A065120 -id:A065120 - cp's OEIS frontend

A358135 Difference of first and last parts of the n-th composition in standard order.

0, 0, 0, 0, -1, 1, 0, 0, -2, 0, -1, 2, 0, 1, 0, 0, -3, -1, -2, 1, -1, 0, -1, 3, 0, 1, 0, 2, 0, 1, 0, 0, -4, -2, -3, 0, -2, -1, -2, 2, -1, 0, -1, 1, -1, 0, -1, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, -5, -3, -4, -1, -3, -2, -3, 1, -2, -1, -2, 0, -2

Offset: 1

Gus Wiseman, Oct 31 2022

sign

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.

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

See link for sequences related to standard compositions.
The first and last parts are A065120 and A001511.
This is the first minus last part of row n of A066099.
The version for Heinz numbers of partitions is A243055.
Row sums of A358133.
The partial sums of standard compositions are A358134, adjusted A242628.
A011782 counts compositions.
A333766 and A333768 give max and min in standard compositions, diff A358138.
A351014 counts distinct runs in standard compositions.
Cf. A000120, A001511, A029837, A029931, A048896, A058891, A070939, A133494, A329395, A357187.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[-First[stc[n]]+Last[stc[n]],{n,1,100}]

a(n) = A001511(n) - A065120(n).

A228350 Triangle read by rows: T(j,k) is the k-th part in nonincreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 6, 5, 4, 4, 3, 3

Offset: 1

Omar E. Pol, Aug 20 2013

Triangle read by rows in which row n lists the A006519(n) elements of the row A001511(n) of triangle A065120, n >= 1.

The equivalent sequence for integer partitions is A206437.

			---------------------------------------------------------
.              Diagram                Triangle
Compositions     of            of compositions (rows)
.   of 5       regions          and regions (columns)
----------------------------------------------------------
.             _ _ _ _ _
.         5  |_        |                                5
.       1+4  |_|_      |                              1 4
.       2+3  |_  |     |                            2   3
.     1+1+3  |_|_|_    |                          1 1   3
.       3+2  |_    |   |                        3       2
.     1+2+2  |_|_  |   |                      1 2       2
.     2+1+2  |_  | |   |                    2   1       2
.   1+1+1+2  |_|_|_|_  |                  1 1   1       2
.       4+1  |_      | |                4               1
.     1+3+1  |_|_    | |              1 3               1
.     2+2+1  |_  |   | |            2   2               1
.   1+1+2+1  |_|_|_  | |          1 1   2               1
.     3+1+1  |_    | | |        3       1               1
.   1+2+1+1  |_|_  | | |      1 2       1               1
.   2+1+1+1  |_  | | | |    2   1       1               1
. 1+1+1+1+1  |_|_|_|_|_|  1 1   1       1               1
.
Also the structure could be represented by an isosceles triangle in which the n-th diagonal gives the n-th region. For the composition of 4 see below:
.             _ _ _ _
.         4  |_      |                  4
.       1+3  |_|_    |                1   3
.       2+2  |_  |   |              2       2
.     1+1+2  |_|_|_  |            1   1       2
.       3+1  |_    | |          3               1
.     1+2+1  |_|_  | |        1   2               1
.     2+1+1  |_  | | |      2       1               1
.   1+1+1+1  |_|_|_|_|    1   1       1               1
.
Illustration of the four sections of the set of compositions of 4:
.                                      _ _ _ _
.                                     |_      |     4
.                                     |_|_    |   1+3
.                                     |_  |   |   2+2
.                       _ _ _         |_|_|_  | 1+1+2
.                      |_    |   3          | |     1
.             _ _      |_|_  | 1+2          | |     1
.     _      |_  | 2       | |   1          | |     1
.    |_| 1     |_| 1       |_|   1          |_|     1
.
.
Illustration of initial terms. The parts of the eight regions of the set of compositions of 4:
--------------------------------------------------------
\j:  1      2    3        4     5      6    7          8
k
--------------------------------------------------------
.  _    _ _    _    _ _ _     _    _ _    _    _ _ _ _
1 |_|1 |_  |2 |_|1 |_    |3  |_|1 |_  |2 |_|1 |_      |4
2        |_|1        |_  |2         |_|1        |_    |3
3                      | |1                       |   |2
4                      |_|1                       |_  |2
5                                                   | |1
6                                                   | |1
7                                                   | |1
8                                                   |_|1
.
Triangle begins:
1;
2,1;
1;
3,2,1,1;
1;
2,1;
1;
4,3,2,2,1,1,1,1;
1;
2,1;
1;
3,2,1,1;
1;
2,1;
1;
5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1;
...
.
Also triangle read by rows T(n,m) in which row n lists the parts of the n-th section of the set of compositions of the integers >= n, ordered by regions. Row lengths give A045623. Row sums give A001792 (see below):
[1];
[2,1];
[1],[3,2,1,1];
[1],[2,1],[1],[4,3,2,2,1,1,1,1];
[1],[2,1],[1],[3,2,1,1],[1],[2,1],[1],[5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1];

Main triangle: column 1 is A001511. Row j has length A006519(j). Row sums give A038712.

Cf. A001787, A001792, A011782, A029837, A045623, A065120, A070939, A135010, A141285, A187816, A187818, A193870, A206437, A228347, A228348, A228349, A228351, A228366, A228367, A228370, A228371, A228525, A228526.

T(j,k) = A065120(A001511(j)),k) = A001511(j) - A029837(k), 1<=k<=A006519(j), j>=1.

A374516 Sum of leaders of maximal anti-runs in the n-th composition in standard order.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 1, 3, 4, 3, 4, 3, 1, 1, 2, 4, 5, 4, 3, 4, 2, 4, 2, 4, 1, 1, 3, 2, 2, 2, 3, 5, 6, 5, 4, 5, 6, 3, 3, 5, 2, 2, 6, 5, 2, 2, 3, 5, 1, 1, 1, 2, 1, 3, 1, 3, 2, 2, 4, 3, 3, 3, 4, 6, 7, 6, 5, 6, 4, 4, 4, 6, 3, 6, 5, 4, 3, 3, 4, 6, 2, 2, 2, 3, 4, 6, 4

Offset: 0

Gus Wiseman, Jul 31 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 1234567th composition in standard order is (3,2,1,2,2,1,2,5,1,1,1), with maximal anti-runs ((3,2,1,2),(2,1,2,5,1),(1),(1)), so a(1234567) is 3 + 2 + 1 + 1 = 7.

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

For length instead of sum we have A333381.
Row-sums of A374515.
Other types of runs (instead of anti-):
- For identical runs we have A373953, row-sums of A374251.
- For weakly increasing runs we have A374630, row-sums of A374629.
- For strictly increasing runs we have A374684, row-sums of A374683.
- For weakly decreasing runs we have A374741, row-sums of A374740.
- For strictly decreasing runs we have A374758, row-sums of A374757.
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.
- Ranks of non-contiguous compositions are A374253, counted by A335548.
Six types of maximal runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A029931, A228351, A233564, A238343, A272919, A373949, A374517, A374518, A374519, A374638.

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

A375138 Numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed pattern 23-1.

Original entry on oeis.org

41, 81, 83, 105, 145, 161, 163, 165, 166, 167, 169, 209, 211, 233, 289, 290, 291, 297, 321, 323, 325, 326, 327, 329, 331, 332, 333, 334, 335, 337, 339, 361, 401, 417, 419, 421, 422, 423, 425, 465, 467, 489, 545, 553, 577, 578, 579, 581, 582, 583, 593, 595, 617

Offset: 1

Gus Wiseman, Aug 09 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 the maximal weakly increasing runs in the reverse of the k-th composition in standard order do not have weakly decreasing leaders, where the leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.
The reverse version (A375137) ranks compositions matching the dashed pattern 1-32.

			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:
   41: (2,3,1)
   81: (2,4,1)
   83: (2,3,1,1)
  105: (1,2,3,1)
  145: (3,4,1)
  161: (2,5,1)
  163: (2,4,1,1)
  165: (2,3,2,1)
  166: (2,3,1,2)
  167: (2,3,1,1,1)
  169: (2,2,3,1)
  209: (1,2,4,1)
  211: (1,2,3,1,1)
  233: (1,1,2,3,1)

Wikipedia, Permutation pattern.

The complement is too dense, but counted by A189076.
The non-dashed version is A335482, reverse A335480.
For leaders of identical runs we have A335486, reverse A335485.
Compositions of this type are counted by A374636.
The reverse version is A375137, counted by A374636.
Matching 12-1 also gives A375296, counted by A375140 (complement A188920).
A003242 counts anti-runs, ranks A333489.
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).
- 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, A238343, A333213, A335466, A373948, A373953, A374633, A375123, A375139, A374768.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],MatchQ[stc[#],{_,y_,z_,_,x_,_}/;x

A187816 Triangle read by rows in which row n lists the first 2^(n-1) terms of A006519 in nonincreasing order, n >= 1.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 1, 8, 4, 2, 2, 1, 1, 1, 1, 16, 8, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 32, 16, 8, 8, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 64, 32, 16, 16, 8, 8, 8, 8, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2

Offset: 1

Omar E. Pol, Sep 10 2013

T(n,k) is also the number of parts in the k-th largest region of the diagram of regions of the set of compositions of n, n >= 1, k >= 1, see example.
Row lengths is A000079.
Row sums give A001792(n-1).

			For n = 5 the diagram of regions of the set of compositions of 5 has 2^(5-1) regions, see below:
------------------------------------------------------
.          A006519
.         as a tree
.         of number        Diagram
Region    of parts       of regions     Composition
------------------------------------------------------
.                         _ _ _ _ _
1      | 1          |    |_| | | | |    1, 1, 1, 1, 1
2      |   2        |    |_ _| | | |    2, 1, 1, 1
3      | 1          |    |_|   | | |    1, 2, 1, 1
4      |      4     |    |_ _ _| | |    3, 1, 1
5      | 1          |    |_| |   | |    1, 1, 2, 1
6      |   2        |    |_ _|   | |    2, 2, 1
7      | 1          |    |_|     | |    1, 3, 1
8      |        8   |    |_ _ _ _| |    4, 1
9      | 1          |    |_| | |   |    1, 1, 1, 2
10     |   2        |    |_ _| |   |    2, 1, 2
11     | 1          |    |_|   |   |    1, 2, 2
12     |      4     |    |_ _ _|   |    3, 2
13     | 1          |    |_| |     |    1, 1, 3
14     |   2        |    |_ _|     |    2, 3
15     | 1          |    |_|       |    1, 4
16     |         16 |    |_ _ _ _ _|    5
.
The first largest region in the diagram is the 16th region which contains 16 parts, so T(5,1) = 16. The second largest region is the 8th region which contains 8 parts, so T(5,2) = 8. The third and the fourth largest regions are both the 4th region and the 12th region, each contains 4 parts, so T(5,3) = 4 and T(5,4) = 4. And so on. The sequence of the number of parts of the k-th largest region of the diagram is [16, 8, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1], the same as the 5th row of triangle, as shown below.
Triangle begins:
1;
2,1;
4,2,1,1;
8,4,2,2,1,1,1,1;
16,8,4,4,2,2,2,2,1,1,1,1,1,1,1,1;
32,16,8,8,4,4,4,4,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
...

Cf. A000079, A001511, A001792, A006519, A011782, A065120, A187818, A228525, A228369.

A187818 Triangle read by rows in which row n lists the first 2^(n-1) terms of A038712 in nonincreasing order, n >= 1.

Original entry on oeis.org

1, 3, 1, 7, 3, 1, 1, 15, 7, 3, 3, 1, 1, 1, 1, 31, 15, 7, 7, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 63, 31, 15, 15, 7, 7, 7, 7, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 127, 63, 31, 31, 15, 15, 15, 15, 7, 7, 7, 7, 7, 7, 7, 7, 3, 3

Offset: 1

Omar E. Pol, Sep 10 2013

T(n,k) is also the sum of all parts of the k-th largest region of the diagram of regions of the set of compositions of n, n >= 1, k >= 1, see example.
Row lengths is A000079.
Row sums give A001787, n >= 1.

			For n = 5 the diagram of regions of the set of compositions of 5 has 2^(5-1) regions, see below:
------------------------------------------------------
.         A038712 as
.       a tree of sum      Diagram
Region   of all parts    of regions     Composition
------------------------------------------------------
.                         _ _ _ _ _
1      | 1          |    |_| | | | |    1, 1, 1, 1, 1
2      |   3        |    |_ _| | | |    2, 1, 1, 1
3      | 1          |    |_|   | | |    1, 2, 1, 1
4      |      7     |    |_ _ _| | |    3, 1, 1
5      | 1          |    |_| |   | |    1, 1, 2, 1
6      |   3        |    |_ _|   | |    2, 2, 1
7      | 1          |    |_|     | |    1, 3, 1
8      |       15   |    |_ _ _ _| |    4, 1
9      | 1          |    |_| | |   |    1, 1, 1, 2
10     |   3        |    |_ _| |   |    2, 1, 2
11     | 1          |    |_|   |   |    1, 2, 2
12     |      7     |    |_ _ _|   |    3, 2
13     | 1          |    |_| |     |    1, 1, 3
14     |   3        |    |_ _|     |    2, 3
15     | 1          |    |_|       |    1, 4
16     |         31 |    |_ _ _ _ _|    5
.
The first largest region in the diagram is the 16th region which contains 16 parts and the sum of parts is 31, so T(5,1) = 31. The second largest region is the 8th region which contains 8 parts and the sum of parts is 15, so T(5,2) = 15. The third and the fourth largest regions are both the 4th region and the 12th region, each contains 4 parts and the sum of parts is 7, so T(5,3) = 7 and T(5,4) = 7. And so on. The sequence of the sum of all parts of the k-th largest region of the diagram is [31, 15, 7, 7, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1], the same as the 5th row of triangle, as shown below.
Triangle begins:
1;
3,1;
7,3,1,1;
15,7,3,3,1,1,1,1;
31,15,7,7,3,3,3,3,1,1,1,1,1,1,1,1;
63,31,15,15,7,7,7,7,3,3,3,3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
...

Cf. A000079, A000225, A001511, A001787, A001792, A006519, A011782, A038712, A065120, A187816, A228525, A228369.

A358170 Heinz number of the partial sums of the n-th composition in standard order (A066099).

Original entry on oeis.org

1, 2, 3, 6, 5, 15, 10, 30, 7, 35, 21, 105, 14, 70, 42, 210, 11, 77, 55, 385, 33, 231, 165, 1155, 22, 154, 110, 770, 66, 462, 330, 2310, 13, 143, 91, 1001, 65, 715, 455, 5005, 39, 429, 273, 3003, 195, 2145, 1365, 15015, 26, 286, 182, 2002, 130, 1430, 910, 10010

Offset: 0

Gus Wiseman, Dec 20 2022

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 Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
           1: {}
           2: {1}
           3: {2}
           6: {1,2}
           5: {3}
          15: {2,3}
          10: {1,3}
          30: {1,2,3}
           7: {4}
          35: {3,4}
          21: {2,4}
         105: {2,3,4}
          14: {1,4}
          70: {1,3,4}
          42: {1,2,4}
         210: {1,2,3,4}

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

See link for sequences related to standard compositions.
Applying A001221 or A001222 gives A000120.
The image is A005117 (squarefree numbers).
The reverse version is A019565, triangular version A048793.
Greatest prime index of a(n) is A029837 or A070939.
Least prime index of a(n) is A065120.
The adjusted version is A253565, inverse A253566, reverse A005940.
These are the Heinz numbers of the rows of A358134.
Sum of prime indices of a(n) is A359042.
A066099 lists standard compositions.
A112798 list prime indices, sum A056239.
Cf. A000720, A001511, A029931, A059893, A061395, A241916, A242628, A355536, A358133, A358135, A358137.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Times@@Prime/@#&/@Table[Accumulate[stc[n]],{n,0,100}]

A374639 Numbers k such that the leaders of maximal anti-runs in the k-th composition in standard order (A066099) are not distinct.

Original entry on oeis.org

3, 7, 10, 14, 15, 21, 23, 27, 28, 29, 30, 31, 36, 39, 42, 43, 47, 51, 55, 56, 57, 58, 59, 60, 61, 62, 63, 71, 73, 79, 84, 85, 86, 87, 90, 94, 95, 99, 103, 106, 107, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 135

Offset: 1

Gus Wiseman, Aug 06 2024

nonn

The leaders of maximal 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 sequence of terms together with the corresponding compositions begins:
   3: (1,1)
   7: (1,1,1)
  10: (2,2)
  14: (1,1,2)
  15: (1,1,1,1)
  21: (2,2,1)
  23: (2,1,1,1)
  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)

First differs from A335466 in lacking 166, complement A335467.
The complement for leaders of identical runs is A374249, counted by A274174.
For leaders of identical runs we have A374253, counted by A335548.
Positions of non-distinct (or non-strict) rows in A374515.
The complement is A374638, counted by A374518.
For identical instead of non-distinct we have A374519, counted by A374517.
For identical instead of distinct we have A374520, counted by A374640.
Compositions of this type are counted by A374678.
Other functional neighbors are A374768, A374698, A374701, A374767.
A065120 gives leaders of standard compositions.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
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, A335466, A373949.

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

A375295 Numbers k such that the leaders of maximal weakly increasing runs in the k-th composition in standard order (row k of A066099) are not strictly decreasing.

Original entry on oeis.org

13, 25, 27, 29, 45, 49, 50, 51, 53, 54, 55, 57, 59, 61, 77, 82, 89, 91, 93, 97, 98, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121, 123, 125, 141, 153, 155, 157, 162, 165, 173, 177, 178, 179, 181, 182, 183, 185, 187, 189

Offset: 1

Gus Wiseman, Aug 12 2024

nonn

First differs from the non-dashed version in lacking 166, corresponding to the composition (2,3,1,2).
The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.
Also numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed patterns 1-32 or 1-21.

			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)
  45: (2,1,2,1)
  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)
  61: (1,1,1,2,1)
  77: (3,1,2,1)
  82: (2,3,2)
  89: (2,1,3,1)
  91: (2,1,2,1,1)
  93: (2,1,1,2,1)

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 A335485.
Positions of non-strictly decreasing rows in A374629 (sums A374630).
For identical leaders we have A374633, counted by A374631.
Matching 1-32 only gives A375137, reverse A375138, both counted by A374636.
Interchanging weak/strict gives A375139, counted by A375135.
Compositions of this type are counted by A375140, complement A188920.
The reverse version is A375296.
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, A261982, A333213, A335480, A335482, A373948, A374746, A374768, A375123.

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

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.

Original entry on oeis.org

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

cp's OEIS Frontend

A358135 Difference of first and last parts of the n-th composition in standard order.

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A228350 Triangle read by rows: T(j,k) is the k-th part in nonincreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).

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A374516 Sum of leaders of maximal anti-runs in the n-th composition in standard order.

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A375138 Numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed pattern 23-1.

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A187816 Triangle read by rows in which row n lists the first 2^(n-1) terms of A006519 in nonincreasing order, n >= 1.

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A187818 Triangle read by rows in which row n lists the first 2^(n-1) terms of A038712 in nonincreasing order, n >= 1.

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A358170 Heinz number of the partial sums of the n-th composition in standard order (A066099).

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A374639 Numbers k such that the leaders of maximal anti-runs in the k-th composition in standard order (A066099) are not distinct.

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A375295 Numbers k such that the leaders of maximal weakly increasing runs in the k-th composition in standard order (row k of A066099) are not strictly decreasing.

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