A333382 -id:A333382 - cp's OEIS frontend

A337504 Number of compositions of 2*n with n maximal anti-runs.

1, 1, 3, 8, 13, 33, 112, 286, 769, 2288, 6695, 18745, 54654, 160888, 467402, 1362378, 4016517, 11807966, 34708018, 102451390, 302870005, 895207191, 2650590597, 7859253320, 23316653154, 69231883374, 205773157904, 612021943421, 1821435719846, 5424528040529, 16165017705176

Offset: 0

Gus Wiseman, Sep 04 2020

nonn

An anti-run is a sequence with no adjacent equal parts.

			The a(0) = 1 through a(4) = 13 compositions:
  ()  (2)  (2,2)    (2,2,2)      (2,2,2,2)
           (1,1,2)  (1,1,1,3)    (1,1,1,1,4)
           (2,1,1)  (1,1,2,2)    (1,1,2,2,2)
                    (2,2,1,1)    (2,2,2,1,1)
                    (3,1,1,1)    (4,1,1,1,1)
                    (1,1,1,2,1)  (1,1,1,1,3,1)
                    (1,1,2,1,1)  (1,1,1,2,2,1)
                    (1,2,1,1,1)  (1,1,1,3,1,1)
                                 (1,1,2,2,1,1)
                                 (1,1,3,1,1,1)
                                 (1,2,2,1,1,1)
                                 (1,3,1,1,1,1)
                                 (2,1,1,1,1,2)

Andrew Howroyd, Table of n, a(n) for n = 0..200

A106356 has this as main diagonal n = 2*k.
A336108 is the version for runs.
A337505 is the version for patterns.
A337564 is the version for runs in patterns.
A003242 counts anti-run compositions.
A011782 counts compositions.
A124767 counts runs in standard compositions.
A238343 counts compositions by descents.
A333213 counts compositions by weak ascents.
A333381 counts anti-runs in standard compositions.
A333382 counts adjacent unequal pairs in standard compositions.
A333489 ranks anti-runs.
A333755 counts compositions by number of runs.
A333769 gives run-lengths in standard compositions.
A337565 gives anti-run lengths in standard compositions.
Cf. A106351, A124762, A233564, A235998, A238130, A238279, A333214, A333216.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[2*n],Length[Split[#,UnsameQ]]==n&]],{n,0,10}]

a(n)={polcoef(polcoef(1 - y + y*(y-1)/(y - 1 - sum(d=1, 2*n, (y-1)^d*x^d/(1 - x^d) + O(x^(2*n+1)))), 2*n, x), n, y)} \\ Andrew Howroyd, Feb 02 2021

a(n) = [x^(2*n)*y^n] 1 - y + y*(y-1)/(y - 1 - Sum_{d>=1} (y-1)^d*x^d/(1 - x^d)). - Andrew Howroyd, Feb 02 2021

Terms a(11) and beyond from Andrew Howroyd, Feb 02 2021

A386578 Irregular triangle read by rows where T(n,k) is the number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent equal parts.

Original entry on oeis.org

1, 0, 1, 2, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 1, 6, 0, 0, 2, 2, 2, 0, 0, 2, 2, 0, 0, 0, 0, 0, 1, 6, 6, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 3, 2, 0, 1, 4, 3, 2, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 12, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 12, 6, 0, 0, 0, 3, 6, 4, 2, 0

Offset: 2

Gus Wiseman, Aug 04 2025

Row 1 is empty, so offset is 2.
Same as A386579 with rows reversed.
This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			Row n = 21 counts the following permutations:
  .  112121  111212  111221  111122  .
     121121  112112  112211  221111
     121211  121112  122111
             211121  211112
             211211
             212111
Triangle begins
   .
   1
   0  1
   2  0
   0  0  1
   1  2  0
   0  0  0  1
   6  0  0
   2  2  2  0
   0  2  2  0
   0  0  0  0  1
   6  6  0  0
   0  0  0  0  0  1
   0  0  3  2  0
   1  4  3  2  0
  24  0  0  0
   0  0  0  0  0  0  1
  12 12  6  0  0
   0  0  0  0  0  0  0  1
   2 12  6  0  0
   0  3  6  4  2  0

Column k = last is A010051.
Row lengths are A056239.
Initial zeros are counted by A252736 = A001222 - 1.
Row sums are A318762.
Column k = 0 is A335125.
For prime indices we have A386577.
Reversing all rows gives A386579.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.
A124762 gives inseparability of standard compositions, separability A333382.
A305936 is a multiset whose multiplicities are the prime indices of n.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A336106 counts partitions of separable type, ranks A335127, sums of A386585.
Cf. A001221, A003963, A051903, A051904, A106351, A112798, A238130, A336102, A373957, A374246, A382525, A386581.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
aqt[c_,x_]:=Select[Permutations[c],Function[q,Length[Select[Range[Length[q]-1],q[[#]]==q[[#+1]]&]]==x]];
Table[Table[Length[aqt[nrmptn[n],k]],{k,0,Length[nrmptn[n]]-1}],{n,30}]

A337506 Triangle read by rows where T(n,k) is the number of length-n sequences covering an initial interval of positive integers with k maximal anti-runs.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Sep 06 2020

An anti-run is a sequence with no adjacent equal parts. The number of maximal anti-runs is one more than the number of adjacent equal parts.

			Triangle begins:
  1
  0      1
  0      2      1
  0      8      4      1
  0     44     24      6      1
  0    308    176     48      8      1
  0   2612   1540    440     80     10      1
  0  25988  15672   4620    880    120     12      1
  0 296564 181916  54852  10780   1540    168     14      1
Row n = 3 counts the following sequences (empty column indicated by dot):
  .  (1,2,1)  (1,1,2)  (1,1,1)
     (1,2,3)  (1,2,2)
     (1,3,2)  (2,1,1)
     (2,1,2)  (2,2,1)
     (2,1,3)
     (2,3,1)
     (3,1,2)
     (3,2,1)

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

A000670 gives row sums.
A005649 gives column k = 1.
A337507 gives column k = 2.
A337505 gives the diagonal n = 2*k.
A106356 is the version for compositions.
A238130/A238279/A333755 is the version for runs in compositions.
A335461 has the reversed rows (except zeros).
A003242 counts anti-run compositions.
A124762 counts adjacent equal terms in standard compositions.
A124767 counts maximal runs in standard compositions.
A333381 counts maximal anti-runs in standard compositions.
A333382 counts adjacent unequal terms in standard compositions.
A333489 ranks anti-run compositions.
A333769 gives maximal run-lengths in standard compositions.
A337565 gives maximal anti-run lengths in standard compositions.
Cf. A019472, A052841, A060223, A106351, A269134, A325535, A337564.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],Length[Split[#,UnsameQ]]==k&]],{n,0,5},{k,0,n}]

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

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

Terms a(45) and beyond from Andrew Howroyd, Dec 31 2020

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 14 2024

nonn

Such a composition cannot be strict.

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

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

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

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

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

Equals A333489 /\ A374253.

A337507 Number of length-n sequences covering an initial interval of positive integers with exactly two maximal anti-runs, or with one pair of adjacent equal parts.

Original entry on oeis.org

0, 0, 1, 4, 24, 176, 1540, 15672, 181916, 2372512, 34348932, 546674120, 9486840748, 178285201008, 3607174453844, 78177409231768, 1806934004612220, 44367502983673664, 1153334584544496676, 31643148872573831016

Offset: 0

Gus Wiseman, Sep 06 2020

nonn

An anti-run is a sequence with no adjacent equal parts. For example, the maximal anti-runs in (3,1,1,2,2,2,1) are ((3,1),(1,2),(2),(2,1)). In general, there is one more maximal anti-run than the number of pairs of adjacent equal parts.

			The a(4) = 24 sequences:
  (2,1,2,2)  (2,1,3,3)  (3,1,2,2)
  (2,2,1,2)  (2,3,3,1)  (3,2,2,1)
  (1,2,2,1)  (3,3,1,2)  (1,1,2,3)
  (2,1,1,2)  (3,3,2,1)  (1,1,3,2)
  (1,1,2,1)  (1,2,2,3)  (2,1,1,3)
  (1,2,1,1)  (1,3,2,2)  (2,3,1,1)
  (1,2,3,3)  (2,2,1,3)  (3,1,1,2)
  (1,3,3,2)  (2,2,3,1)  (3,2,1,1)

A002133 is the version for runs in partitions.
A106357 is the version for compositions.
A337506 has this as column k = 2.
A000670 counts patterns.
A005649 counts anti-run patterns.
A003242 counts anti-run compositions.
A106356 counts compositions by number of maximal anti-runs.
A124762 counts adjacent equal terms in standard compositions.
A124767 counts maximal runs in standard compositions.
A238130/A238279/A333755 count maximal runs in compositions.
A333381 counts maximal anti-runs in standard compositions.
A333382 counts adjacent unequal terms in standard compositions.
A333489 ranks anti-run compositions.
A333769 gives maximal run lengths in standard compositions.
A337565 gives maximal anti-run lengths in standard compositions.
Cf. A019472, A052841, A060223, A106351, A269134, A335461, A337505, A337564.

kv=2;
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],Length[Split[#,UnsameQ]]==kv&]],{n,0,6}]

a(n > 0) = (n - 1)*A005649(n - 2).

A373955 Numbers k such that the k-th integer composition in standard order contains two adjacent ones and no other runs.

Original entry on oeis.org

3, 11, 14, 19, 27, 28, 29, 35, 46, 51, 56, 57, 67, 75, 78, 83, 91, 92, 93, 99, 110, 112, 113, 114, 116, 118, 131, 139, 142, 155, 156, 157, 163, 179, 184, 185, 195, 203, 206, 211, 219, 220, 221, 224, 225, 226, 229, 230, 232, 233, 236, 237, 259, 267, 270, 275

Offset: 1

Gus Wiseman, Jun 29 2024

nonn

Also numbers k such that the excess compression of the k-th integer composition in standard order 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.
postn of 1 in

			The terms and corresponding compositions begin:
    3: (1,1)
   11: (2,1,1)
   14: (1,1,2)
   19: (3,1,1)
   27: (1,2,1,1)
   28: (1,1,3)
   29: (1,1,2,1)
   35: (4,1,1)
   46: (2,1,1,2)
   51: (1,3,1,1)
   56: (1,1,4)
   57: (1,1,3,1)
   67: (5,1,1)
   75: (3,2,1,1)
   78: (3,1,1,2)
   83: (2,3,1,1)
   91: (2,1,2,1,1)
   92: (2,1,1,3)
   93: (2,1,1,2,1)
   99: (1,4,1,1)

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

These compositions are counted by A373950.
Positions of ones in A373954.
A003242 counts compressed compositions (or anti-runs).
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 encodes compression using compositions in standard order.
A373949 counts compositions by compression-sum.
A373953 gives compression-sum of standard compositions.
Cf. A106356, A124762, A238130, A238279, A238343, A285981, A333382, A333489, A373951, A373952.

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

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 23 2024

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
These are also numbers k such that:
(1) the maximal weakly increasing runs in the reverse of the k-th composition in standard order do not have weakly decreasing leaders, and
(2) the maximal weakly increasing runs in the k-th composition in standard order do not have weakly decreasing leaders.

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

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

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

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

Intersection of A375138 and A375137.

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 18 2024

nonn

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

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

Eric Weisstein's World of Mathematics, Unimodal Sequence.

The version for run-lengths of compositions is A332833.
Compositions of this type are counted by A332834, complement maybe A329398.
A001523 counts unimodal compositions, ranks too dense.
A011782 counts compositions.
A114994 ranks weakly decreasing compositions, complement A335485.
A115981 counts non-unimodal compositions, ranked by A335373.
A225620 ranks weakly increasing compositions, complement A335486.
A238130, A238279, A333755 count compositions by number of runs.
A332835 counts compositions with weakly incr. or weakly decr. run-lengths.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of strict compositions are A233564.
- Ranks of constant compositions are A272919.
- Anti-runs are ranked by A333489, counted by A003242.
- Run-length transform is A333627, sum A070939.
Cf. A001511, A002051, A065120, A329744, A332745, A332836, A332870, A333218.

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

Intersection of A335485 and A335486.

A382288 Number of records in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

John Tyler Rascoe, Mar 20 2025

Here a record is a part of the composition that is greater than all parts before it, reading left to right. The first part of any nonempty composition is a record so a(n) >= 1 for n > 0. See A066099 for the standard order of integer compositions.

The first appearance of k occurs at n = A164894(k) for k > 0.

			The 883rd composition is (1, 2, 1, 1, 3, 1, 1) with records 1, 2, and 3; so a(883) = 3.
                          ^  ^        ^

Cf. A066099, A124762, A124767, A124768, A164894, A333381, A333382, A382312.

def comp(n):
    return # see A357625
def A382288(n):
    r,c = 0,0
    for i in comp(n):
        if i > r:
            c += 1
            r = i
    return c

a(A164894(n)) = n for n > 0.

cp's OEIS Frontend

A337504 Number of compositions of 2*n with n maximal anti-runs.

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A386578 Irregular triangle read by rows where T(n,k) is the number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent equal parts.

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A337506 Triangle read by rows where T(n,k) is the number of length-n sequences covering an initial interval of positive integers with k maximal anti-runs.

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

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A337507 Number of length-n sequences covering an initial interval of positive integers with exactly two maximal anti-runs, or with one pair of adjacent equal parts.

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A373955 Numbers k such that the k-th integer composition in standard order contains two adjacent ones and no other runs.

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

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

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