A106356 -id:A106356 - cp's OEIS frontend

A124762 Number of levels for compositions in standard order.

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

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

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.

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.

A345192 Number of non-alternating compositions of n.

Original entry on oeis.org

0, 0, 1, 1, 4, 9, 20, 45, 99, 208, 437, 906, 1862, 3803, 7732, 15659, 31629, 63747, 128258, 257722, 517339, 1037652, 2079984, 4167325, 8346204, 16710572, 33449695, 66944254, 133959021, 268028868, 536231903, 1072737537, 2145905285, 4292486690, 8586035993, 17173742032, 34350108745, 68704342523, 137415168084

Offset: 0

Gus Wiseman, Jun 17 2021

nonn

First differs from A261983 at a(6) = 20, A261983(6) = 18.

			The a(2) = 1 through a(6) = 20 compositions:
  (11)  (111)  (22)    (113)    (33)
               (112)   (122)    (114)
               (211)   (221)    (123)
               (1111)  (311)    (222)
                       (1112)   (321)
                       (1121)   (411)
                       (1211)   (1113)
                       (2111)   (1122)
                       (11111)  (1131)
                                (1221)
                                (1311)
                                (2112)
                                (2211)
                                (3111)
                                (11112)
                                (11121)
                                (11211)
                                (12111)
                                (21111)
                                (111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The complement is counted by A025047 (ascend: A025048, descend: A025049).
Dominates A261983 (non-anti-run compositions), ranked by A348612.
These compositions are ranked by A345168, complement A345167.
The case without twins is A348377.
The version for factorizations is A348613.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A011782 counts compositions.
A032020 counts strict compositions.
A106356 counts compositions by number of maximal anti-runs.
A114901 counts compositions where each part is adjacent to an equal part.
A274174 counts compositions with equal parts contiguous.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts alternating compositions with twins.
A344605 counts alternating patterns with twins.
A344654 counts non-twin partitions with no alternating permutation.
A345162 counts normal partitions with no alternating permutation.
A345164 counts alternating permutations of prime indices.
A345170 counts partitions w/ alternating permutation, ranked by A345172.
A345165 counts partitions w/o alternating permutation, ranked by A345171.
Patterns:
- A128761 avoiding (1,2,3) adjacent.
- A344614 avoiding (1,2,3) and (3,2,1) adjacent.
- A344615 weakly avoiding (1,2,3) adjacent.
Cf. A000070, A008965, A178470, A238279, A333755, A335126, A344606, A344653, A344740, A345163, A345166, A345169, A345173, A348380.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!wigQ[#]&]],{n,0,15}]

a(n) = A011782(n) - A025047(n).

A261982 Number of compositions of n with some part repeated.

Original entry on oeis.org

0, 0, 1, 1, 5, 11, 21, 51, 109, 229, 455, 959, 1947, 3963, 7999, 16033, 32333, 64919, 130221, 260967, 522733, 1045825, 2093855, 4189547, 8382315, 16768455, 33543127, 67093261, 134193413, 268404995, 536829045, 1073686083, 2147408773, 4294869253, 8589803783

Offset: 0

Alois P. Heinz, Sep 07 2015

nonn

Also compositions matching the pattern (1,1). - Gus Wiseman, Jun 23 2020

			a(2) = 1: 11.
a(3) = 1: 111.
a(4) = 5: 22, 211, 121, 112, 1111.

Alois P. Heinz, Table of n, a(n) for n = 0..3322
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Row sums of A261981 and of A262191.
Cf. A262047.
The version for patterns is A019472.
The (1,1)-avoiding version is A032020.
The case of partitions is A047967.
(1,1,1)-matching compositions are counted by A335455.
Patterns matched by compositions are counted by A335456.
(1,1)-matching compositions are ranked by A335488.
Cf. A011782, A056986, A106356, A181796, A238279, A269134, A333755.

b:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
      `if`(k=0, `if`(n=0, 1, 0), b(n-k, k) +k*b(n-k, k-1)))
    end:
a:= n-> ceil(2^(n-1))-add(b(n, k), k=0..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..40);

b[n_, k_] := b[n, k] = If[k<0 || n<0, 0, If[k==0, If[n==0, 1, 0], b[n-k, k] + k*b[n-k, k-1]]]; a[n_] := Ceiling[2^(n-1)]-Sum[b[n, k], {k, 0, Floor[ (Sqrt[8n+1]-1)/2]}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 08 2017, translated from Maple *)
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],Length[#]>Length[Split[#]]&]],{n,0,10}] (* Gus Wiseman, Jun 24 2020 *)

a(n) = A011782(n) - A032020(n).

G.f.: (1 - x) / (1 - 2*x) - Sum_{k>=0} k! * x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Jan 30 2020

A373948 Run-compression encoded as a transformation of compositions in standard order.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 6, 1, 8, 9, 2, 5, 12, 13, 6, 1, 16, 17, 18, 9, 20, 5, 22, 5, 24, 25, 6, 13, 12, 13, 6, 1, 32, 33, 34, 17, 4, 37, 38, 9, 40, 41, 2, 5, 44, 45, 22, 5, 48, 49, 50, 25, 52, 13, 54, 13, 24, 25, 6, 13, 12, 13, 6, 1, 64, 65, 66, 33, 68, 69, 70, 17, 72

Offset: 0

Gus Wiseman, Jun 24 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.
We define the (run-) compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has compression (1,2,1).
For the present sequence, the a(n)-th composition in standard order is obtained by compressing the n-th composition in standard order.

			The standard compositions and their compressions begin:
   0: ()        -->  0: ()
   1: (1)       -->  1: (1)
   2: (2)       -->  2: (2)
   3: (1,1)     -->  1: (1)
   4: (3)       -->  4: (3)
   5: (2,1)     -->  5: (2,1)
   6: (1,2)     -->  6: (1,2)
   7: (1,1,1)   -->  1: (1)
   8: (4)       -->  8: (4)
   9: (3,1)     -->  9: (3,1)
  10: (2,2)     -->  2: (2)
  11: (2,1,1)   -->  5: (2,1)
  12: (1,3)     --> 12: (1,3)
  13: (1,2,1)   --> 13: (1,2,1)
  14: (1,1,2)   -->  6: (1,2)
  15: (1,1,1,1) -->  1: (1)

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

Positions of 1's are A000225.
The image is A333489, counted by A003242.
Sum of standard composition for a(n) is given by A373953, length A124767.
A037201 gives compression of first differences of primes, halved A373947.
A066099 lists the parts of all compositions in standard order.
A114901 counts compositions with no isolated parts.
A116861 counts partitions by compressed sum, by length A116608.
A240085 counts compositions with no unique parts.
A333755 counts compositions by compressed length.
A373949 counts compositions by compressed sum, opposite A373951.
Cf. A106356, A124762, A238130, A238279, A272919, A333381, A333382, A333627, A373952, A373954.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[First/@Split[stc[n]]],{n,0,30}]

A029837(a(n)) = A373953(n).

A000120(a(n)) = A124767(n).

A351014 Number of distinct runs in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Feb 07 2022

nonn

The n-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of n, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The number 3310 has binary expansion 110011101110 and standard composition (1,3,1,1,2,1,1,2), with runs (1), (3), (1,1), (2), (1,1), (2), of which 4 are distinct, so a(3310) = 4.

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

Counting not necessarily distinct runs gives A124767.
Using binary expansions instead of standard compositions gives A297770.
Positions of first appearances are A351015.
A005811 counts runs in binary expansion.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse A085208.
A333489 ranks anti-runs, complement A348612.
A345167 ranks alternating compositions, counted by A025047.
A351204 counts partitions where every permutation has all distinct runs.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Selected statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
Cf. A098859, A106356, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351201.

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

A344604 Number of alternating compositions of n, including twins (x,x).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 13, 19, 30, 48, 76, 118, 187, 293, 461, 725, 1140, 1789, 2815, 4422, 6950, 10924, 17169, 26979, 42405, 66644, 104738, 164610, 258708, 406588, 639010, 1004287, 1578364, 2480606, 3898600, 6127152, 9629624, 15134213, 23785389, 37381849, 58750469

Offset: 0

Gus Wiseman, May 27 2021

nonn

We define a composition to be alternating including twins (x,x) if there are no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z. Except in the case of twins (x,x), all such compositions are anti-runs (A003242). These compositions avoid the weak consecutive patterns (1,2,3) and (3,2,1), the strict version being A344614.

The version without twins (x,x) is A025047 (alternating compositions).

			The a(1) = 1 through a(7) = 19 compositions:
  (1)  (2)   (3)   (4)    (5)    (6)     (7)
       (11)  (12)  (13)   (14)   (15)    (16)
             (21)  (22)   (23)   (24)    (25)
                   (31)   (32)   (33)    (34)
                   (121)  (41)   (42)    (43)
                          (131)  (51)    (52)
                          (212)  (132)   (61)
                                 (141)   (142)
                                 (213)   (151)
                                 (231)   (214)
                                 (312)   (232)
                                 (1212)  (241)
                                 (2121)  (313)
                                         (412)
                                         (1213)
                                         (1312)
                                         (2131)
                                         (3121)
                                         (12121)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

A001250 counts alternating permutations.
A005649 counts anti-run patterns.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A106356 counts compositions by number of maximal anti-runs.
A114901 counts compositions where each part is adjacent to an equal part.
A325534 counts separable partitions.
A325535 counts inseparable partitions.
A344605 counts alternating patterns including twins.
A344606 counts alternating permutations of prime factors including twins.
Counting compositions by patterns:
- A011782 no conditions.
- A003242 avoiding (1,1) adjacent.
- A102726 avoiding (1,2,3).
- A106351 avoiding (1,1) adjacent by sum and length.
- A128695 avoiding (1,1,1) adjacent.
- A128761 avoiding (1,2,3) adjacent.
- A232432 avoiding (1,1,1).
- A335456 all patterns.
- A335457 all patterns adjacent.
- A335514 matching (1,2,3).
- A344614 avoiding (1,2,3) and (3,2,1) adjacent.
- A344615 weakly avoiding (1,2,3) adjacent.
Cf. A000041, A006330, A008965, A238279, A239830, A333213, A238279/A333755, A344612, A344616, A344617, A344618.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]],{n,0,15}]

a(n > 0) = A025047(n) + 1 if n is even, otherwise A025047(n). - Gus Wiseman, Nov 03 2021

a(21)-a(40) from Alois P. Heinz, Nov 04 2021

A344614 Number of compositions of n with no adjacent triples (..., x, y, z, ...) where x < y < z or x > y > z.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 30, 58, 110, 209, 397, 753, 1429, 2711, 5143, 9757, 18511, 35117, 66621, 126389, 239781, 454897, 863010, 1637260, 3106138, 5892821, 11179603, 21209446, 40237641, 76337091, 144823431, 274752731, 521249018, 988891100, 1876081530, 3559220898, 6752400377

Offset: 0

Gus Wiseman, May 27 2021

nonn

These compositions avoid the strict consecutive patterns (1,2,3) and (3,2,1), the weak version being A344604.

			The a(6) = 30 compositions are:
  (6)  (15)  (114)  (1113)  (11112)  (111111)
       (24)  (132)  (1122)  (11121)
       (33)  (141)  (1131)  (11211)
       (42)  (213)  (1212)  (12111)
       (51)  (222)  (1221)  (21111)
             (231)  (1311)
             (312)  (2112)
             (411)  (2121)
                    (2211)
                    (3111)
Missing are: (123), (321).

A001250 counts alternating permutations.
A005649 counts anti-run patterns.
A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
A106356 counts compositions by number of maximal anti-runs.
A114901 counts compositions where each part is adjacent to an equal part.
A325534 counts separable partitions.
A325535 counts inseparable partitions.
A344604 counts wiggly compositions with twins.
A344605 counts wiggly patterns with twins.
A344606 counts wiggly permutations of prime factors with twins.
Counting compositions by patterns:
- A003242 avoiding (1,1) adjacent.
- A011782 no conditions.
- A106351 avoiding (1,1) adjacent by sum and length.
- A128695 avoiding (1,1,1) adjacent.
- A128761 avoiding (1,2,3).
- A232432 avoiding (1,1,1).
- A335456 all patterns.
- A335457 all patterns adjacent.
- A335514 matching (1,2,3).
- A344604 weakly avoiding (1,2,3) and (3,2,1) adjacent.
- A344614 avoiding (1,2,3) and (3,2,1) adjacent.
- A344615 weakly avoiding (1,2,3) adjacent.
Cf. A000041, A006330, A008965, A049774, A056986, A238279/A333755, A333213, A335515, A344612, A344652.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,y_,z_,_}/;xy>z]&]],{n,0,15}]

More terms from Bert Dobbelaere, Jun 12 2021

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cp's OEIS Frontend

A124762 Number of levels for compositions in standard order.

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

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A345192 Number of non-alternating compositions of n.

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A261982 Number of compositions of n with some part repeated.

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A373948 Run-compression encoded as a transformation of compositions in standard order.

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A351014 Number of distinct runs in the n-th composition in standard order.

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