A066099 -id:A066099 - cp's OEIS frontend

A353847 Composition run-sum transformation in terms of standard composition numbers. The a(k)-th composition in standard order is the sequence of run-sums of the k-th composition in standard order. Takes each index of a row of A066099 to the index of the row consisting of its run-sums.

0, 1, 2, 2, 4, 5, 6, 4, 8, 9, 8, 10, 12, 13, 10, 8, 16, 17, 18, 18, 20, 17, 22, 20, 24, 25, 24, 26, 20, 21, 18, 16, 32, 33, 34, 34, 32, 37, 38, 36, 40, 41, 32, 34, 44, 45, 42, 40, 48, 49, 50, 50, 52, 49, 54, 52, 40, 41, 40, 42, 36, 37, 34, 32, 64, 65, 66, 66

Offset: 0

Gus Wiseman, May 30 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 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.

			As a triangle:
   0
   1
   2  2
   4  5  6  4
   8  9  8 10 12 13 10  8
  16 17 18 18 20 17 22 20 24 25 24 26 20 21 18 16
These are the standard composition numbers of the following compositions (transposed):
  ()  (1)  (2)  (3)    (4)      (5)
           (2)  (2,1)  (3,1)    (4,1)
                (1,2)  (4)      (3,2)
                (3)    (2,2)    (3,2)
                       (1,3)    (2,3)
                       (1,2,1)  (4,1)
                       (2,2)    (2,1,2)
                       (4)      (2,3)
                                (1,4)
                                (1,3,1)
                                (1,4)
                                (1,2,2)
                                (2,3)
                                (2,2,1)
                                (3,2)
                                (5)

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

Standard compositions are listed by A066099.
The version for partitions is A353832.
The run-sums themselves are listed by A353932, with A353849 distinct terms.
A005811 counts runs in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353840-A353846 pertain to partition run-sum trajectory.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
A353863 counts run-sum-complete partitions.
Cf. A003242. A175413, A181819, A238279, A274174, A333381, A333489, A333755, A353835, A353839, A353864.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Total/@Split[stc[n]]],{n,0,100}]

A335465 Number of minimal normal patterns avoided by the n-th composition in standard order (A066099).

Original entry on oeis.org

1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 12, 4, 3, 3, 3, 3, 4, 3, 4, 12, 4, 3, 12, 4, 12, 4, 12, 4, 3, 3, 3, 3, 4, 3, 3, 6, 4, 3, 6, 3, 3, 6, 10, 10, 4, 3, 12, 6, 12, 3, 10, 10, 12, 4, 12, 3, 12, 4, 12, 4, 3, 3, 3, 3, 4, 3, 3, 6

Offset: 0

Gus Wiseman, Jun 20 2020

These patterns comprise the basis of the class of patterns generated by this composition.
We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).
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 bases of classes generated by (), (1), (2,1,1), (3,1,2), (2,1,2,1), and (1,2,1), corresponding to n = 0, 1, 11, 38, 45, 13, are the respective columns below.
  (1)  (1,1)  (1,2)    (1,1)    (1,1,1)    (1,1,1)
       (1,2)  (1,1,1)  (1,2,3)  (1,1,2)    (1,1,2)
       (2,1)  (2,2,1)  (1,3,2)  (1,2,2)    (1,2,2)
              (3,2,1)  (2,1,3)  (1,2,3)    (1,2,3)
                       (2,3,1)  (1,3,2)    (1,3,2)
                       (3,2,1)  (2,1,3)    (2,1,1)
                                (2,3,1)    (2,1,2)
                                (3,1,2)    (2,1,3)
                                (3,2,1)    (2,2,1)
                                (2,2,1,1)  (2,3,1)
                                           (3,1,2)
                                           (3,2,1)

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

Patterns matched by standard compositions are counted by A335454.
Patterns matched by compositions of n are counted by A335456(n).
The version for Heinz numbers of partitions is A335550.
Patterns are counted by A000670 and ranked by A333217.
Knapsack compositions are counted by A325676 and ranked by A333223.
The n-th composition has A334299(n) distinct subsequences.
Cf. A056986, A124767, A124770, A124771, A269134, A333218, A333257, A335549.

A344618 Reverse-alternating sums of standard compositions (A066099). Alternating sums of the compositions ranked by A228351.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 03 2021

Up to sign, same as A124754.
The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
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 nonnegative integers together with the corresponding standard compositions and their reverse-alternating sums begins:
  0:     () ->  0    15: (1111) ->  0    30:  (1112) ->  1
  1:    (1) ->  1    16:    (5) ->  5    31: (11111) ->  1
  2:    (2) ->  2    17:   (41) -> -3    32:     (6) ->  6
  3:   (11) ->  0    18:   (32) -> -1    33:    (51) -> -4
  4:    (3) ->  3    19:  (311) ->  3    34:    (42) -> -2
  5:   (21) -> -1    20:   (23) ->  1    35:   (411) ->  4
  6:   (12) ->  1    21:  (221) ->  1    36:    (33) ->  0
  7:  (111) ->  1    22:  (212) ->  3    37:   (321) ->  2
  8:    (4) ->  4    23: (2111) -> -1    38:   (312) ->  4
  9:   (31) -> -2    24:   (14) ->  3    39:  (3111) -> -2
  10:  (22) ->  0    25:  (131) -> -1    40:    (24) ->  2
  11: (211) ->  2    26:  (122) ->  1    41:   (231) ->  0
  12:  (13) ->  2    27: (1211) ->  1    42:   (222) ->  2
  13: (121) ->  0    28:  (113) ->  3    43:  (2211) ->  0
  14: (112) ->  2    29: (1121) -> -1    44:   (213) ->  4
Triangle begins (row lengths A011782):
  0
  1
  2  0
  3 -1  1  1
  4 -2  0  2  2  0  2  0
  5 -3 -1  3  1  1  3 -1  3 -1  1  1  3 -1  1  1

Up to sign, same as the reverse version A124754.
The version for Heinz numbers of partitions is A344616.
Positions of zeros are A344619.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A116406 counts compositions with alternating sum >= 0.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
All of the following pertain to compositions in standard order:
- The length is A000120.
- Converting to reversed ranking gives A059893.
- The rows are A066099.
- The sum is A070939.
- The runs are counted by A124767.
- The reversed version is A228351.
- Strict compositions are ranked by A233564.
- Constant compositions are ranked by A272919.
- The Heinz number is A333219.
- Anti-run compositions are ranked by A333489.
Cf. A000070, A000097, A003242, A028260, A119899, A239830, A344605, A344607, A344608, A344650, A344739.

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

A335454 Number of normal patterns matched by the n-th composition in standard order (A066099).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 5, 3, 6, 5, 5, 2, 3, 3, 5, 3, 5, 6, 7, 3, 6, 5, 9, 5, 9, 7, 6, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 4, 7, 5, 10, 9, 9, 3, 6, 5, 9, 4, 9, 10, 12, 5, 9, 7, 13, 7, 12, 9, 7, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 5, 7, 6, 10, 9, 9, 3, 5, 6, 8, 5

Offset: 0

Gus Wiseman, Jun 14 2020

We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

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 a(n) patterns for n = 0, 1, 3, 7, 11, 13, 23, 83, 27, 45:
  0:  1:   11:   111:   211:   121:   2111:   2311:   1211:   2121:
---------------------------------------------------------------------
  ()  ()   ()    ()     ()     ()     ()      ()      ()      ()
      (1)  (1)   (1)    (1)    (1)    (1)     (1)     (1)     (1)
           (11)  (11)   (11)   (11)   (11)    (11)    (11)    (11)
                 (111)  (21)   (12)   (21)    (12)    (12)    (12)
                        (211)  (21)   (111)   (21)    (21)    (21)
                               (121)  (211)   (211)   (111)   (121)
                                      (2111)  (231)   (121)   (211)
                                              (2311)  (211)   (212)
                                                      (1211)  (221)
                                                              (2121)

John Tyler Rascoe, Table of n, a(n) for n = 0..8192
Wikipedia, Permutation pattern
Gus Wiseman, Statistics, classes, and transformations of standard compositions
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

References found in the links are not all included here.
Summing over indices with binary length n gives A335456(n).
The contiguous case is A335458.
The version for Heinz numbers of partitions is A335549.
Patterns are counted by A000670 and ranked by A333217.
The n-th composition has A124771(n) distinct consecutive subsequences.
Knapsack compositions are counted by A325676 and ranked by A333223.
The n-th composition has A333257(n) distinct subsequence-sums.
The n-th composition has A334299(n) distinct subsequences.
Minimal avoided patterns are counted by A335465.
Cf. A034691, A056986, A108917, A124767, A124770, A158005, A269134, A333218, A333222, A333224, A334030.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Length[Union[mstype/@Subsets[stc[n]]]],{n,0,30}]

from itertools import combinations
def comp(n):
    # see A357625
    return
def A335465(n):
    A,B,C = set(),set(),comp(n)
    c = range(len(C))
    for j in c:
        for k in combinations(c, j):
            A.add(tuple(C[i] for i in k))
    for i in A:
        D = {v: rank + 1 for rank, v in enumerate(sorted(set(i)))}
        B.add(tuple(D[v] for v in i))
    return len(B)+1 # John Tyler Rascoe, Mar 12 2025

A344619 The a(n)-th composition in standard order (A066099) has alternating sum 0.

Original entry on oeis.org

0, 3, 10, 13, 15, 36, 41, 43, 46, 50, 53, 55, 58, 61, 63, 136, 145, 147, 150, 156, 162, 165, 167, 170, 173, 175, 180, 185, 187, 190, 196, 201, 203, 206, 210, 213, 215, 218, 221, 223, 228, 233, 235, 238, 242, 245, 247, 250, 253, 255, 528, 545, 547, 550, 556, 568

Offset: 1

Gus Wiseman, Jun 03 2021

nonn

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

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:
    0: ()
    3: (1,1)
   10: (2,2)
   13: (1,2,1)
   15: (1,1,1,1)
   36: (3,3)
   41: (2,3,1)
   43: (2,2,1,1)
   46: (2,1,1,2)
   50: (1,3,2)
   53: (1,2,2,1)
   55: (1,2,1,1,1)
   58: (1,1,2,2)
   61: (1,1,1,2,1)
   63: (1,1,1,1,1,1)
  136: (4,4)
  145: (3,4,1)
  147: (3,3,1,1)
  150: (3,2,1,2)
  156: (3,1,1,3)

The version for Heinz numbers of partitions is A000290, counted by A000041.
These are the positions of zeros in A344618.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A116406 counts compositions with alternating sum >= 0.
A124754 gives the alternating sum of standard compositions.
A316524 is the alternating sum of the prime indices of n.
A344604 counts wiggly compositions with twins.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A344616 gives the alternating sum of reversed prime indices.
All of the following pertain to compositions in standard order:
- The length is A000120.
- Converting to reversed ranking gives A059893.
- The rows are A066099.
- The sum is A070939.
- The runs are counted by A124767.
- The reversed version is A228351.
- Strict compositions are ranked by A233564.
- Constant compositions are ranked by A272919.
- The Heinz number is A333219.
- Anti-run compositions are ranked by A333489.
Cf. A000070, A000097, A003242, A006330, A028260, A119899, A239830, A344605, A344607, A344650, A344739.

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

A353848 Numbers k such that the k-th composition in standard order (row k of A066099) has all equal run-sums.

Original entry on oeis.org

0, 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, 143, 168, 170, 175, 187, 238, 248, 250, 255, 256, 292, 316, 487, 511, 512, 528, 543, 682, 750, 955, 1008, 1023, 1024, 2047, 2048, 2080, 2084, 2090, 2111, 2184

Offset: 0

Gus Wiseman, May 30 2022

nonn

Every sequence can be uniquely split into non-overlapping runs, read left-to-right. 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 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:
     0:       0  ()
     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)
For example:
- The 59th composition in standard order is (1,1,2,1,1), with run-sums (2,2,2), so 59 is in the sequence.
- The 2298th composition in standard order is (4,1,1,1,1,2,2), with run-sums (4,4,4), so 2298 is in the sequence.
- The 2346th composition in standard order is (3,3,2,2,2), with run-sums (6,6), so 2346 is in the sequence.

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

Standard compositions are listed by A066099.
For equal lengths instead of sums we have A353744, counted by A329738.
The version for partitions is A353833, counted by A304442.
These compositions are counted by A353851.
The distinct instead of equal version is A353852, counted by A353850.
The run-sums themselves are listed by A353932, with A353849 distinct terms.
A005811 counts runs in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents the run-sum transformation for compositions.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
A353863 counts run-sum-complete partitions.
Cf. A003242, A047966, A106356, A140690, A238279, A274174, A333381, A333489, A333755, A353832, A353864.

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

A353849(a(n)) = 1.

A353852 Numbers k such that the k-th composition in standard order (row k of A066099) has all distinct run-sums.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 47, 48, 50, 51, 52, 55, 56, 57, 58, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 79, 80, 81, 84, 85, 86, 87, 88

Offset: 0

Gus Wiseman, May 31 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.

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 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)
  12:     1100  (1,3)
  15:     1111  (1,1,1,1)
  16:    10000  (5)
  17:    10001  (4,1)
  18:    10010  (3,2)
  19:    10011  (3,1,1)
  20:    10100  (2,3)
  21:    10101  (2,2,1)
  23:    10111  (2,1,1,1)

The version for runs in binary expansion is A175413.
The version for parts instead of run-sums is A233564, counted A032020.
The version for run-lengths instead of run-sums is A351596, counted A329739.
The version for runs instead of run-sums is A351290, counted by A351013.
The version for partitions is A353838, counted A353837, complement A353839.
The equal instead of distinct version is A353848, counted by A353851.
These compositions are counted by A353850.
The weak version (rucksack compositions) is A354581, counted by A354580.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A242882 counts composition with distinct multiplicities, partitions A098859.
A304442 counts partitions with all equal run-sums.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353853-A353859 pertain to composition run-sum trajectory.
A353864 counts rucksack partitions, perfect A353865.
A353929 counts distinct runs in binary expansion, firsts A353930.
Cf. A044813, A238279, A333755, A351016, A351017, A353832, A353847, A353849, A353860, A353863, A353932.

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

A333228 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are pairwise coprime, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Gus Wiseman, May 28 2020

nonn

First differs from A291166 in lacking 69, which corresponds to the composition (4,2,1).
We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).
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 together with the corresponding compositions begins:
   1: (1)          21: (2,2,1)        39: (3,1,1,1)
   3: (1,1)        22: (2,1,2)        41: (2,3,1)
   5: (2,1)        23: (2,1,1,1)      43: (2,2,1,1)
   6: (1,2)        24: (1,4)          44: (2,1,3)
   7: (1,1,1)      25: (1,3,1)        45: (2,1,2,1)
   9: (3,1)        26: (1,2,2)        46: (2,1,1,2)
  11: (2,1,1)      27: (1,2,1,1)      47: (2,1,1,1,1)
  12: (1,3)        28: (1,1,3)        48: (1,5)
  13: (1,2,1)      29: (1,1,2,1)      49: (1,4,1)
  14: (1,1,2)      30: (1,1,1,2)      50: (1,3,2)
  15: (1,1,1,1)    31: (1,1,1,1,1)    51: (1,3,1,1)
  17: (4,1)        33: (5,1)          52: (1,2,3)
  18: (3,2)        35: (4,1,1)        53: (1,2,2,1)
  19: (3,1,1)      37: (3,2,1)        54: (1,2,1,2)
  20: (2,3)        38: (3,1,2)        55: (1,2,1,1,1)

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

Pairwise coprime or singleton partitions are A051424.
Coprime or singleton sets are ranked by A087087.
The version for relatively prime instead of coprime appears to be A291166.
Numbers whose binary indices are pairwise coprime are A326675.
Coprime partitions are counted by A327516.
Not ignoring repeated parts gives A333227.
The complement is A335238.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Product is A124758.
- Reverse is A228351
- GCD is A326674.
- Heinz number is A333219.
- LCM is A333226.
- Number of distinct parts is A334028.
Cf. A007360, A048793, A101268, A233564, A272919, A291166, A302569, A335235, A335236, A335237, A335239.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,120],CoprimeQ@@Union[stc[#]]&]

A335458 Number of normal patterns contiguously matched by the n-th composition in standard order (A066099).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 5, 3, 5, 5, 5, 2, 3, 3, 5, 3, 5, 5, 7, 3, 5, 5, 8, 5, 8, 7, 6, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 4, 7, 5, 7, 8, 9, 3, 5, 5, 8, 4, 8, 7, 11, 5, 8, 7, 11, 7, 11, 9, 7, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 5, 7, 5, 7, 8, 9, 3, 5, 5, 8, 5, 7

Offset: 0

Gus Wiseman, Jun 21 2020

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.

We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(180) = 7 patterns are: (), (1), (1,2), (2,1), (1,2,3), (2,1,2), (2,1,2,3).

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

The non-contiguous version is A335454.
Summing over indices with binary length n gives A335457(n).
The nonempty version is A335474.
Patterns are counted by A000670 and ranked by A333217.
The n-th composition has A124771(n) distinct consecutive subsequences.
Knapsack compositions are counted by A325676 and ranked by A333223.
The n-th composition has A333257(n) distinct subsequence-sums.
The n-th composition has A334299(n) distinct subsequences.
Minimal avoided patterns are counted by A335465.
Cf. A108917, A124767, A124770, A181796, A269134, A333218, A333222, A333628, A334030, A335456.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Length[Union[mstype/@ReplaceList[stc[n],{_,s___,_}:>{s}]]],{n,0,30}]

a(n) = A335474(n) + 1.

A335373 Numbers k such that the k-th composition in standard order (A066099) is not unimodal.

Original entry on oeis.org

22, 38, 44, 45, 46, 54, 70, 76, 77, 78, 86, 88, 89, 90, 91, 92, 93, 94, 102, 108, 109, 110, 118, 134, 140, 141, 142, 148, 150, 152, 153, 154, 155, 156, 157, 158, 166, 172, 173, 174, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 198

Offset: 1

Gus Wiseman, Jun 03 2020

nonn

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

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 together with the corresponding compositions begins:
  22: (2,1,2)
  38: (3,1,2)
  44: (2,1,3)
  45: (2,1,2,1)
  46: (2,1,1,2)
  54: (1,2,1,2)
  70: (4,1,2)
  76: (3,1,3)
  77: (3,1,2,1)
  78: (3,1,1,2)
  86: (2,2,1,2)
  88: (2,1,4)
  89: (2,1,3,1)
  90: (2,1,2,2)
  91: (2,1,2,1,1)
  92: (2,1,1,3)
  93: (2,1,1,2,1)
  94: (2,1,1,1,2)

The dual version (non-co-unimodal compositions) is A335374.
The case that is not co-unimodal either is A335375.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Unimodal permutations are A011782.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers with non-unimodal unsorted prime signature are A332282.
Partitions with non-unimodal 0-appended first differences are A332284.
Non-unimodal permutations of the multiset of prime indices of n are A332671.
Cf. A000120, A029931, A048793, A066099, A070939, A334299.
Cf. A072704, A332281, A332286, A332287, A332639, A332642, A332669, A332672.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,200],!unimodQ[stc[#]]&]

cp's OEIS Frontend

A353847 Composition run-sum transformation in terms of standard composition numbers. The a(k)-th composition in standard order is the sequence of run-sums of the k-th composition in standard order. Takes each index of a row of A066099 to the index of the row consisting of its run-sums.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A335465 Number of minimal normal patterns avoided by the n-th composition in standard order (A066099).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A344618 Reverse-alternating sums of standard compositions (A066099). Alternating sums of the compositions ranked by A228351.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A335454 Number of normal patterns matched by the n-th composition in standard order (A066099).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

A344619 The a(n)-th composition in standard order (A066099) has alternating sum 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A353848 Numbers k such that the k-th composition in standard order (row k of A066099) has all equal run-sums.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A353852 Numbers k such that the k-th composition in standard order (row k of A066099) has all distinct run-sums.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A333228 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are pairwise coprime, where a singleton is not considered coprime unless it is (1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A335458 Number of normal patterns contiguously matched by the n-th composition in standard order (A066099).

Views