A333627 -id:A333627 - cp's OEIS frontend

A375296 Numbers k such that the leaders of maximal weakly increasing runs in the reverse of the k-th composition in standard order (row k of A228351) are not strictly decreasing.

13, 25, 27, 29, 41, 45, 49, 51, 53, 54, 55, 57, 59, 61, 77, 81, 82, 83, 89, 91, 93, 97, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 115, 117, 118, 119, 121, 123, 125, 141, 145, 153, 155, 157, 161, 162, 163, 165, 166, 167, 169, 173, 177, 179, 181, 182

Offset: 1

Gus Wiseman, Aug 13 2024

nonn

The leaders of maximal weakly increasing runs in a sequence are obtained by splitting it into maximal weakly 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.
Also numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed patterns 23-1 or 12-1.

			The sequence together with corresponding compositions begins:
  13: (1,2,1)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  41: (2,3,1)
  45: (2,1,2,1)
  49: (1,4,1)
  51: (1,3,1,1)
  53: (1,2,2,1)
  54: (1,2,1,2)
  55: (1,2,1,1,1)
  57: (1,1,3,1)
  59: (1,1,2,1,1)
  61: (1,1,1,2,1)

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

For leaders of identical runs we have A335486, reverse A335485.
Matching 1-32 only gives A375138, reverse A375137, both counted by A374636.
Compositions of this type are counted by A375140, complement A188920.
The reverse version is A375295.
A003242 counts anti-runs, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A374637 counts compositions by sum of leaders of weakly increasing runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Leader is A065120.
- Parts are listed by A066099, reverse A228351.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Run-length transform is A333627, sum A070939.
- Run-counts: A124766, A124765, A124768, A124769, A333381, A124767.
- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
Cf. A056823, A106356, A188919, A189076, A238343, A333213, A335480, A335482, A373948, A374630, A374633, A374768, A375123.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!Greater@@First/@Split[Reverse[stc[#]],LessEqual]&]
- or -
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,300],MatchQ[stc[#],{_,y_,z_,_,x_,_}/;x<=y

A335461 Triangle read by rows where T(n,k) is the number of patterns of length n with k runs.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 4, 8, 0, 1, 6, 24, 44, 0, 1, 8, 48, 176, 308, 0, 1, 10, 80, 440, 1540, 2612, 0, 1, 12, 120, 880, 4620, 15672, 25988, 0, 1, 14, 168, 1540, 10780, 54852, 181916, 296564, 0, 1, 16, 224, 2464, 21560, 146272, 727664, 2372512, 3816548

Offset: 0

Gus Wiseman, Jul 03 2020

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.

			Triangle begins:
     1
     0     1
     0     1     2
     0     1     4     8
     0     1     6    24    44
     0     1     8    48   176   308
     0     1    10    80   440  1540  2612
     0     1    12   120   880  4620 15672 25988
Row n = 3 counts the following patterns:
  (1,1,1)  (1,1,2)  (1,2,1)
           (1,2,2)  (1,2,3)
           (2,1,1)  (1,3,2)
           (2,2,1)  (2,1,2)
                    (2,1,3)
                    (2,3,1)
                    (3,1,2)
                    (3,2,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A000670.
Column n = k is A005649 (anti-run patterns).
Central coefficients are A337564.
The version for compositions is A333755.
Runs of standard compositions are counted by A124767.
Run-lengths of standard compositions are A333769.
Cf. A003242, A052841, A060223, A106351, A106356, A269134, A325535, A333489, A333627, A335838.

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],Length[Split[#]]==k&]],{n,0,5},{k,0,n}]

\\ here b(n) is A005649.
b(n) = {sum(k=0, n, stirling(n,k,2)*(k + 1)!)}
T(n,k)=if(n==0, k==0, b(k-1)*binomial(n-1,k-1)) \\ Andrew Howroyd, Dec 31 2020

T(n,k) = A005649(k-1) * binomial(n-1,k-1) for k > 0. - Andrew Howroyd, Dec 31 2020

A353856 Triangle read by rows where T(n,k) is the number of integer compositions of n with run-sum trajectory (condensation) ending in a composition of length k.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 5, 2, 1, 0, 0, 2, 12, 2, 0, 0, 0, 8, 10, 12, 2, 0, 0, 0, 2, 32, 23, 6, 1, 0, 0, 0, 20, 26, 51, 28, 3, 0, 0, 0, 0, 5, 66, 109, 52, 22, 2, 0, 0, 0, 0, 8, 108, 144, 188, 53, 10, 1, 0, 0, 0, 0, 2, 134, 358, 282, 196, 48, 4, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Jun 01 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 run-sum trajectory is obtained by repeatedly taking the run-sums transformation (or condensation, represented by A353847) until an anti-run is reached. For example, the trajectory (2,1,1,3,1,1,2,1,1,2,1) -> (2,2,3,2,2,2,2,1) -> (4,3,8,1) is counted under T(15,4).

			Triangle begins:
   1
   0   1
   0   2   0
   0   2   2   0
   0   5   2   1   0
   0   2  12   2   0   0
   0   8  10  12   2   0   0
   0   2  32  23   6   1   0   0
   0  20  26  51  28   3   0   0   0
   0   5  66 109  52  22   2   0   0   0
   0   8 108 144 188  53  10   1   0   0   0
   0   2 134 358 282 196  48   4   0   0   0   0
For example, row n = 6 counts the following compositions:
  .  (6)       (15)     (123)    (1212)  .  .
     (33)      (24)     (132)    (2121)
     (222)     (42)     (141)
     (1113)    (51)     (213)
     (2112)    (114)    (231)
     (3111)    (411)    (312)
     (11211)   (1122)   (321)
     (111111)  (2211)   (1131)
               (11112)  (1221)
               (21111)  (1311)
                        (11121)
                        (12111)

Row sums are A011782.
Row-lengths without zeros appear to be A131737.
The version for partitions is A353843.
The length of the trajectory is A353854, firsts A072639, partitions A353841.
The last part of the same trajectory is A353855.
Column k = 1 is A353858.
A066099 lists compositions in standard order.
A318928 gives runs-resistance of binary expansion.
A325268 counts partitions by omicron, rank statistic A304465.
A333489 ranks anti-runs, counted by A003242 (complement A261983).
A333627 ranks the run-lengths of standard compositions.
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents the run-sums of a composition, partitions A353832.
A353853-A353859 pertain to composition run-sum trajectory.
A353932 lists run-sums of standard compositions.
Cf. A237685, A238279, A304442, A325277, A333755, A353848, A353850, A353852.

Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],Length[FixedPoint[Total/@Split[#]&,#]]==k&]],{n,0,15},{k,0,n}]

A375139 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are not weakly decreasing.

Original entry on oeis.org

26, 50, 53, 58, 90, 98, 100, 101, 106, 107, 114, 117, 122, 154, 164, 178, 181, 186, 194, 196, 197, 201, 202, 203, 210, 212, 213, 214, 215, 218, 226, 228, 229, 234, 235, 242, 245, 250, 282, 306, 309, 314, 324, 329, 346, 354, 356, 357, 362, 363, 370, 373, 378

Offset: 1

Gus Wiseman, Aug 12 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 terms together with corresponding compositions begin:
   26: (1,2,2)
   50: (1,3,2)
   53: (1,2,2,1)
   58: (1,1,2,2)
   90: (2,1,2,2)
   98: (1,4,2)
  100: (1,3,3)
  101: (1,3,2,1)
  106: (1,2,2,2)
  107: (1,2,2,1,1)
  114: (1,1,3,2)
  117: (1,1,2,2,1)
  122: (1,1,1,2,2)
  154: (3,1,2,2)
  164: (2,3,3)
  178: (2,1,3,2)
  181: (2,1,2,2,1)
  186: (2,1,1,2,2)

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

For leaders of identical runs we have A335485.
Ranked by positions of non-weakly decreasing rows in A374683.
For identical leaders we have A374685, counted by A374686.
The complement is counted by A374697.
For distinct leaders we have A374698, counted by A374687.
Compositions of this type are counted by A375135.
Weakly increasing leaders: A375137, counts A374636, complement A189076.
Interchanging weak/strict: A375295, counted by A375140, complement A188920.
A003242 counts anti-run compositions, ranks A333489.
A374700 counts compositions by sum of leaders of strictly increasing runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Leader is A065120.
- Parts are listed by A066099.
- Strict compositions are A233564.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Run-count: A124766, A124765, A124768, A124769, A333381, A124767.
- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Run-rank: A375123, A375124, A375125, A375126, A375127.
Cf. A000009, A261982, A333213, A374520, A374630, A374699, A374768.

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

A353857 Numbers k such that the k-th composition in standard order has run-sum trajectory ending in a singleton.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 10, 11, 14, 15, 16, 31, 32, 36, 39, 42, 46, 59, 60, 63, 64, 127, 128, 136, 138, 139, 142, 143, 168, 170, 174, 175, 184, 186, 187, 232, 238, 239, 248, 250, 251, 255, 256, 292, 316, 487, 511, 512, 528, 543, 682, 750, 955, 1008, 1023, 1024, 2047

Offset: 1

Gus Wiseman, Jun 01 2022

nonn

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 run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353847) until the rank of an anti-run is reached. For example, the trajectory 11 -> 10 -> 8 corresponds to the trajectory (2,1,1) -> (2,2) -> (4).

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 their binary expansions and corresponding compositions begin:
   1:        1  (1)
   2:       10  (2)
   3:       11  (1,1)
   4:      100  (3)
   7:      111  (1,1,1)
   8:     1000  (4)
  10:     1010  (2,2)
  11:     1011  (2,1,1)
  14:     1110  (1,1,2)
  15:     1111  (1,1,1,1)
  16:    10000  (5)
  31:    11111  (1,1,1,1,1)
  32:   100000  (6)
  36:   100100  (3,3)
  39:   100111  (3,1,1,1)
  42:   101010  (2,2,2)
  46:   101110  (2,1,1,2)
  59:   111011  (1,1,2,1,1)
  60:   111100  (1,1,1,3)
  63:   111111  (1,1,1,1,1,1)

The version for partitions is A353844.
The trajectory length is A353854, firsts A072639, for partitions A353841.
The last part of the trajectory is A353855, for partitions A353842.
These compositions are counted by A353858.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order.
A318928 gives runs-resistance of binary expansion.
A325268 counts partitions by omicron, rank statistic A304465.
A333627 ranks the run-lengths of standard compositions.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents composition run-sum transformation, partitions A353832.
A353853-A353859 pertain to composition run-sum trajectory.
A353932 lists run-sums of standard compositions.
Cf. A237685, A238279, A304442, A325277, A333381, A333755, A353848, A353849, A353850, A353852.

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

A354579 Number of distinct lengths of runs in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 11 2022

nonn

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 lengths (2,3,1,2).

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 positions of first appearances together with the corresponding compositions begin:
       1: (1)
      11: (2,1,1)
     119: (1,1,2,1,1,1)
    5615: (2,2,1,1,1,2,1,1,1,1)
  251871: (1,1,1,2,2,1,1,1,1,2,1,1,1,1,1)

Standard compositions are listed by A066099.
The version for partitions is A071625.
For runs instead of run-lengths we have A351014, firsts A351015.
Positions of 0's and 1's are A353744, counted by A329738.
For sums instead of lengths we have A353849, ones at A353848.
Positions of first appearances are A354906.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A333627 ranks the run-lengths of standard compositions.
A351596 ranks compositions with distinct run-lengths, counted by A329739.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353847 ranks the run-sums of standard compositions.
A353852 ranks compositions with distinct run-sums, counted by A353850.
A353860 counts collapsible compositions.
Cf. A029837, A124767, A181819, A238279, A333381, A333755, A353839, A353851.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[Length/@Split[stc[n]]]],{n,0,100}]

A374254 Numbers k such that the k-th composition in standard order is an anti-run and matches the patterns (1,2,1) or (2,1,2).

Original entry on oeis.org

13, 22, 25, 45, 49, 54, 76, 77, 82, 89, 97, 101, 102, 105, 108, 109, 141, 148, 150, 153, 162, 165, 166, 177, 178, 180, 182, 193, 197, 198, 204, 205, 209, 210, 216, 217, 269, 278, 280, 281, 297, 300, 301, 305, 306, 308, 310, 322, 325, 326, 332, 333, 353, 354

Offset: 1

Gus Wiseman, Jul 14 2024

nonn

Such a composition cannot be strict.

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 their standard compositions begin:
   13: (1,2,1)
   22: (2,1,2)
   25: (1,3,1)
   45: (2,1,2,1)
   49: (1,4,1)
   54: (1,2,1,2)
   76: (3,1,3)
   77: (3,1,2,1)
   82: (2,3,2)
   89: (2,1,3,1)
   97: (1,5,1)
  101: (1,3,2,1)
  102: (1,3,1,2)
  105: (1,2,3,1)
  108: (1,2,1,3)
  109: (1,2,1,2,1)
  141: (4,1,2,1)
  148: (3,2,3)
  150: (3,2,1,2)
  153: (3,1,3,1)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Compositions of this type are counted by A285981.
Permutations of prime indices of this type are counted by A335460.
This is the anti-run complement case of A374249, counted by A274174.
This is the anti-run case of A374253, counted by A335548.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A025047 counts wiggly compositions, ranks A345167.
A066099 lists compositions in standard order.
A124767 counts runs in standard compositions, anti-runs A333381.
A233564 ranks strict compositions, counted by A032020.
A333755 counts compositions by number of runs.
A335454 counts patterns matched by standard compositions.
A335456 counts patterns matched by compositions.
A335462 counts (1,2,1)- and (2,1,2)-matching permutations of prime indices.
A335465 counts minimal patterns avoided by a standard composition.
- A335470 counts (1,2,1)-matching compositions, ranks A335466.
- A335471 counts (1,2,1)-avoiding compositions, ranks A335467.
- A335472 counts (2,1,2)-matching compositions, ranks A335468.
- A335473 counts (2,1,2)-avoiding compositions, ranks A335469.
A373948 encodes run-compression using compositions in standard order.
A373949 counts compositions by run-compressed sum, opposite A373951.
A373953 gives run-compressed sum of standard compositions, excess A373954.
Cf. A106356, A124762, A238130, A238279, A261982, A333175, A333382, A333627, A335463, A335524, A335525.

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

Equals A333489 /\ A374253.

A354906 Position of first appearance of n in A354579 = Number of distinct run-lengths of standard compositions.

Original entry on oeis.org

0, 1, 11, 119, 5615, 251871

Offset: 0

Gus Wiseman, Jun 23 2022

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 their corresponding compositions begin:
       0: ()
       1: (1)
      11: (2,1,1)
     119: (1,1,2,1,1,1)
    5615: (2,2,1,1,1,2,1,1,1,1)
  251871: (1,1,1,2,2,1,1,1,1,2,1,1,1,1,1)

The standard compositions used here are A066099, run-sums A353847/A353932.
The version for partitions is A006939, for run-sums A002110.
For run-sums instead of run-lengths we have A246534 (firsts in A353849).
For runs instead of run-lengths we have A351015 (firsts in A351014).
These are the positions of first appearances in A354579.
A005811 counts runs in binary expansion.
A333627 ranks the run-lengths of standard compositions.
A351596 ranks compositions with distinct run-lengths, counted by A329739.
A353744 ranks compositions with equal run-lengths, counted by A329738.
A353852 ranks compositions with distinct run-sums, counted by A353850.
A353853-A353859 are sequences pertaining to composition run-sum trajectory.
A353860 counts collapsible compositions.
Cf. A003242, A029837, A071625, A124767, A182857, A238279/A333755, A325278, A333381, A333489, A333629.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pd=Table[Length[Union[Length/@Split[stc[n]]]],{n,0,10000}];
Table[Position[pd,n][[1,1]]-1,{n,0,Max@@pd}]

A375407 Numbers k such that the k-th composition in standard order (row k of A066099) matches both of the dashed patterns 23-1 and 1-32.

Original entry on oeis.org

421, 649, 802, 809, 837, 843, 933, 1289, 1299, 1330, 1445, 1577, 1602, 1605, 1617, 1619, 1669, 1673, 1675, 1685, 1686, 1687, 1701, 1826, 1833, 1861, 1867, 1957, 2469, 2569, 2577, 2579, 2597, 2598, 2599, 2610, 2658, 2661, 2674, 2697, 2850, 2857, 2885, 2891

Offset: 1

Gus Wiseman, Aug 23 2024

nonn

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.
These are also numbers k such that:
(1) the maximal weakly increasing runs in the reverse of the k-th composition in standard order do not have weakly decreasing leaders, and
(2) the maximal weakly increasing runs in the k-th composition in standard order do not have weakly decreasing leaders.

			Composition 89 is (2,1,3,1), which matches 2-3-1 but not 23-1.
Composition 165 is (2,3,2,1), which matches 23-1 but not 231.
Composition 358 is (2,1,3,1,2), which matches 2-3-1 and 1-3-2 but not 23-1 or 1-32.
The sequence together with corresponding compositions begins:
   421: (1,2,3,2,1)
   649: (2,4,3,1)
   802: (1,3,4,2)
   809: (1,3,2,3,1)
   837: (1,2,4,2,1)
   843: (1,2,3,2,1,1)
   933: (1,1,2,3,2,1)
  1289: (2,5,3,1)
  1299: (2,4,3,1,1)
  1330: (2,3,1,3,2)
  1445: (2,1,2,3,2,1)
  1577: (1,4,2,3,1)
  1602: (1,3,5,2)
  1605: (1,3,4,2,1)
  1617: (1,3,2,4,1)
  1619: (1,3,2,3,1,1)

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

The non-dashed version is the intersection of A335482 and A335480.
Compositions of this type are counted by A375297.
For leaders of identical runs we have A375408, counted by A332834.
A003242 counts anti-runs, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335486 ranks compositions matching 21, reverse A335485.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Leader is A065120.
- Parts are listed by A066099, reverse A228351.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Run-length transform is A333627, sum A070939.
- Run-counts: A124766, A124765, A124768, A124769, A333381, A124767.
- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
Cf. A188919, A189076, A238343, A333213, A335466, A373948, A373953, A374633, A375123, A375137, A375139, A374768.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],MatchQ[stc[#],{_,y_,z_,_,x_,_}/;x_,x_,_,z_,y_,_}/;x

Intersection of A375138 and A375137.

A375408 Numbers k such that the k-th composition in standard order is not weakly increasing or weakly decreasing.

Original entry on oeis.org

13, 22, 25, 27, 29, 38, 41, 44, 45, 46, 49, 50, 51, 53, 54, 55, 57, 59, 61, 70, 76, 77, 78, 81, 82, 83, 86, 88, 89, 90, 91, 92, 93, 94, 97, 98, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121, 123, 125, 134, 140, 141, 142

Offset: 1

Gus Wiseman, Sep 18 2024

nonn

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 and corresponding compositions begin:
  13: (1,2,1)
  22: (2,1,2)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  38: (3,1,2)
  41: (2,3,1)
  44: (2,1,3)
  45: (2,1,2,1)
  46: (2,1,1,2)
  49: (1,4,1)
  50: (1,3,2)
  51: (1,3,1,1)
  53: (1,2,2,1)
  54: (1,2,1,2)
  55: (1,2,1,1,1)
  57: (1,1,3,1)
  59: (1,1,2,1,1)

Eric Weisstein's World of Mathematics, Unimodal Sequence.

The version for run-lengths of compositions is A332833.
Compositions of this type are counted by A332834, complement maybe A329398.
A001523 counts unimodal compositions, ranks too dense.
A011782 counts compositions.
A114994 ranks weakly decreasing compositions, complement A335485.
A115981 counts non-unimodal compositions, ranked by A335373.
A225620 ranks weakly increasing compositions, complement A335486.
A238130, A238279, A333755 count compositions by number of runs.
A332835 counts compositions with weakly incr. or weakly decr. run-lengths.
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.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of strict compositions are A233564.
- Ranks of constant compositions are A272919.
- Anti-runs are ranked by A333489, counted by A003242.
- Run-length transform is A333627, sum A070939.
Cf. A001511, A002051, A065120, A329744, A332745, A332836, A332870, A333218.

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

Intersection of A335485 and A335486.

cp's OEIS Frontend

A375296 Numbers k such that the leaders of maximal weakly increasing runs in the reverse of the k-th composition in standard order (row k of A228351) are not strictly decreasing.

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A335461 Triangle read by rows where T(n,k) is the number of patterns of length n with k runs.

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A353856 Triangle read by rows where T(n,k) is the number of integer compositions of n with run-sum trajectory (condensation) ending in a composition of length k.

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A375139 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are not weakly decreasing.

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A353857 Numbers k such that the k-th composition in standard order has run-sum trajectory ending in a singleton.

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A354579 Number of distinct lengths of runs in the n-th composition in standard order.

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A374254 Numbers k such that the k-th composition in standard order is an anti-run and matches the patterns (1,2,1) or (2,1,2).

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A354906 Position of first appearance of n in A354579 = Number of distinct run-lengths of standard compositions.

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