A333213 -id:A333213 - cp's OEIS frontend

A374759 Numbers k such that the leaders of strictly decreasing runs in the k-th composition in standard order are identical.

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 15, 16, 17, 18, 21, 22, 31, 32, 33, 34, 36, 37, 42, 45, 63, 64, 65, 66, 68, 69, 73, 76, 85, 86, 90, 127, 128, 129, 130, 132, 133, 136, 137, 146, 148, 153, 170, 173, 181, 182

Offset: 1

Gus Wiseman, Jul 29 2024

nonn

The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.

			The 18789th composition in standard order is (3,3,2,1,3,2,1), with strictly decreasing runs ((3),(3,2,1),(3,2,1)), with leaders (3,3,3), so 18789 is in the sequence.
The terms together with the corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   5: (2,1)
   7: (1,1,1)
   8: (4)
   9: (3,1)
  10: (2,2)
  15: (1,1,1,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  21: (2,2,1)
  22: (2,1,2)
  31: (1,1,1,1,1)
  32: (6)
  33: (5,1)
  34: (4,2)
  36: (3,3)
  37: (3,2,1)

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

For leaders of anti-runs we have A374519 (counted by A374517).
For leaders of weakly increasing runs we have A374633, counted by A374631.
The opposite version is A374685 (counted by A374686).
The weak version is A374744.
Compositions of this type are counted by A374760.
For distinct instead of identical runs we have A374767 (counted by A374761).
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.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Ranks of non-contiguous compositions are A374253, counted by A335548.
Six types of runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A000961, A065120, A106356, A188920, A189076, A238343, A272919, A333213, A374698, A374706, A374758, A375128.

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

A188900 Number of compositions of n that avoid the pattern 12-3.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 31, 60, 114, 215, 402, 746, 1375, 2520, 4593, 8329, 15036, 27027, 48389, 86314, 153432, 271853, 480207, 845804, 1485703, 2603018, 4549521, 7933239, 13803293, 23966682, 41530721, 71830198, 124010381, 213725823, 367736268, 631723139, 1083568861

Offset: 0

Nathaniel Johnston, Apr 17 2011

nonn

First differs from the non-dashed version A102726 at a(9) = 215, A102726(9) = 214, due to the composition (1,3,2,3).
The value a(11) = 7464 in Heubach et al. is a typo.
Theorem: A composition avoids 3-12 iff its leaders of maximal weakly decreasing runs are weakly increasing. For example, the composition q = (1,1,2,1,2,2,1,3) has maximal weakly decreasing runs ((1,1),(2,1),(2,2,1),(3)), with leaders (1,2,2,3), which are weakly increasing, so q is counted under a(13); also q avoids 3-12, as required. On the other hand, the composition q = (3,2,1,2,2,1,2) has maximal weakly decreasing runs ((3,2,1),(2,2,1),(2)), with leaders (3,2,2), which are not weakly increasing, so q is not counted under a(13); also q matches 3-12, as required. - Gus Wiseman, Aug 21 2024

			The initial terms are too dense, but see A375406 for the complement. - _Gus Wiseman_, Aug 21 2024

Nathaniel Johnston, Table of n, a(n) for n = 0..200
S. Heubach, T. Mansour and A. O. Munagi, Avoiding Permutation Patterns of Type (2,1) in Compositions, Online Journal of Analytic Combinatorics, 4 (2009).

The non-dashed version A102726, non-ranks A335483.
For 23-1 we have A189076.
The non-ranks are a subset of A335479 and do not include 404, 788, 809, ...
For strictly increasing leaders we have A358836, ranks A326533.
The strict version is A374762.
The complement is counted by A375406.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
Cf. A106356, A188920, A238343, A261982, A333175, A333213, A374630, A374679, A374688, A374740, A374741.

with(PolynomialTools):n:=20:taypoly:=taylor(mul(1/(1 - x^i/mul(1-x^j,j=1..i-1)),i=1..n),x=0,n+1):seq(coeff(taypoly,x,m),m=0..n);

m = 35;
Product[1/(1 - x^i/Product[1 - x^j, {j, 1, i - 1}]), {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Mar 31 2020 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], LessEqual@@First/@Split[#,GreaterEqual]&]],{n,0,15}] (* Gus Wiseman, Aug 21 2024 *)

G.f.: Product_{i>=1} (1/(1 - x^i/Product_{j=1..i-1} (1 - x^j))).

a(n) = 2^(n-1) - A375406(n). - Gus Wiseman, Aug 22 2024

A375128 Irregular triangle read by rows where row n lists the minima of maximal strictly increasing runs in the weakly increasing prime indices of n.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 4, 1, 1, 1, 2, 2, 1, 5, 1, 1, 6, 1, 2, 1, 1, 1, 1, 7, 1, 2, 8, 1, 1, 2, 1, 9, 1, 1, 1, 3, 3, 1, 2, 2, 2, 1, 1, 10, 1, 11, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 12, 1, 2, 1, 1, 1, 13, 1, 14, 1, 1, 2, 2, 1, 15, 1, 1, 1, 1, 4, 4, 1, 3, 2, 1, 1, 16

Offset: 1

Gus Wiseman, Aug 04 2024

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.

The minima of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.

			The prime indices of 540 are {1,1,2,2,2,3}, with strictly increasing runs ({1},{1,2},{2},{2,3}), with minima (1,1,2,2), which is row 540.
Triangle begins:
   1:
   2:  1
   3:  2
   4:  1  1
   5:  3
   6:  1
   7:  4
   8:  1  1  1
   9:  2  2
  10:  1
  11:  5
  12:  1  1
  13:  6
  14:  1
  15:  2
  16:  1  1  1  1

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

Row-minima are A055396.
Row-sums are A374706.
Row-lengths are A375136.
For leaders of constant runs we have A304038, row-sums A066328.
For compositions we have A374683, row-sums of A374684 (length A124768).
A112798 lists prime indices:
- length A001222, distinct A001221
- leader A055396
- sum A056239
- reverse A296150
Cf. A034296, A141199, A218482, A279790, A320324, A333213, A358836, A374634, A374700, A374758, A375133.

Table[If[n==1,{},First/@Split[Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]],Less]],{n,100}]

A333230 Positions of weak ascents in the sequence of differences between primes.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 10, 13, 14, 15, 17, 20, 22, 23, 26, 28, 29, 31, 33, 35, 36, 38, 39, 41, 43, 45, 46, 49, 50, 52, 54, 55, 57, 60, 61, 64, 65, 67, 69, 70, 71, 73, 75, 76, 78, 79, 81, 83, 85, 86, 89, 90, 93, 95, 96, 98, 100, 102, 104, 105, 107, 109, 110, 113

Offset: 1

Gus Wiseman, Mar 18 2020

Partial sums of A333252.

			The prime gaps split into the following strictly decreasing subsequences: (1), (2), (2), (4,2), (4,2), (4), (6,2), (6,4,2), (4), (6), (6,2), (6,4,2), (6,4), (6), (8,4,2), ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Thomas Bloom, Let d_n = p_(n+1)-p_n. The set of n such that d_(n+1) >= d_n has density 1/2, and similarly for d_(n+1) <= d_n. Furthermore, there are infinitely many n such that d_(n+1) = d_n, Erdős Problems.
Terence Tao, Erdős problem database, see no. 218.

The version for the Kolakoski sequence is A022297.
The version for equal differences is A064113.
The version for strict ascents is A258025.
The version for strict descents is A258026.
The version for distinct differences is A333214.
The version for weak descents is A333231.
First differences are A333252 (if the first term is 0).
Prime gaps are A001223.
Weakly decreasing runs of standard compositions are counted by A124765.
Weakly increasing runs of standard compositions are counted by A124766.
Strictly increasing runs of standard compositions are counted by A124768.
Strictly decreasing runs of standard compositions are counted by A124769.
Runs of prime gaps with nonzero differences are A333216.
Cf. A000040, A000720, A036263, A054819, A084758, A333212, A333213, A333215.

Accumulate[Length/@Split[Differences[Array[Prime,100]],#1>#2&]]//Most
(* or *)
Select[Range[100],Prime[#+1]-Prime[#]<=Prime[#+2]-Prime[#+1]&]

Numbers k such that prime(k+2) - 2*prime(k+1) + prime(k) >= 0.

A374679 Number of integer compositions of n whose leaders of anti-runs are strictly increasing.

Original entry on oeis.org

1, 1, 1, 3, 4, 8, 15, 24, 45, 84, 142, 256, 464, 817, 1464, 2621, 4649, 8299, 14819, 26389, 47033, 83833, 149325, 266011, 473867, 843853

Offset: 0

Gus Wiseman, Aug 01 2024

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.

			The a(0) = 1 through a(6) = 15 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)
                (12)  (13)   (14)   (15)
                (21)  (31)   (23)   (24)
                      (121)  (32)   (42)
                             (41)   (51)
                             (122)  (123)
                             (131)  (132)
                             (212)  (141)
                                    (213)
                                    (231)
                                    (312)
                                    (321)
                                    (1212)
                                    (1221)
                                    (2121)

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

For distinct but not necessarily increasing leaders we have A374518.
For partitions instead of compositions we have A375134.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A000041.
- For leaders of weakly increasing runs we have A374634.
- For leaders of strictly increasing runs we have A374688.
- For leaders of strictly decreasing runs we have A374762.
Other types of run-leaders (instead of strictly increasing):
- For identical leaders we have A374517.
- For distinct leaders we have A374518.
- For weakly increasing leaders we have A374681.
- For weakly decreasing leaders we have A374682.
- For strictly decreasing leaders we have A374680.
A003242 counts anti-runs, ranks A333489.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs.
A238424 counts partitions whose first differences are an anti-run.
A274174 counts contiguous compositions, ranks A374249.
Cf. A188920, A238343, A333213, A333381, A373949, A374515, A374632, A374635, A374678, A374700, A374706, A375133.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Less@@First/@Split[#,UnsameQ]&]],{n,0,15}]

A344652 Number of permutations of the prime indices of n with no adjacent triples (..., x, y, z, ...) such that x <= y <= z.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 2, 1, 2, 2, 0, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 0, 2, 1, 5, 1, 0, 2, 2, 2, 3, 1, 2, 2, 1, 1, 5, 1, 2, 2, 2, 1, 0, 1, 2, 2, 2, 1, 1, 2, 1, 2, 2, 1, 7, 1, 2, 2, 0, 2, 5, 1, 2, 2, 5, 1, 2, 1, 2, 2, 2, 2, 5, 1, 0, 0, 2, 1, 7, 2, 2, 2

Offset: 1

Gus Wiseman, Jun 17 2021

nonn

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.

			The permutations for n = 2, 6, 8, 30, 36, 60, 180, 210, 360:
  (1)  (12)  (132)  (1212)  (1213)  (12132)  (1324)  (121213)
       (21)  (213)  (2121)  (1312)  (13212)  (1423)  (121312)
             (231)  (2211)  (1321)  (13221)  (1432)  (121321)
             (312)          (2131)  (21213)  (2143)  (131212)
             (321)          (2311)  (21312)  (2314)  (132121)
                            (3121)  (21321)  (2413)  (132211)
                            (3211)  (22131)  (2431)  (212131)
                                    (23121)  (3142)  (213121)
                                    (23211)  (3214)  (213211)
                                    (31212)  (3241)  (221311)
                                    (32121)  (3412)  (231211)
                                    (32211)  (3421)  (312121)
                                             (4132)  (321211)
                                             (4213)
                                             (4231)
                                             (4312)
                                             (4321)

All permutations of prime indices are counted by A008480.
The case of permutations is A049774.
Avoiding (3,2,1) also gives A344606.
The wiggly case is A345164.
A001250 counts wiggly permutations.
A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
A056239 adds up prime indices, row sums of A112798.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A335452 counts anti-run permutations of prime indices.
A345170 counts partitions with a wiggly permutation, ranked by A345172.
A345192 counts non-wiggly compositions, ranked by A345168.
Counting compositions by patterns:
- A102726 avoiding (1,2,3).
- A128761 avoiding (1,2,3) adjacent.
- A335514 matching (1,2,3).
- A344614 avoiding (1,2,3) and (3,2,1) adjacent.
- A344615 weakly avoiding (1,2,3) adjacent.
Cf. A001222, A003242, A056986, A316524, A333213, A335511, A344604, A344653, A344654, A345167, A345173.

Table[Length[Select[Permutations[Flatten[ ConstantArray@@@FactorInteger[n]]],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z]&]],{n,100}]

A374744 Numbers k such that the leaders of weakly decreasing runs in the k-th composition in standard order (A066099) are identical.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 21, 22, 23, 31, 32, 33, 34, 35, 36, 37, 39, 42, 43, 45, 46, 47, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 79, 85, 86, 87, 90, 91, 93, 94, 95, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138

Offset: 1

Gus Wiseman, Jul 24 2024

nonn

The leaders of weakly decreasing runs in a sequence are obtained by splitting into maximal weakly decreasing 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 terms together with the corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   5: (2,1)
   7: (1,1,1)
   8: (4)
   9: (3,1)
  10: (2,2)
  11: (2,1,1)
  15: (1,1,1,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  19: (3,1,1)
  21: (2,2,1)
  22: (2,1,2)
  23: (2,1,1,1)
  31: (1,1,1,1,1)

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

Other types of runs and their counts: A272919 (A000005), A374519 (A374517), A374685 (A374686), A374759 (A374760).
The opposite is A374633, counted by A374631.
For distinct (instead of identical) leaders we have A374701, count A374743.
Positions of constant rows in A374740, opposite A374629, cf. A374630.
Compositions of this type are counted by A374742.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A374748 counts compositions by sum of leaders of weakly decreasing runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) (or sometimes A070939).
- Parts are listed by A066099.
- Adjacent equal pairs are counted by 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.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
Cf. A065120, A106356, A238343, A333175, A333213, A335456, A373949, A374635, A374634, A374768.

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

A333215 Lengths of maximal weakly increasing subsequences in the sequence of prime gaps (A001223).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 14 2020

nonn

Prime gaps are differences between adjacent prime numbers.

			The prime gaps split into the following weakly increasing subsequences: (1,2,2,4), (2,4), (2,4,6), (2,6), (4), (2,4,6,6), (2,6), (4), (2,6), (4,6,8), (4), (2,4), (2,4,14), ...

Prime gaps are A001223.
Ones correspond to strong prime quartets A054804.
Weakly increasing runs of compositions in standard order are A124766.
First differences of A258026 (with zero prepended).
The version for the Kolakoski sequence is A332875.
The weakly decreasing version is A333212.
The unequal version is A333216.
Positions of weak ascents in prime gaps are A333230.
The strictly decreasing version is A333252.
The strictly increasing version is A333253.
The equal version is A333254.
Cf. A000040, A000720, A036263, A054819, A064113, A084758, A124765, A124768, A258025, A333213, A333214.

Length/@Split[Differences[Array[Prime,100]],#1<=#2&]//Most

Ones correspond to strong prime quartets (A054804), so the sum of terms up to but not including the n-th one is A000720(A054804(n - 1)).

A349054 Number of alternating strict compositions of n. Number of alternating (up/down or down/up) permutations of strict integer partitions of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 9, 11, 15, 21, 35, 41, 59, 75, 103, 155, 193, 255, 339, 443, 569, 841, 1019, 1365, 1743, 2295, 2879, 3785, 5151, 6417, 8301, 10625, 13567, 17229, 21937, 27509, 37145, 45425, 58345, 73071, 93409, 115797, 147391, 182151, 229553, 297061, 365625

Offset: 0

Gus Wiseman, Dec 21 2021

nonn

A strict composition of n is a finite sequence of distinct positive integers summing to n.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either.
The case starting with an increase (or decrease, it doesn't matter in the enumeration) is counted by A129838.

			The a(1) = 1 through a(7) = 11 compositions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)
            (1,2)  (1,3)  (1,4)  (1,5)    (1,6)
            (2,1)  (3,1)  (2,3)  (2,4)    (2,5)
                          (3,2)  (4,2)    (3,4)
                          (4,1)  (5,1)    (4,3)
                                 (1,3,2)  (5,2)
                                 (2,1,3)  (6,1)
                                 (2,3,1)  (1,4,2)
                                 (3,1,2)  (2,1,4)
                                          (2,4,1)
                                          (4,1,2)

Wikipedia, Alternating permutation

Ranking sequences are put in parentheses below.
This is the strict case of A025047/A025048/A025049 (A345167).
This is the alternating case of A032020 (A233564).
The unordered case (partitions) is A065033.
The directed case is A129838.
A001250 = alternating permutations (A349051), complement A348615 (A350250).
A003242 = Carlitz (anti-run) compositions, complement A261983.
A011782 = compositions, unordered A000041.
A345165 = partitions without an alternating permutation (A345171).
A345170 = partitions with an alternating permutation (A345172).
A345192 = non-alternating compositions (A345168).
A345195 = non-alternating anti-run compositions (A345169).
A349800 = weakly but not strongly alternating compositions (A349799).
A349052 = weakly alternating compositions, complement A349053 (A349057).
Cf. A000111, A008965, A015723, A114901, A128761, A129852, A129853, A218074, A333213, A344614, A345164, A345194, A349060, A349795.

g:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(g(o-1+j, u-j), j=1..u))
    end:
b:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
      `if`(k=0, `if`(n=0, 2, 0), b(n-k, k)+b(n-k, k-1)))
    end:
a:= n-> add(b(n, k)*g(k, 0), k=0..floor((sqrt(8*n+1)-1)/2))-1:
seq(a(n), n=0..46);  # Alois P. Heinz, Dec 22 2021

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

a(n) = 2 * A129838(n) - 1.

G.f.: Sum_{n>0} A001250(n)*x^(n*(n+1)/2)/Product_{k=1..n}(1-x^k).

A349800 Number of integer compositions of n that are weakly alternating and have at least two adjacent equal parts.

Original entry on oeis.org

0, 0, 1, 1, 4, 9, 16, 33, 62, 113, 205, 373, 664, 1190, 2113, 3744, 6618, 11683, 20564, 36164, 63489, 111343, 195042, 341357, 596892, 1042976, 1821179, 3178145, 5543173, 9663545, 16839321, 29332231, 51075576, 88908912, 154722756, 269186074, 468221264

Offset: 0

Gus Wiseman, Dec 16 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

This sequence counts compositions that are weakly but not strongly alternating; also weakly alternating non-anti-run compositions.

			The a(2) = 1 through a(6) = 16 compositions:
  (1,1)  (1,1,1)  (2,2)      (1,1,3)      (3,3)
                  (1,1,2)    (1,2,2)      (1,1,4)
                  (2,1,1)    (2,2,1)      (2,2,2)
                  (1,1,1,1)  (3,1,1)      (4,1,1)
                             (1,1,1,2)    (1,1,1,3)
                             (1,1,2,1)    (1,1,2,2)
                             (1,2,1,1)    (1,1,3,1)
                             (2,1,1,1)    (1,3,1,1)
                             (1,1,1,1,1)  (2,2,1,1)
                                          (3,1,1,1)
                                          (1,1,1,1,2)
                                          (1,1,1,2,1)
                                          (1,1,2,1,1)
                                          (1,2,1,1,1)
                                          (2,1,1,1,1)
                                          (1,1,1,1,1,1)

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

This is the weakly alternating case of A345192, ranked by A345168.
The case of partitions is A349795, ranked by A350137.
The version counting permutations of prime indices is A349798.
These compositions are ranked by A349799.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A003242 = Carlitz (anti-run) compositions, ranked by A333489.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A261983 = non-anti-run compositions, ranked by A348612.
A345165 = partitions without an alternating permutation, ranked by A345171.
A345170 = partitions with an alternating permutation, ranked by A345172.
A345173 = non-alternating anti-run partitions, ranked by A345166.
A345195 = non-alternating anti-run compositions, ranked by A345169.
A348377 = non-alternating non-twin compositions.
A349801 = non-alternating partitions, ranked by A289553.
Weakly alternating:
- A349052 = compositions, directed A129852/A129853, complement A349053.
- A349056 = permutations of prime indices, complement A349797.
- A349057 = complement of standard composition numbers (too dense).
- A349058 = patterns, complement A350138.
- A349059 = ordered factorizations, complement A350139.
- A349060 = partitions, complement A349061.
Cf. A008965, A011782, A027383, A096441, A274230, A333213, A344614, A344615, A348382, A348613, A349796, A350140.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y] &&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],(whkQ[#]||whkQ[-#])&&!wigQ[#]&]],{n,0,10}]

a(n) = A349052(n) - A025047(n). - Andrew Howroyd, Jan 31 2024

a(21) onwards from Andrew Howroyd, Jan 31 2024

cp's OEIS Frontend

A374759 Numbers k such that the leaders of strictly decreasing runs in the k-th composition in standard order are identical.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A188900 Number of compositions of n that avoid the pattern 12-3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A375128 Irregular triangle read by rows where row n lists the minima of maximal strictly increasing runs in the weakly increasing prime indices of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A333230 Positions of weak ascents in the sequence of differences between primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A374679 Number of integer compositions of n whose leaders of anti-runs are strictly increasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A344652 Number of permutations of the prime indices of n with no adjacent triples (..., x, y, z, ...) such that x <= y <= z.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A374744 Numbers k such that the leaders of weakly decreasing runs in the k-th composition in standard order (A066099) are identical.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A333215 Lengths of maximal weakly increasing subsequences in the sequence of prime gaps (A001223).

Views

Author

Keywords

Comments

Examples

Crossrefs