A333489 -id:A333489 - cp's OEIS frontend

A333381 Number of maximal anti-runs of the n-th composition in standard order.

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

Offset: 0

Gus Wiseman, Mar 24 2020

nonn

Anti-runs are sequences without any adjacent equal terms.
A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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.
For n > 0, also one plus the number of adjacent equal pairs in the n-th composition in standard order.

			The 46th composition in standard order is (2,1,1,2), with maximal anti-runs ((2,1),(1,2)), so a(46) = 2.

Wikipedia, Longest increasing subsequence

Anti-runs summing to n are counted by A003242(n).
A triangle counting maximal anti-runs of compositions is A106356.
A triangle counting maximal runs of compositions is A238279.
Partitions whose first differences are an anti-run are A238424.
All of the following pertain to compositions in standard order (A066099):
- Adjacent equal pairs are counted by A124762.
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769.
- Strict compositions are ranked by A233564.
- Constant compositions are ranked by A272919.
- Normal compositions are ranked by A333217.
- Adjacent unequal pairs are counted by A333382.
- Anti-runs are ranked by A333489.
Cf. A000120, A029931, A048793, A059893, A070939, A114994, A225620, A228351.

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

For n > 0, a(n) = A124762(n) + 1.

A351295 Numbers whose multiset of prime factors has no permutation with all distinct run-lengths.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140

Offset: 1

Gus Wiseman, Feb 16 2022

nonn

First differs from A130092 (non-Wilf partitions) in lacking 216.

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:
      6: (2,1)         46: (9,1)         84: (4,2,1,1)
     10: (3,1)         51: (7,2)         85: (7,3)
     14: (4,1)         55: (5,3)         86: (14,1)
     15: (3,2)         57: (8,2)         87: (10,2)
     21: (4,2)         58: (10,1)        90: (3,2,2,1)
     22: (5,1)         60: (3,2,1,1)     91: (6,4)
     26: (6,1)         62: (11,1)        93: (11,2)
     30: (3,2,1)       65: (6,3)         94: (15,1)
     33: (5,2)         66: (5,2,1)       95: (8,3)
     34: (7,1)         69: (9,2)        100: (3,3,1,1)
     35: (4,3)         70: (4,3,1)      102: (7,2,1)
     36: (2,2,1,1)     74: (12,1)       105: (4,3,2)
     38: (8,1)         77: (5,4)        106: (16,1)
     39: (6,2)         78: (6,2,1)      110: (5,3,1)
     42: (4,2,1)       82: (13,1)       111: (12,2)
For example, the prime indices of 150 are {1,2,3,3}, with permutations and run-lengths (right):
  (3,3,2,1) -> (2,1,1)
  (3,3,1,2) -> (2,1,1)
  (3,2,3,1) -> (1,1,1,1)
  (3,2,1,3) -> (1,1,1,1)
  (3,1,3,2) -> (1,1,1,1)
  (3,1,2,3) -> (1,1,1,1)
  (2,3,3,1) -> (1,2,1)
  (2,3,1,3) -> (1,1,1,1)
  (2,1,3,3) -> (1,1,2)
  (1,3,3,2) -> (1,2,1)
  (1,3,2,3) -> (1,1,1,1)
  (1,2,3,3) -> (1,1,2)
Since none have all distinct run-lengths, 150 is in the sequence.

Wilf partitions are counted by A098859, ranked by A130091.
Non-Wilf partitions are counted by A336866, ranked by A130092.
A variant for runs is A351201, counted by A351203 (complement A351204).
These partitions appear to be counted by A351293.
The complement is A351294, apparently counted by A239455.
A032020 = number of binary expansions with distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A056239 = sum of prime indices, row sums of A112798.
A165413 = number of distinct run-lengths in binary expansion.
A181819 = Heinz number of prime signature (prime shadow).
A182850/A323014 = frequency depth, counted by A225485/A325280.
A297770 = number of distinct runs in binary expansion.
A320922 ranks graphical partitions, complement A339618, counted by A000569.
A329739 = compositions with all distinct run-lengths, for all runs A351013.
A329747 = runs-resistance, counted by A329746.
A333489 ranks anti-runs, complement A348612.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
Cf. A000961, A001221, A001222, A175413, A182857, A304660, A320924, A328592, A351202, A351290.

Select[Range[100],Select[Permutations[Join@@ ConstantArray@@@FactorInteger[#]],UnsameQ@@Length/@Split[#]&]=={}&]

Name edited by Gus Wiseman, Aug 13 2025

A374629 Irregular triangle listing the leaders of maximal weakly increasing runs in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jul 20 2024

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.

			The 58654th composition in standard order is (1,1,3,2,4,1,1,1,2), with maximal weakly increasing runs ((1,1,3),(2,4),(1,1,1,2)), so row 58654 is (1,2,1).
The nonnegative integers, corresponding compositions, and leaders of maximal weakly increasing runs begin:
    0:      () -> ()      15: (1,1,1,1) -> (1)
    1:     (1) -> (1)     16:       (5) -> (5)
    2:     (2) -> (2)     17:     (4,1) -> (4,1)
    3:   (1,1) -> (1)     18:     (3,2) -> (3,2)
    4:     (3) -> (3)     19:   (3,1,1) -> (3,1)
    5:   (2,1) -> (2,1)   20:     (2,3) -> (2)
    6:   (1,2) -> (1)     21:   (2,2,1) -> (2,1)
    7: (1,1,1) -> (1)     22:   (2,1,2) -> (2,1)
    8:     (4) -> (4)     23: (2,1,1,1) -> (2,1)
    9:   (3,1) -> (3,1)   24:     (1,4) -> (1)
   10:   (2,2) -> (2)     25:   (1,3,1) -> (1,1)
   11: (2,1,1) -> (2,1)   26:   (1,2,2) -> (1)
   12:   (1,3) -> (1)     27: (1,2,1,1) -> (1,1)
   13: (1,2,1) -> (1,1)   28:   (1,1,3) -> (1)
   14: (1,1,2) -> (1)     29: (1,1,2,1) -> (1,1)

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

Row-leaders are A065120.
Row-lengths are A124766.
Row-sums are A374630.
Positions of constant rows are A374633, counted by A374631.
Positions of strict rows are A374768, counted by A374632.
For other types of runs we have A374251, A374515, A374683, A374740, A374757.
Positions of non-weakly decreasing rows are A375137.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
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.
- 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, length A124767, 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.
Cf. A046660, A106356, A188920, A189076, A238343, A272919, A333213, A373949, A374634, A374635, A374637, A374701, A375123.

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

A373949 Triangle read by rows where T(n,k) is the number of integer compositions of n such that replacing each run of repeated parts with a single part (run-compression) yields a composition of k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 1, 2, 4, 0, 1, 0, 4, 4, 7, 0, 1, 1, 5, 6, 5, 14, 0, 1, 0, 6, 10, 10, 14, 23, 0, 1, 1, 6, 14, 12, 29, 26, 39, 0, 1, 0, 9, 16, 19, 40, 54, 46, 71, 0, 1, 1, 8, 22, 22, 64, 82, 96, 92, 124, 0, 1, 0, 10, 26, 30, 82, 137, 144, 204, 176, 214

Offset: 0

Gus Wiseman, Jun 28 2024

			Triangle begins:
   1
   0   1
   0   1   1
   0   1   0   3
   0   1   1   2   4
   0   1   0   4   4   7
   0   1   1   5   6   5  14
   0   1   0   6  10  10  14  23
   0   1   1   6  14  12  29  26  39
   0   1   0   9  16  19  40  54  46  71
   0   1   1   8  22  22  64  82  96  92 124
   0   1   0  10  26  30  82 137 144 204 176 214
   0   1   1  11  32  31 121 186 240 331 393 323 378
Row n = 6 counts the following compositions:
  .  (111111)  (222)  (33)     (3111)   (411)   (6)
                      (2211)   (1113)   (114)   (51)
                      (1122)   (1221)   (1311)  (15)
                      (21111)  (12111)  (1131)  (42)
                      (11112)  (11211)  (2112)  (24)
                               (11121)          (141)
                                                (321)
                                                (312)
                                                (231)
                                                (213)
                                                (132)
                                                (123)
                                                (2121)
                                                (1212)
For example, the composition (1,2,2,1) with compression (1,2,1) is counted under T(6,4).

John Tyler Rascoe, Rows n = 0..130, flattened

Column k = n is A003242 (anti-runs or compressed compositions).
Row-sums are A011782.
Same as A373951 with rows reversed.
Column k = 3 is A373952.
This statistic is represented by A373953, difference A373954.
A114901 counts compositions with no isolated parts.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs in standard compositions, anti-runs A333381.
A240085 counts compositions with no unique parts.
A333755 counts compositions by compressed length.
A373948 represents the run-compression transformation.
Cf. A037201 (halved A373947), A106356, A124762, A238130, A238279, A238343, A285981, A333213, A333489, A373950.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#]]==k&]], {n,0,10},{k,0,n}]

T_xy(row_max) = {my(N=row_max+1, x='x+O('x^N), h=1/(1-sum(i=1,N, (y^i*x^i)/(1+x^i*(y^i-1))))); vector(N, n, Vecrev(polcoeff(h, n-1)))}
T_xy(13) \\ John Tyler Rascoe, Mar 20 2025

G.f.: 1/(1 - Sum_{i>0} (y^i * x^i)/(1 + x^i * (y^i - 1))). - John Tyler Rascoe, Mar 20 2025

A124762 Number of levels for compositions in standard order.

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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. This sequence gives the number of adjacent equal terms in the n-th composition in standard order. Alternatively, a(n) is one fewer than the number of maximal anti-runs in the same composition, where anti-runs are sequences without any adjacent equal terms. For example, the 1234567th composition in standard order is (3,2,1,2,2,1,2,5,1,1,1) with anti-runs ((3,2,1,2),(2,1,2,5,1),(1),(1)), so a(1234567) = 4 - 1 = 3. - Gus Wiseman, Apr 08 2020

			Composition number 11 is 2,1,1; 2>1=1, so a(11) = 1.
The table starts:
  0
  0
  0 1
  0 0 0 2
  0 0 1 1 0 0 1 3
  0 0 0 1 0 1 0 2 0 0 1 1 1 1 2 4
  0 0 0 1 1 0 0 2 0 0 2 2 0 0 1 3 0 0 0 1 0 1 0 2 1 1 2 2 2 2 3 5

Cf. A066099, A124760, A124761, A124763, A124764, A011782 (row lengths), A059570 (row sums).
Anti-runs summing to n are counted by A003242(n).
A triangle counting maximal anti-runs of compositions is A106356.
A triangle counting maximal runs of compositions is A238279.
Partitions whose first differences are an anti-run are A238424.
All of the following pertain to compositions in standard order (A066099):
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Adjacent unequal pairs are counted by A333382.
- Anti-runs are A333489.
Cf. A000120, A029931, A048793, A059893, A070939, A114994, A225620, A228351.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Select[Partition[stc[n],2,1],SameQ@@#&]],{n,0,100}] (* Gus Wiseman, Apr 08 2020 *)

For a composition b(1),...,b(k), a(n) = Sum_{1<=i=1

For n > 0, a(n) = A333381(n) - 1. - Gus Wiseman, Apr 08 2020

A335452 Number of separations (Carlitz compositions or anti-runs) of the prime indices of n.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 6, 1, 0, 2, 2, 2, 2, 1, 2, 2, 0, 1, 6, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 6, 1, 2, 1, 0, 2, 6, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 2, 6, 1, 0, 0, 2, 1, 6, 2, 2, 2

Offset: 1

Gus Wiseman, Jun 21 2020

nonn

The first term that is not a factorial number is a(180) = 12.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A separation (or Carlitz composition) of a multiset is a permutation with no adjacent equal parts.
a(n) depends only on the prime signature of n. - Andrew Howroyd, Feb 03 2021

			The a(n) separations for n = 2, 6, 30, 180:
  (1)  (12)  (123)  (12123)
       (21)  (132)  (12132)
             (213)  (12312)
             (231)  (12321)
             (312)  (13212)
             (321)  (21213)
                    (21231)
                    (21312)
                    (21321)
                    (23121)
                    (31212)
                    (32121)

Andrew Howroyd, Table of n, a(n) for n = 1..4096
Wikipedia, Permutation pattern

Separations are counted by A003242 and ranked by A333489.
Patterns are counted by A000670 and ranked by A333217.
Permutations of prime indices are counted by A008480.
Inseparable partitions are counted by A325535 and ranked by A335448.
Cf. A000961, A005117, A056239, A112798, A181796, A261962, A333221, A335451, A335454, A335465, A335489.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],!MatchQ[#,{_,x_,x_,_}]&]],{n,100}]

F(i, j, r, t) = {sum(k=max(0, i-j), min(min(i,t), (i-j+t)\2), binomial(i, k)*binomial(r-i+1, t+i-j-2*k)*binomial(t-1, k-i+j))}
count(sig)={my(s=vecsum(sig), r=0, v=[1]); for(p=1, #sig, my(t=sig[p]); v=vector(s-r-t+1, j, sum(i=1, #v, v[i]*F(i-1, j-1, r, t))); r += t); v[1]}
a(n)={count(factor(n)[,2])} \\ Andrew Howroyd, Feb 03 2021

A345167 Numbers k such that the k-th composition in standard order is alternating.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 12, 13, 16, 17, 18, 20, 22, 24, 25, 32, 33, 34, 38, 40, 41, 44, 45, 48, 49, 50, 54, 64, 65, 66, 68, 70, 72, 76, 77, 80, 81, 82, 88, 89, 96, 97, 98, 102, 108, 109, 128, 129, 130, 132, 134, 140, 141, 144, 145, 148, 152, 153, 160, 161, 162

Offset: 1

Gus Wiseman, Jun 15 2021

nonn

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

			The terms together with their binary indices begin:
      1: (1)         25: (1,3,1)       66: (5,2)
      2: (2)         32: (6)           68: (4,3)
      4: (3)         33: (5,1)         70: (4,1,2)
      5: (2,1)       34: (4,2)         72: (3,4)
      6: (1,2)       38: (3,1,2)       76: (3,1,3)
      8: (4)         40: (2,4)         77: (3,1,2,1)
      9: (3,1)       41: (2,3,1)       80: (2,5)
     12: (1,3)       44: (2,1,3)       81: (2,4,1)
     13: (1,2,1)     45: (2,1,2,1)     82: (2,3,2)
     16: (5)         48: (1,5)         88: (2,1,4)
     17: (4,1)       49: (1,4,1)       89: (2,1,3,1)
     18: (3,2)       50: (1,3,2)       96: (1,6)
     20: (2,3)       54: (1,2,1,2)     97: (1,5,1)
     22: (2,1,2)     64: (7)           98: (1,4,2)
     24: (1,4)       65: (6,1)        102: (1,3,1,2)

Wikipedia, Alternating permutation

These compositions are counted by A025047, complement A345192.
The complement is A345168.
Partitions with a permutation of this type: A345170, complement A345165.
Factorizations with a permutation of this type: A348379.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A345164 counts alternating permutations of prime indices.
A345194 counts alternating patterns, with twins A344605.
Statistics of standard compositions:
- Length is A000120.
- Constant runs are A124767.
- Heinz number is A333219.
- Number of maximal anti-runs is A333381.
- Runs-resistance is A333628.
- Number of distinct parts is A334028.
Classes of standard compositions:
- Weakly decreasing compositions (partitions) are A114994.
- Weakly increasing compositions (multisets) are A225620.
- Anti-runs are A333489.
- Non-alternating anti-runs are A345169.
Cf. A025048, A025049, A059893, A106356, A238279, A335448, A344604, A344615, A344653, A344742, A345163, A348377.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]] ==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Select[Range[0,100],wigQ@*stc]

A374249 Numbers k such that the k-th composition in standard order has its equal parts contiguous.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 50, 52, 56, 58, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 78, 79, 80, 81, 83, 84, 85

Offset: 1

Gus Wiseman, Jul 13 2024

nonn

These are compositions avoiding the patterns (1,2,1) and (2,1,2).

			The terms together with their standard compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   5: (2,1)
   6: (1,2)
   7: (1,1,1)
   8: (4)
   9: (3,1)
  10: (2,2)
  11: (2,1,1)
  12: (1,3)
  14: (1,1,2)
  15: (1,1,1,1)
  16: (5)
See A374253 for the complement: 13, 22, 25, 27, 29, ...

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

The strict (also anti-run) case is A233564, counted by A032020.
Compositions of this type are counted by A274174.
Permutations of prime indices of this type are counted by A333175.
The complement is A374253 (anti-run A374254), counted by A335548.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A066099 lists compositions in standard order.
A124767 counts runs in standard compositions, anti-runs A333381.
A333755 counts compositions by number of runs.
A335454 counts patterns matched by standard compositions.
A335462 counts (1,2,1)- and (2,1,2)-matching permutations of prime indices.
- 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.
Cf. A106356, A124762, A238130, A238279, A261982, A272919, A333382, A335450, A335460, A335524, A335525.

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

Equals A335467 /\ A335469.

A333382 Number of adjacent unequal parts in the n-th composition in standard-order.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 2, 1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 0, 0, 1, 1, 1, 0, 2, 2, 1, 1, 2, 0, 1, 2, 3, 2, 1, 1, 2, 2, 2, 2, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 0, 0, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 3, 2, 1, 1, 2, 2, 2, 1, 1, 2

Offset: 0

Gus Wiseman, Mar 24 2020

nonn

For n > 0, a(n) is one fewer than the number of maximal runs of the n-th composition in standard-order.

			The 46th composition in standard order is (2,1,1,2), with maximal runs ((2),(1,1),(2)), so a(46) = 3 - 1 = 2.

Wikipedia, Longest increasing subsequence

Indices of first appearances (not counting 0) are A113835.
Partitions whose 0-appended first differences are a run are A007862.
Partitions whose first differences are a run are A049988.
A triangle counting maximal anti-runs of compositions is A106356.
A triangle counting maximal runs of compositions is A238279.
All of the following pertain to compositions in standard order (A066099):
- Adjacent equal pairs are counted by A124762.
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769.
- Strict compositions are ranked by A233564.
- Constant compositions are ranked by A272919.
- Normal compositions are ranked by A333217.
- Anti-runs are ranked by A333489.
- Anti-runs are counted by A333381.
Cf. A000005, A000120, A003242, A029931, A048793, A059893, A070939, A114994, A225620, A228351, A238424.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Select[Partition[stc[n],2,1],UnsameQ@@#&]],{n,0,100}]

For n > 0, a(n) = A124767(n) - 1.

A124768 Number of strictly increasing runs for compositions in standard order.

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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. a(n) is the number of maximal strictly increasing runs in this composition. Alternatively, a(n) is one plus the number of weak descents in the same composition. For example, the strictly increasing runs of the 1234567th composition are ((3),(2),(1,2),(2),(1,2,5),(1),(1),(1)), so a(1234567) = 8. The 7 weak descents together with the strict ascents are: 3 >= 2 >= 1 < 2 >= 2 >= 1 < 2 < 5 >= 1 >= 1 >= 1. - Gus Wiseman, Apr 08 2020

			Composition number 11 is 2,1,1; the strictly increasing runs are 2; 1; 1; so a(11) = 3.
The table starts:
  0
  1
  1 2
  1 2 1 3
  1 2 2 3 1 2 2 4
  1 2 2 3 1 3 2 4 1 2 2 3 2 3 3 5
  1 2 2 3 2 3 2 4 1 2 3 4 2 3 3 5 1 2 2 3 1 3 2 4 2 3 3 4 3 4 4 6

Cf. A066099, A124763, A011782 (row lengths).
Compositions of n with k weak descents are A333213.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Partial sums from the right are A048793.
- Sum is A070939.
- Weakly decreasing compositions are A114994.
- Adjacent equal pairs are counted by A124762.
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768 (this sequence).
- Strictly decreasing runs are counted by A124769.
- Weakly increasing compositions are A225620.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Initial intervals are A246534.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Permutations are A333218.
- Heinz number is A333219.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
- Anti-runs are A333489.
Cf. A029931, A124760, A124761, A124764, A238343, A333220.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Split[stc[n],Less]],{n,0,100}] (* Gus Wiseman, Apr 08 2020 *)

a(0) = 0, a(n) = A124763(n) + 1 for n > 0.

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cp's OEIS Frontend

A333381 Number of maximal anti-runs of the n-th composition in standard order.

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A351295 Numbers whose multiset of prime factors has no permutation with all distinct run-lengths.

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A374629 Irregular triangle listing the leaders of maximal weakly increasing runs in the n-th composition in standard order.

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A373949 Triangle read by rows where T(n,k) is the number of integer compositions of n such that replacing each run of repeated parts with a single part (run-compression) yields a composition of k.

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A124762 Number of levels for compositions in standard order.

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A335452 Number of separations (Carlitz compositions or anti-runs) of the prime indices of n.

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A345167 Numbers k such that the k-th composition in standard order is alternating.

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A374249 Numbers k such that the k-th composition in standard order has its equal parts contiguous.

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