1)
  12: (1,3)
  16: (5)
  17: (4,1)
  18: (3,2)
  20: (2,3)
  22: (2,1,2)
  24: (1,4)
  26: (1,2,2)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Positions of distinct (strict) rows in A374683.
For identical leaders we have A374685, counted by A374761.
Compositions of this type are counted by A374687.
The opposite version is A374767, counted by A374760.
The weak version is A374768, counted by A374632.
Other types of runs: A374249 (counts A274174), A374638 (counts A374518), A374701 (counts A374743).
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) (or sometimes A070939).
- Parts are listed by A066099.
- Adjacent equal pairs are counted by A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of anti-run compositions are A333489, counted by A003242.
- Run-length transform is A333627.
- Run-compression transform is A373948, sum A373953, excess A373954.
Cf. A065120, A106356, A238343, A333213, A373949, A374520, A374629, A374630, A374635, A374740.

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

A374742 Number of integer compositions of n whose leaders of weakly decreasing runs are identical.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 34, 54, 87, 138, 220, 349, 556, 881, 1403, 2229, 3551, 5653, 9019, 14387, 22988, 36739, 58785, 94100, 150765, 241658, 387617, 622002, 998658, 1604032, 2577512, 4143243, 6662520, 10716931, 17243904, 27753518, 44680121, 71947123, 115880662

Offset: 0

Gus Wiseman, Jul 25 2024

nonn

The weakly decreasing run-leaders of a sequence are obtained by splitting into maximal weakly decreasing subsequences and taking the first term of each.

			The composition (3,1,3,2,1,3,3) has maximal weakly decreasing subsequences ((3,1),(3,2,1),(3,3)), with leaders (3,3,3), so is counted under a(16).
The a(0) = 1 through a(6) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (21)   (22)    (32)     (33)
                 (111)  (31)    (41)     (42)
                        (211)   (212)    (51)
                        (1111)  (221)    (222)
                                (311)    (321)
                                (2111)   (411)
                                (11111)  (2112)
                                         (2121)
                                         (2211)
                                         (3111)
                                         (21111)
                                         (111111)

John Tyler Rascoe, Table of n, a(n) for n = 0..200
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Ranked by A374744 = positions of identical rows in A374740, cf. A374629.
Types of runs (instead of weakly decreasing):
- For leaders of identical runs we have A000005 for n > 0, ranks A272919.
- For leaders of anti-runs we have A374517, ranks A374519.
- For leaders of strictly increasing runs we have A374686, ranks A374685.
- For leaders of weakly increasing runs we have A374631, ranks A374633.
- For leaders of strictly decreasing runs we have A374760, ranks A374759.
Types of run-leaders (instead of identical):
- For strictly decreasing leaders we have A374746.
- For weakly decreasing leaders we have A374747.
- For distinct leaders we have A374743, ranks A374701.
- For weakly increasing leaders we appear to have A188900.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A335456 counts patterns matched by compositions.
A335548 counts non-contiguous compositions, ranks A374253.
A373949 counts compositions by run-compressed sum, opposite A373951.
A374748 counts compositions by sum of leaders of weakly decreasing runs.
Cf. A000009, A106356, A188920, A189076, A238343, A261982, A333213, A374632, A374634, A374635, A374741.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],SameQ@@First/@Split[#,GreaterEqual]&]],{n,0,15}]

B(i) = x^i/(1-x^i) * sum(j=1,i-1, x^j*prod(k=1,j, (1-x^k)^(-1)))
A_x(N) = {my(x='x+O('x^N)); Vec(1+sum(i=1,N,-1+(1+x^i/(1-x^i))/(1-B(i))))}
A_x(30) \\ John Tyler Rascoe, Apr 29 2025

G.f.: 1 + Sum_{i>0} -1 + (1 + x^i/(1 - x^i))/(1 - B(i,x)) where B(i,x) = x^i/(1 - x^i) * Sum_{j=1..i-1} x^j * Product_{k=1..j} (1 - x^k)^(-1). - John Tyler Rascoe, Apr 29 2025

a(24)-a(40) from Alois P. Heinz, Jul 26 2024

A374767 Numbers k such that the leaders of strictly decreasing runs in the k-th composition in standard order are distinct.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 24, 25, 32, 33, 34, 35, 37, 38, 40, 41, 44, 48, 49, 50, 52, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 77, 78, 80, 81, 82, 83, 88, 89, 92, 96, 97, 98, 101, 102, 104, 105, 108, 128, 129, 130, 131, 132, 133

Offset: 1

Gus Wiseman, Jul 29 2024

nonn

The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.

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 10000000th composition in standard order is (3,1,4,3,2,1,2,8), with strictly decreasing runs ((3,1),(4,3,2,1),(2),(8)), with leaders (3,4,2,1) so 10000000 is in the sequence.
The terms together with the corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   4: (3)
   5: (2,1)
   6: (1,2)
   8: (4)
   9: (3,1)
  11: (2,1,1)
  12: (1,3)
  13: (1,2,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  19: (3,1,1)
  20: (2,3)
  24: (1,4)
  25: (1,3,1)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

The opposite version is A374698, counted by A374687.
The weak version is A374701, counted by A374743.
For identical instead of distinct runs we have A374759, counted by A374760.
Compositions of this type are counted by A374761.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
Six types of runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A065120, A106356, A188920, A233564, A238343, A272919, A333213, A373949, A374633, A374685, A374744, A374758, A375128.

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

A374638 Numbers k such that the leaders of anti-runs in the k-th composition in standard order (A066099) are distinct.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 22, 24, 25, 26, 32, 33, 34, 35, 37, 38, 40, 41, 44, 45, 46, 48, 49, 50, 52, 53, 54, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83, 88, 89, 91, 92, 93, 96, 97, 98, 100, 101, 102, 104

Offset: 1

Gus Wiseman, Aug 01 2024

nonn

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.

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 terms together with corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   4: (3)
   5: (2,1)
   6: (1,2)
   8: (4)
   9: (3,1)
  11: (2,1,1)
  12: (1,3)
  13: (1,2,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  19: (3,1,1)
  20: (2,3)
  22: (2,1,2)
  24: (1,4)
  25: (1,3,1)
  26: (1,2,2)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Positions of distinct (strict) rows in A374515.
Compositions of this type are counted by A374518.
For identical instead of distinct we have A374519, counted by A374517.
The complement is A374639.
Other types of runs (instead of anti-):
- For identical runs we have A374249, counted by A274174.
- For weakly increasing runs we have A374768, counted by A374632.
- For strictly increasing runs we have A374698, counted by A374687.
- For weakly decreasing runs we have A374701, counted by A374743.
- For strictly decreasing runs we have A374767, counted by A374761.
A065120 gives leaders of standard compositions.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A238424 counts partitions whose first differences are an anti-run.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Anti-runs are ranked by A333489, counted by A003242.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
Six types of maximal runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A029931, A114994, A228351, A233564, A238343, A272919, A335467, A373949.

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

A374685 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are identical.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 10, 12, 13, 14, 15, 16, 20, 24, 25, 27, 28, 29, 30, 31, 32, 36, 40, 42, 48, 49, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 72, 80, 82, 84, 96, 97, 99, 102, 103, 104, 105, 108, 109, 110, 111, 112, 113, 115, 116, 118, 119, 120, 121

Offset: 1

Gus Wiseman, Jul 27 2024

nonn

The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.

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 maximal strictly increasing subsequences of the 6560th composition in standard order are ((1,3),(1,2,6)), with leaders (1,1), so 6560 is in the sequence.
The terms together with corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   6: (1,2)
   7: (1,1,1)
   8: (4)
  10: (2,2)
  12: (1,3)
  13: (1,2,1)
  14: (1,1,2)
  15: (1,1,1,1)
  16: (5)
  20: (2,3)
  24: (1,4)
  25: (1,3,1)
  27: (1,2,1,1)
  28: (1,1,3)
  29: (1,1,2,1)
  30: (1,1,1,2)
  31: (1,1,1,1,1)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

The weak version is A374633, counted by A374631.
Positions of constant rows in A374683.
Compositions of this type are counted by A374686.
For distinct leaders we have A374698, counted by A374687.
The opposite version is A374759, counted by A374760.
Other types of runs: A272919 (counts A000005), A374519 (counts A374517), A374744 (counts A374742).
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A374748 counts compositions by sum of leaders of weakly decreasing runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) (or sometimes A070939).
- Parts are listed by A066099.
- Adjacent equal pairs are counted by A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of anti-run compositions are A333489, counted by A003242.
- Run-length transform is A333627.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
Cf. A065120, A106356, A238343, A333213, A373949, A374520, A374629, A374630, A374701, A374740, A374768.

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

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[#]]&]

A374519 Numbers k such that the leaders of anti-runs in the k-th composition in standard order (A066099) are identical.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 72, 73, 76, 77, 80, 81, 82, 84, 85

Offset: 1

Gus Wiseman, Aug 01 2024

nonn

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.

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 346th composition in standard order is (2,2,1,2,2), with anti-runs ((2),(2,1,2),(2)), with leaders (2,2,2), so 346 is in the sequence.

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Positions of constant rows in A374515.
Compositions of this type are counted by A374517.
The complement is A374520.
For distinct instead of identical leaders we have A374638, counted by A374518.
Other types of runs (instead of anti-):
- For identical runs we have A272919, counted by A000005.
- For weakly increasing runs we have A374633, counted by A374631.
- For strictly increasing runs we have A374685, counted by A374686.
- For weakly decreasing runs we have A374744, counted by A374742.
- For strictly decreasing runs we have A374759, counted by A374760.
A065120 gives leaders of standard compositions.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs.
A238424 counts partitions whose first differences are an anti-run.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Anti-runs are ranked by A333489, counted by A003242.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
Six types of maximal runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A029931, A114994, A228351, A233564, A238343, A373949.

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

cp's OEIS Frontend

A351290 Numbers k such that the k-th composition in standard order has all distinct runs.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A374698 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are distinct.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A374742 Number of integer compositions of n whose leaders of weakly decreasing runs are identical.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A374767 Numbers k such that the leaders of strictly decreasing runs in the k-th composition in standard order are distinct.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A374638 Numbers k such that the leaders of anti-runs in the k-th composition in standard order (A066099) are distinct.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A374685 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are identical.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author