A128761 -id:A128761 - cp's OEIS frontend

A345192 Number of non-alternating compositions of n.

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.

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 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).

A025048 Number of up/down (initially ascending) compositions of n.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 7, 11, 16, 26, 41, 64, 100, 158, 247, 389, 612, 960, 1509, 2372, 3727, 5858, 9207, 14468, 22738, 35737, 56164, 88268, 138726, 218024, 342652, 538524, 846358, 1330160, 2090522, 3285526, 5163632, 8115323, 12754288, 20045027, 31503382

Offset: 0

David W. Wilson

nonn

Original name was: Ascending wiggly sums: number of sums adding to n in which terms alternately increase and decrease.

A composition is up/down if it is alternately strictly increasing and strictly decreasing, starting with an increase. For example, the partition (3,2,2,2,1) has no up/down permutations, even though it does have the anti-run permutation (2,3,2,1,2). - Gus Wiseman, Jan 15 2022

			From _Gus Wiseman_, Jan 15 2022: (Start)
The a(1) = 1 through a(7) = 11 up/down compositions:
  (1)  (2)  (3)    (4)      (5)      (6)        (7)
            (1,2)  (1,3)    (1,4)    (1,5)      (1,6)
                   (1,2,1)  (2,3)    (2,4)      (2,5)
                            (1,3,1)  (1,3,2)    (3,4)
                                     (1,4,1)    (1,4,2)
                                     (2,3,1)    (1,5,1)
                                     (1,2,1,2)  (2,3,2)
                                                (2,4,1)
                                                (1,2,1,3)
                                                (1,3,1,2)
                                                (1,2,1,2,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Alternating permutation

The case of permutations is A000111.
The undirected version is A025047, ranked by A345167.
The down/up version is A025049, ranked by A350356.
The strict case is A129838, undirected A349054.
The weak version is A129852, down/up A129853.
The version for patterns is A350354.
These compositions are ranked by A350355.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz compositions, complement A261983.
A011782 counts compositions, unordered A000041.
A325534 counts separable partitions, complement A325535.
A345192 counts non-alternating compositions, ranked by A345168.
A345194 counts alternating patterns, complement A350252.
A349052 counts weakly alternating compositions, complement A349053.
Cf. A008965, A049774, A128761, A344604, A344605, A344614, A344615, A345195, A349057, A349800.

updoQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]>y[[m+1]],y[[m]]Gus Wiseman, Jan 15 2022 *)

a(n) = 1 + A025047(n) - A025049(n) = Sum_k A059882(n,k). - Henry Bottomley, Feb 05 2001

a(n) ~ c * d^n, where d = 1.571630806607064114100138865739690782401305155950789062725011227781640624..., c = 0.4408955566119650057730070154620695491718230084159159991449729825619... . - Vaclav Kotesovec, Sep 12 2014

Name and offset changed by Gus Wiseman, Jan 15 2022

A025049 Number of down/up (initially descending) compositions of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 6, 9, 14, 23, 35, 55, 87, 136, 214, 337, 528, 830, 1306, 2051, 3223, 5067, 7962, 12512, 19667, 30908, 48574, 76343, 119982, 188565, 296358, 465764, 732006, 1150447, 1808078, 2841627, 4465992, 7018891, 11031101, 17336823, 27247087, 42822355

Offset: 0

David W. Wilson

nonn

Original name was: Descending wiggly sums: number of sums adding to n in which terms alternately decrease and increase.

A composition is down/up if it is alternately strictly decreasing and strictly increasing, starting with a decrease. For example, the partition (3,2,2,2,1) has no down/up permutations, even though it does have the anti-run permutation (2,1,2,3,2). - Gus Wiseman, Jan 28 2022

			From _Gus Wiseman_, Jan 28 2022: (Start)
The a(1) = 1 through a(8) = 14 down/up compositions:
  (1)  (2)  (3)    (4)    (5)      (6)        (7)        (8)
            (2,1)  (3,1)  (3,2)    (4,2)      (4,3)      (5,3)
                          (4,1)    (5,1)      (5,2)      (6,2)
                          (2,1,2)  (2,1,3)    (6,1)      (7,1)
                                   (3,1,2)    (2,1,4)    (2,1,5)
                                   (2,1,2,1)  (3,1,3)    (3,1,4)
                                              (4,1,2)    (3,2,3)
                                              (2,1,3,1)  (4,1,3)
                                              (3,1,2,1)  (5,1,2)
                                                         (2,1,3,2)
                                                         (2,1,4,1)
                                                         (3,1,3,1)
                                                         (4,1,2,1)
                                                         (2,1,2,1,2)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Mohammed L. Nadji, Moussa Ahmia, Daniel F. Checa, and José L. Ramírez, Arndt Compositions with Restricted Parts, Palindromes, and Colored Variants, J. Int. Seq. (2025) Vol. 28, Issue 3, Article 25.3.6. See p. 12.
Wikipedia, Alternating permutation

The case of permutations is A000111.
The undirected version is A025047, ranked by A345167.
The up/down version is A025048, ranked by A350355.
The strict case is A129838, undirected A349054.
The weak version is A129853, up/down A129852.
The version for patterns is A350354.
These compositions are ranked by A350356.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz compositions, complement A261983.
A011782 counts compositions, unordered A000041.
A325534 counts separable partitions, complement A325535.
A345192 counts non-alternating compositions, ranked by A345168.
A345194 counts alternating patterns, complement A350252.
A349052 counts weakly alternating compositions, complement A349053.
Cf. A008965, A049774, A128761, A344604, A344605, A344614, A344615, A345195, A349057, A349800.

doupQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],doupQ]],{n,0,15}] (* Gus Wiseman, Jan 28 2022 *)

a(n) = 1 + A025047(n) - A025048(n) = Sum_{k=1..n} A059883(n,k). - Henry Bottomley, Feb 05 2001

a(0)=1 prepended by Alois P. Heinz, Jan 20 2022

Name changed by Gus Wiseman, Jan 28 2022

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

A344615 Number of compositions of n with no adjacent triples (..., x, y, z, ...) where x <= y <= z.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 17, 29, 50, 84, 143, 241, 408, 688, 1162, 1959, 3305, 5571, 9393, 15832, 26688, 44980, 75812, 127769, 215338, 362911, 611620, 1030758, 1737131, 2927556, 4933760, 8314754, 14012668, 23615198, 39798098, 67070686, 113032453, 190490542, 321028554

Offset: 0

Gus Wiseman, May 27 2021

nonn

These compositions avoid the weak consecutive pattern (1,2,3), the strict version being A128761.

			The a(1) = 1 through a(6) = 17 compositions:
  (1)  (2)    (3)    (4)      (5)        (6)
       (1,1)  (1,2)  (1,3)    (1,4)      (1,5)
              (2,1)  (2,2)    (2,3)      (2,4)
                     (3,1)    (3,2)      (3,3)
                     (1,2,1)  (4,1)      (4,2)
                     (2,1,1)  (1,3,1)    (5,1)
                              (2,1,2)    (1,3,2)
                              (2,2,1)    (1,4,1)
                              (3,1,1)    (2,1,3)
                              (1,2,1,1)  (2,3,1)
                                         (3,1,2)
                                         (3,2,1)
                                         (4,1,1)
                                         (1,2,1,2)
                                         (1,3,1,1)
                                         (2,1,2,1)
                                         (2,2,1,1)

The case of permutations is A049774.
The strict non-adjacent version is A102726.
The case of permutations of prime indices is A344652.
A001250 counts alternating permutations.
A005649 counts anti-run patterns.
A106356 counts compositions by number of maximal anti-runs.
A114901 counts compositions where each part is adjacent to an equal part.
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, A027187, A238279/A333755, A333213, A335464, A335515, A344612, A344619.

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

More terms from Bert Dobbelaere, Jun 12 2021

A344605 Number of alternating patterns of length n, including pairs (x,x).

Original entry on oeis.org

1, 1, 3, 6, 22, 102, 562, 3618, 26586, 219798, 2018686, 20393790, 224750298, 2683250082, 34498833434, 475237879950, 6983085189454, 109021986683046, 1802213242949602, 31447143854808378, 577609702827987882, 11139837273501641502, 225075546284489412854

Offset: 0

Gus Wiseman, May 27 2021

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. A sequence is alternating (cf. A025047) including pairs (x,x) if there are no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z. These sequences avoid the weak consecutive patterns (1,2,3) and (3,2,1).
An alternating pattern of length > 2 is necessarily an anti-run (A005649).
The version without pairs (x,x) is identical to this sequence except a(2) = 2 instead of 3.

			The a(0) = 1 through a(4) = 22 patterns:
  ()  (1)  (1,1)  (1,2,1)  (1,2,1,2)
           (1,2)  (1,3,2)  (1,2,1,3)
           (2,1)  (2,1,2)  (1,3,1,2)
                  (2,1,3)  (1,3,2,3)
                  (2,3,1)  (1,3,2,4)
                  (3,1,2)  (1,4,2,3)
                           (2,1,2,1)
                           (2,1,3,1)
                           (2,1,3,2)
                           (2,1,4,3)
                           (2,3,1,2)
                           (2,3,1,3)
                           (2,3,1,4)
                           (2,4,1,3)
                           (3,1,2,1)
                           (3,1,3,2)
                           (3,1,4,2)
                           (3,2,3,1)
                           (3,2,4,1)
                           (3,4,1,2)
                           (4,1,3,2)
                           (4,2,3,1)

Martin Ehrenstein, Table of n, a(n) for n = 0..25

The version for permutations is A001250.
The version for compositions is A344604.
The version for permutations of prime indices is A344606.
A000670 counts patterns (ranked by A333217).
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A019536 counts necklace patterns.
A025047 counts alternating or wiggly compositions, complement A345192.
A226316 counts patterns avoiding (1,2,3) (weakly: A052709).
A335515 counts patterns matching (1,2,3).
Cf. A000041, A006330, A049774, A102726, A103919, A124754, A128761, A335456, A335457, A335517, A344612, A344614, A344615, A348377.

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],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]],{n,0,6}]

a(10) and beyond from Martin Ehrenstein, Jun 10 2021

A349053 Number of non-weakly alternating integer compositions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 12, 37, 95, 232, 533, 1198, 2613, 5619, 11915, 25011, 52064, 107694, 221558, 453850, 926309, 1884942, 3825968, 7749312, 15667596, 31628516, 63766109, 128415848, 258365323, 519392582, 1043405306, 2094829709, 4203577778, 8431313237, 16904555958

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. Then a sequence is (strongly) alternating iff it is a weakly alternating anti-run.

			The a(6) = 12 compositions:
  (1,1,2,2,1)  (1,1,2,3)  (1,2,4)
  (1,2,1,1,2)  (1,2,3,1)  (4,2,1)
  (1,2,2,1,1)  (1,3,2,1)
  (2,1,1,2,1)  (2,1,1,3)
               (3,1,1,2)
               (3,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000 (terms 0..55 from Martin Ehrenstein)
Wikipedia, Alternating permutation

Complementary directed versions are A129852/A129853, strong A025048/A025049.
The strong version is A345192.
The complement is counted by A349052.
These compositions are ranked by A349057, strong A345168.
The complementary version for patterns is A349058, strong A345194.
The complementary multiplicative version is A349059, strong A348610.
An unordered version (partitions) is A349061, complement A349060.
The version for ordered prime factorizations is A349797, complement A349056.
The version for patterns is A350138, strong A350252.
The version for ordered factorizations is A350139.
A001250 counts alternating permutations, complement A348615.
A001700 counts compositions of 2n with alternating sum 0.
A003242 counts Carlitz (anti-run) compositions.
A011782 counts compositions, unordered A000041.
A025047 counts alternating compositions, ranked by A345167.
A106356 counts compositions by number of maximal anti-runs.
A344604 counts alternating compositions with twins.
A345164 counts alternating ordered prime factorizations.
A349054 counts strict alternating compositions.
Cf. A102726, A114901, A128761, A261983, A333213, A333755, A344614, A344615, A345165, A345170, A345195, A349799, A349800, A350251, A350252.

wwkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}]||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],!wwkQ[#]&]],{n,0,10}]

a(n) = A011782(n) - A349052(n).

a(21)-a(35) from Martin Ehrenstein, Jan 08 2022

A349052 Number of weakly alternating compositions of n.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 28, 52, 91, 161, 280, 491, 850, 1483, 2573, 4469, 7757, 13472, 23378, 40586, 70438, 122267, 212210, 368336, 639296, 1109620, 1925916, 3342755, 5801880, 10070133, 17478330, 30336518, 52653939, 91389518, 158621355, 275313226, 477850887, 829388075

Offset: 0

Gus Wiseman, Nov 29 2021

nonn

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

			The a(5) = 16 compositions:
  (1,1,1,1,1)  (1,1,1,2)  (1,1,3)  (1,4)  (5)
               (1,1,2,1)  (1,2,2)  (2,3)
               (1,2,1,1)  (1,3,1)  (3,2)
               (2,1,1,1)  (2,1,2)  (4,1)
                          (2,2,1)
                          (3,1,1)
The a(6) = 28 compositions:
  (111111)  (11112)  (1113)  (114)  (15)  (6)
            (11121)  (1122)  (132)  (24)
            (11211)  (1131)  (141)  (33)
            (12111)  (1212)  (213)  (42)
            (21111)  (1311)  (222)  (51)
                     (2121)  (231)
                     (2211)  (312)
                     (3111)  (411)

Andrew Howroyd, Table of n, a(n) for n = 0..1000 (Terms 0..55 from Martin Ehrenstein)

The strong case is A025047, ranked by A345167.
The directed versions are A129852 and A129853, strong A025048 and A025049.
The complement is counted by A349053, strong A345192.
The version for permutations of prime indices is A349056, strong A345164.
The complement is ranked by A349057, strong A345168.
The version for patterns is A349058, strong A345194.
The multiplicative version is A349059, strong A348610.
An unordered version (partitions) is A349060, complement A349061.
The non-alternating case is A349800, ranked by A349799.
A001250 counts alternating permutations, complement A348615.
A001700 counts compositions of 2n with alternating sum 0.
A003242 counts Carlitz (anti-run) compositions.
A011782 counts compositions.
A106356 counts compositions by number of maximal anti-runs.
A344604 counts alternating compositions with twins.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A349054 counts strict alternating compositions.
Cf. A000041, A008965, A102726, A114901, A128761, A261983, A333213, A333755, A344614, A344615, A345165, A345195.

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[-#]&]],{n,0,10}]

C(n,f)={my(M=matrix(n,n,j,k,k>=j), s=M[,n]); for(b=1, n, f=!f; M=matrix(n,n,j,k, if(k1,M[j-k,k-1]) ))); for(k=2, n, M[,k]+=M[,k-1]); s+=M[,n]); s~}
seq(n) = concat([1], C(n,0) + C(n,1) - vector(n,j,numdiv(j))) \\ Andrew Howroyd, Jan 31 2024

a(21)-a(37) from Martin Ehrenstein, Jan 08 2022

A345194 Number of alternating patterns of length n.

Original entry on oeis.org

1, 1, 2, 6, 22, 102, 562, 3618, 26586, 219798, 2018686, 20393790, 224750298, 2683250082, 34498833434, 475237879950, 6983085189454, 109021986683046, 1802213242949602, 31447143854808378, 577609702827987882, 11139837273501641502, 225075546284489412854

Offset: 0

Gus Wiseman, Jun 17 2021

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.
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). An alternating pattern is necessarily an anti-run (A005649).
The version with twins (A344605) is identical to this sequence except with a(2) = 3 instead of 2.
From Gus Wiseman, Jan 16 2022: (Start)
Conjecture: Also the number of weakly up/down patterns of length n, where a sequence is weakly up/down if it is alternately weakly increasing and weakly decreasing, starting with an increase. For example, the a(0) = 1 through a(3) = 6 weakly up/down patterns are:
  ()  (1)  (1,1)  (1,1,1)
           (2,1)  (1,1,2)
                  (2,1,1)
                  (2,1,2)
                  (2,1,3)
                  (3,1,2)
(End)

			The a(0) = 1 through a(3) = 6 alternating patterns:
  ()  (1)  (1,2)  (1,2,1)
           (2,1)  (1,3,2)
                  (2,1,2)
                  (2,1,3)
                  (2,3,1)
                  (3,1,2)

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

The version for permutations is A001250, complement A348615.
The version for compositions is A025047, complement A345192.
The version with twins (x,x) is A344605.
The version for perms of prime indices is A345164, complement A350251.
The version for factorizations is A348610, complement A348613, weak A349059.
The weak version is A349058, complement A350138, compositions A349052.
The complement is counted by A350252.
A000670 = patterns, ranked by A333217.
A003242 = anti-run compositions.
A005649 = anti-run patterns, complement A069321.
A019536 = necklace patterns.
A129852 and A129853 = up/down and down/up compositions.
A226316 = patterns avoiding (1,2,3), weakly A052709, complement A335515.
A345170 = partitions w/ alternating permutation, complement A345165.
A349055 = normal multisets w/ alternating permutation, complement A349050.
Cf. A006330, A049774, A128761, A335456, A335457, A335517, A344614, A344615, A345167, A345168.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
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],wigQ]],{n,0,6}]

F(p,x) = {sum(k=0, p, (-1)^((k+1)\2)*binomial((p+k)\2, k)*x^k)}
R(n,k) = {Vec(if(k==1, x, 2*F(k-2,-x)/F(k-1,x)-2-(k-2)*x) + O(x*x^n))}
seq(n)= {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Feb 04 2022

a(n) = 2*A350354(n) for n >= 2. - Andrew Howroyd, Feb 04 2022

a(10)-a(18) from Alois P. Heinz, Dec 10 2021

Terms a(19) and beyond from Andrew Howroyd, Feb 04 2022

cp's OEIS Frontend

A345192 Number of non-alternating compositions of n.

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A025048 Number of up/down (initially ascending) compositions of n.

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A025049 Number of down/up (initially descending) compositions of n.

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

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

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A344615 Number of compositions of n with no adjacent triples (..., x, y, z, ...) where x <= y <= z.

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A344605 Number of alternating patterns of length n, including pairs (x,x).

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A349053 Number of non-weakly alternating integer compositions of n.