A345171 -id:A345171 - cp's OEIS frontend

A025047 Number of alternating compositions, i.e., compositions with alternating increases and decreases, starting with either an increase or a decrease.

1, 1, 1, 3, 4, 7, 12, 19, 29, 48, 75, 118, 186, 293, 460, 725, 1139, 1789, 2814, 4422, 6949, 10924, 17168, 26979, 42404, 66644, 104737, 164610, 258707, 406588, 639009, 1004287, 1578363, 2480606, 3898599, 6127152, 9629623, 15134213, 23785388, 37381849, 58750468

Offset: 0

David W. Wilson

nonn

Original name: Wiggly sums: number of sums adding to n in which terms alternately increase and decrease or vice versa.

			From _Joerg Arndt_, Dec 28 2012: (Start)
There are a(7)=19 such compositions of 7:
[ 1] +  [ 1 2 1 2 1 ]
[ 2] +  [ 1 2 1 3 ]
[ 3] +  [ 1 3 1 2 ]
[ 4] +  [ 1 4 2 ]
[ 5] +  [ 1 5 1 ]
[ 6] +  [ 1 6 ]
[ 7] -  [ 2 1 3 1 ]
[ 8] -  [ 2 1 4 ]
[ 9] +  [ 2 3 2 ]
[10] +  [ 2 4 1 ]
[11] +  [ 2 5 ]
[12] -  [ 3 1 2 1 ]
[13] -  [ 3 1 3 ]
[14] +  [ 3 4 ]
[15] -  [ 4 1 2 ]
[16] -  [ 4 3 ]
[17] -  [ 5 2 ]
[18] -  [ 6 1 ]
[19] 0  [ 7 ]
For A025048(7)-1=10 of these the first two parts are increasing (marked by '+'),
and for A025049(7)-1=8 the first two parts are decreasing (marked by '-').
The composition into one part is counted by both A025048 and A025049.
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..3333
Edward A. Bender and E. Rodney Canfield, Locally Restricted Compositions III. Adjacent-Part Periodic Inequalities, Electronic Journal of Combinatorics 17 (2010), #R145.

Dominated by A003242 (anti-run compositions), complement A261983.
The ascending case is A025048.
The descending case is A025049.
The version allowing pairs (x,x) is A344604.
These compositions are ranked by A345167, permutations A349051.
The complement is counted by A345192, ranked by A345168.
The version for patterns is A345194 (with twins: A344605).
A001250 counts alternating permutations, complement A348615.
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.
A345164 counts alternating permutations of prime indices.
A345165 counts partitions w/o alternating permutation, ranked by A345171.
A345170 counts partitions w/ alternating permutation, ranked by A345172.
Cf. A000070, A008965, A238279, A333755, A344606, A344614, A344653, A344740, A345163, A345166, A345169.

b:= proc(n, l, t) option remember; `if`(n=0, 1, add(
      b(n-j, j, 1-t), j=`if`(t=1, 1..min(l-1, n), l+1..n)))
    end:
a:= n-> 1+add(add(b(n-j, j, i), i=0..1), j=1..n-1):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 31 2024

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}] (* Gus Wiseman, Jun 17 2021 *)

D(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]), M[j-k,n]-M[j-k,k] ))); for(k=2, n, M[,k]+=M[,k-1]); s+=M[,n]); s~}
seq(n) = concat([1], D(n,0) + D(n,1) - vector(n,j,1)) \\ Andrew Howroyd, Jan 31 2024

a(n) = A025048(n) + A025049(n) - 1 = sum_k[A059881(n, k)] = sum_k[S(n, k) + T(n, k)] - 1 where if n>k>0 S(n, k) = sum_j[T(n - k, j)] over j>k and T(n, k) = sum_j[S(n - k, j)] over k>j (note reversal) and if n>0 S(n, n) = T(n, n) = 1; S(n, k) = A059882(n, k), T(n, k) = A059883(n, k). - Henry Bottomley, Feb 05 2001
a(n) ~ c * d^n, where d = 1.571630806607064114100138865739690782401305155950789062725..., c = 0.82222360450823867604750473815253345888526601460811483897... . - Vaclav Kotesovec, Sep 12 2014
a(n) = A344604(n) + 1 - n mod 2. - Gus Wiseman, Jun 17 2021

Better name using a comment of Franklin T. Adams-Watters by Peter Luschny, Oct 31 2021

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.

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).

A345170 Number of integer partitions of n with an alternating permutation.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 10, 14, 19, 25, 36, 48, 64, 84, 111, 146, 191, 244, 315, 404, 515, 651, 823, 1035, 1295, 1616, 2011, 2492, 3076, 3787, 4650, 5695, 6952, 8463, 10280, 12460, 15059, 18162, 21858, 26254, 31463, 37641, 44933, 53554, 63704, 75653, 89683, 106162, 125445, 148020

Offset: 0

Gus Wiseman, Jun 13 2021

nonn

First differs from A325534 at a(10) = 25, A325534(10) = 26. The first separable partition without an alternating permutation is (3,2,2,2,1).

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

			The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)  (3)   (4)    (5)    (6)     (7)      (8)
            (21)  (31)   (32)   (42)    (43)     (53)
                  (211)  (41)   (51)    (52)     (62)
                         (221)  (321)   (61)     (71)
                         (311)  (411)   (322)    (332)
                                (2211)  (331)    (422)
                                        (421)    (431)
                                        (511)    (521)
                                        (3211)   (611)
                                        (22111)  (3221)
                                                 (3311)
                                                 (4211)
                                                 (22211)
                                                 (32111)

Joseph Likar, Table of n, a(n) for n = 0..1000

Includes all strict partitions A000009.
Including twins (x,x) gives A344740.
The normal case is A345163 (complement: A345162).
The complement is counted by A345165, ranked by A345171.
The Heinz numbers of these partitions are A345172.
The version for factorizations is A348379.
A000041 counts integer partitions.
A001250 counts alternating permutations.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A025047 counts alternating compositions (ascend: A025048, descend: A025049).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts alternating compositions with twins.
Cf. A000070, A103919, A335126, A344605, A344653, A344654, A344742, A345164, A345166, A345167, A345168, A345195.

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

a(26)-a(32) from Robert Price, Jun 23 2021
a(33)-a(48) from Alois P. Heinz, Jun 23 2021
a(49) onwards from Joseph Likar, Sep 05 2023

A344654 Number of integer partitions of n of which every permutation has a consecutive monotone triple, i.e., a triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 7, 11, 16, 20, 28, 37, 50, 65, 84, 106, 140, 175, 222, 277, 350, 432, 539, 663, 819, 999, 1225, 1489, 1816, 2192, 2653, 3191, 3846, 4603, 5516, 6578, 7852, 9327, 11083, 13120, 15532, 18328, 21620, 25430, 29904, 35071, 41110, 48080

Offset: 0

Gus Wiseman, Jun 12 2021

nonn

Such a permutation is characterized by being neither a twin (x,x) nor wiggly (A025047, A345192). A sequence is wiggly if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,3,2,2,2,2,1) has no wiggly permutations, even though it has the anti-run permutations (2,3,2,3,2,1,2), (2,3,2,1,2,3,2), and (2,1,2,3,2,3,2).

			The a(3) = 1 through a(9) = 11 partitions:
  (111)  (1111)  (2111)   (222)     (2221)     (2222)      (333)
                 (11111)  (3111)    (4111)     (5111)      (3222)
                          (21111)   (31111)    (41111)     (6111)
                          (111111)  (211111)   (221111)    (22221)
                                    (1111111)  (311111)    (51111)
                                               (2111111)   (321111)
                                               (11111111)  (411111)
                                                           (2211111)
                                                           (3111111)
                                                           (21111111)
                                                           (111111111)

Joseph Likar, Table of n, a(n) for n = 0..1000

The Heinz numbers of these partitions are A344653, complement A344742.
The complement is counted by A344740.
The normal case starts 0, 0, 0, then becomes A345162, complement A345163.
Allowing twins (x,x) gives A345165, ranked by A345171.
A001250 counts wiggly permutations.
A003242 counts anti-run compositions.
A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts wiggly compositions with twins.
A344605 counts wiggly patterns with twins.
A344606 counts wiggly permutations of prime indices with twins.
A344614 counts compositions with no consecutive strictly monotone triple.
A345164 counts wiggly permutations of prime indices.
A345170 counts partitions with a wiggly permutation, ranked by A345172.
A345192 counts non-wiggly compositions.
Cf. A000041, A000070, A102726, A103919, A333489, A335126, A344607, A344615, A345166, A345168, A345169.

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

a(26)-a(32) from Robert Price, Jun 22 2021

a(33) onwards from Joseph Likar, Sep 06 2023

A344653 Every permutation of the prime factors of n has a consecutive monotone triple, i.e., a triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.

Original entry on oeis.org

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 80, 81, 88, 96, 104, 112, 125, 128, 135, 136, 144, 152, 160, 162, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 270, 272, 288, 296, 297, 304, 320, 324, 328, 336, 343, 344, 351, 352, 368, 375, 376, 378, 384

Offset: 1

Gus Wiseman, Jun 12 2021

nonn

Differs from A335448 in lacking squares and having 270 etc.
First differs from A345193 in having 270.
Such a permutation is characterized by being neither a twin (x,x) nor wiggly (A025047, A345192). A sequence is wiggly if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no wiggly permutations, even though it has anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
   8: {1,1,1}
  16: {1,1,1,1}
  24: {1,1,1,2}
  27: {2,2,2}
  32: {1,1,1,1,1}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  54: {1,2,2,2}
  56: {1,1,1,4}
  64: {1,1,1,1,1,1}
  80: {1,1,1,1,3}
  81: {2,2,2,2}
  88: {1,1,1,5}
  96: {1,1,1,1,1,2}
For example, 36 has prime indices (1,1,2,2), which has the two wiggly permutations (1,2,1,2) and (2,1,2,1), so 36 is not in the sequence.

A superset of A335448, counted by A325535.
Positions of 0's in A344606.
These partitions are counted by A344654.
The complement is A344742, counted by A344740.
The separable case is A345173, counted by A345166.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A001250 counts wiggly permutations.
A003242 counts anti-run compositions.
A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
A325534 counts separable partitions, ranked by A335433.
A344604 counts wiggly compositions with twins.
A345164 counts wiggly permutations of prime indices.
A345165 counts partitions without a wiggly permutation, ranked by A345171.
A345170 counts partitions with a wiggly permutation, ranked by A345172.
A345192 counts non-wiggly compositions.
Cf. A001222, A071321, A071322, A316523, A316524, A335126, A344614, A344615, A344616, A344652, A345163, A345168, A345193.

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

Complement of A001248 in A345171.

A345165 Number of integer partitions of n without an alternating permutation.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 5, 8, 11, 17, 20, 29, 37, 51, 65, 85, 106, 141, 175, 223, 277, 351, 432, 540, 663, 820, 999, 1226, 1489, 1817, 2192, 2654, 3191, 3847, 4603, 5517, 6578, 7853, 9327, 11084, 13120, 15533, 18328, 21621, 25430, 29905, 35071, 41111, 48080, 56206, 65554, 76420, 88918

Offset: 0

Gus Wiseman, Jun 12 2021

nonn

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

			The a(2) = 1 through a(9) = 11 partitions:
  (11)  (111)  (22)    (2111)   (33)      (2221)     (44)        (333)
               (1111)  (11111)  (222)     (4111)     (2222)      (3222)
                                (3111)    (31111)    (5111)      (6111)
                                (21111)   (211111)   (41111)     (22221)
                                (111111)  (1111111)  (221111)    (51111)
                                                     (311111)    (321111)
                                                     (2111111)   (411111)
                                                     (11111111)  (2211111)
                                                                 (3111111)
                                                                 (21111111)
                                                                 (111111111)

Joseph Likar, Table of n, a(n) for n = 0..1000
Joseph Likar, Java Implementation using QBinomials

Excluding twins (x,x) gives A344654, complement A344740.
The normal case is A345162, complement A345163.
The complement is counted by A345170, ranked by A345172.
The Heinz numbers of these partitions are A345171.
The version for factorizations is A348380, complement A348379.
A version for ordered factorizations is A348613, complement A348610.
A000041 counts integer partitions.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A025047 counts alternating or wiggly compositions.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts alternating compositions with twins.
A345164 counts alternating permutations of prime indices, w/ twins A344606.
A345192 counts non-alternating compositions, without twins A348377.
Cf. A000070, A025048, A025049, A103919, A335126, A344605, A344607, A344615, A344653, A345166, A345167, A345168, A345169, A347706, A348609.

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

a(26) onwards by Joseph Likar, Aug 21 2023

A345168 Numbers k such that the k-th composition in standard order is not alternating.

Original entry on oeis.org

3, 7, 10, 11, 14, 15, 19, 21, 23, 26, 27, 28, 29, 30, 31, 35, 36, 37, 39, 42, 43, 46, 47, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 69, 71, 73, 74, 75, 78, 79, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 99, 100, 101, 103, 104, 105, 106, 107, 110

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 sequence of terms together with their binary indices begins:
     3: (1,1)          35: (4,1,1)        59: (1,1,2,1,1)
     7: (1,1,1)        36: (3,3)          60: (1,1,1,3)
    10: (2,2)          37: (3,2,1)        61: (1,1,1,2,1)
    11: (2,1,1)        39: (3,1,1,1)      62: (1,1,1,1,2)
    14: (1,1,2)        42: (2,2,2)        63: (1,1,1,1,1,1)
    15: (1,1,1,1)      43: (2,2,1,1)      67: (5,1,1)
    19: (3,1,1)        46: (2,1,1,2)      69: (4,2,1)
    21: (2,2,1)        47: (2,1,1,1,1)    71: (4,1,1,1)
    23: (2,1,1,1)      51: (1,3,1,1)      73: (3,3,1)
    26: (1,2,2)        52: (1,2,3)        74: (3,2,2)
    27: (1,2,1,1)      53: (1,2,2,1)      75: (3,2,1,1)
    28: (1,1,3)        55: (1,2,1,1,1)    78: (3,1,1,2)
    29: (1,1,2,1)      56: (1,1,4)        79: (3,1,1,1,1)
    30: (1,1,1,2)      57: (1,1,3,1)      83: (2,3,1,1)
    31: (1,1,1,1,1)    58: (1,1,2,2)      84: (2,2,3)

Wikipedia, Alternating permutation

The complement is A345167.
These compositions are counted by A345192.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A025047 counts alternating or wiggly compositions, directed A025048, A025049.
A344604 counts alternating compositions with twins.
A345194 counts alternating patterns (with twins: A344605).
A345164 counts alternating permutations of prime indices (with twins: A344606).
A345165 counts partitions without a alternating permutation, ranked by A345171.
A345170 counts partitions with a alternating permutation, ranked by A345172.
A348610 counts alternating ordered factorizations, complement A348613.
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.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Anti-run compositions are A333489.
- Non-anti-run compositions are A348612.
Cf. A001222, A005649, A008965, A059893, A106356, A238279, A344615, A345162, A345163, A345166, A345169, A348377, A348380.

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],Not@*wigQ@*stc]

A348615 Number of non-alternating permutations of {1...n}.

Original entry on oeis.org

0, 0, 0, 2, 14, 88, 598, 4496, 37550, 347008, 3527758, 39209216, 473596070, 6182284288, 86779569238, 1303866853376, 20884006863710, 355267697410048, 6397563946377118, 121586922638606336, 2432161265800164950, 51081039175603191808, 1123862030028821404198

Offset: 0

Gus Wiseman, Nov 03 2021

nonn

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either.

Also permutations of {1...n} matching the consecutive patterns (1,2,3) or (3,2,1). Matching only one of these gives A065429.

			The a(4) = 14 permutations:
  (1,2,3,4)  (3,1,2,4)
  (1,2,4,3)  (3,2,1,4)
  (1,3,4,2)  (3,4,2,1)
  (1,4,3,2)  (4,1,2,3)
  (2,1,3,4)  (4,2,1,3)
  (2,3,4,1)  (4,3,1,2)
  (2,4,3,1)  (4,3,2,1)

Wikipedia, Alternating permutation

The complement is counted by A001250, ranked by A333218.
The complementary version for compositions is A025047, ranked by A345167.
A directed version is A065429, complement A049774.
The version for compositions is A345192, ranked by A345168.
The version for ordered factorizations is A348613, complement A348610.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A348379 counts factorizations with an alternating permutation.
A348380 counts factorizations without an alternating permutation.
Cf. A056986, A102726, A325534, A325535, A344614, A344653, A344654, A347050, A347706, A348377, A348609.

b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
    end:
a:= n-> n!-`if`(n<2, 1, 2)*b(n, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 04 2021

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]] ==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Permutations[Range[n]],!wigQ[#]&]],{n,0,6}]

from itertools import accumulate, count, islice
def A348615_gen(): # generator of terms
    yield from (0,0)
    blist, f = (0,2), 1
    for n in count(2):
        f *= n
        yield f - (blist := tuple(accumulate(reversed(blist),initial=0)))[-1]
A348615_list = list(islice(A348615_gen(),40)) # Chai Wah Wu, Jun 09-11 2022

a(n) = n! - A001250(n).

A345172 Numbers whose multiset of prime factors has an alternating permutation.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83

Offset: 1

Gus Wiseman, Jun 13 2021

nonn

First differs from A212167 in containing 72.
First differs from A335433 in lacking 270, corresponding to the partition (3,2,2,2,1).
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,3,2,2,2,2,1) has no alternating permutations, even though it has the anti-run permutations (2,3,2,3,2,1,2), (2,3,2,1,2,3,2), and (2,1,2,3,2,3,2).
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      1: {}          20: {1,1,3}       39: {2,6}
      2: {1}         21: {2,4}         41: {13}
      3: {2}         22: {1,5}         42: {1,2,4}
      5: {3}         23: {9}           43: {14}
      6: {1,2}       26: {1,6}         44: {1,1,5}
      7: {4}         28: {1,1,4}       45: {2,2,3}
     10: {1,3}       29: {10}          46: {1,9}
     11: {5}         30: {1,2,3}       47: {15}
     12: {1,1,2}     31: {11}          50: {1,3,3}
     13: {6}         33: {2,5}         51: {2,7}
     14: {1,4}       34: {1,7}         52: {1,1,6}
     15: {2,3}       35: {3,4}         53: {16}
     17: {7}         36: {1,1,2,2}     55: {3,5}
     18: {1,2,2}     37: {12}          57: {2,8}
     19: {8}         38: {1,8}         58: {1,10}

Including squares of primes A001248 gives A344742, counted by A344740.
This is a subset of A335433, which is counted by A325534.
Positions of nonzero terms in A345164.
The partitions with these Heinz numbers are counted by A345170.
The complement is A345171, which is counted by A345165.
A345173 = A345171 /\ A335433 is counted by A345166.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A001250 counts alternating permutations.
A003242 counts anti-run compositions.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts alternating compositions with twins.
A344606 counts alternating permutations of prime indices with twins.
A345192 counts non-alternating compositions.
Cf. A001222, A071321, A071322, A316523, A316524, A335126, A344605, A344614, A344616, A344653, A344654, A345163, A345167, A347706, A348379.

wigQ[y_]:=Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1;
Select[Range[100],Select[Permutations[ Flatten[ConstantArray@@@FactorInteger[#]]],wigQ[#]&]!={}&]

Complement of A001248 (squares of primes) in A344742.

A345164 Number of alternating permutations of the multiset of prime factors of n.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 4, 1, 0, 2, 2, 2, 2, 1, 2, 2, 0, 1, 4, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 4, 1, 2, 1, 0, 2, 4, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 2, 4, 1, 0, 0, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Jun 13 2021

nonn

First differs from A335452 at a(30) = 4, A335452(30) = 6. The anti-runs (2,3,5) and (5,3,2) are not alternating.

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 permutation, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

			The a(n) alternating permutations of prime indices for n = 180, 210, 300, 420, 900:
  (12132)  (1324)  (13132)  (12143)  (121323)
  (21213)  (1423)  (13231)  (13142)  (132312)
  (21312)  (2143)  (21313)  (13241)  (213132)
  (23121)  (2314)  (23131)  (14132)  (213231)
  (31212)  (2413)  (31213)  (14231)  (231213)
           (3142)  (31312)  (21314)  (231312)
           (3241)           (21413)  (312132)
           (3412)           (23141)  (323121)
           (4132)           (24131)
           (4231)           (31214)
                            (31412)
                            (34121)
                            (41213)
                            (41312)

Counting all permutations gives A008480.
Dominated by A335452 (number of separations of prime factors).
Including twins (x,x) gives A344606.
Positions of zeros are A345171, counted by A345165.
Positions of nonzero terms are A345172.
A000041 counts integer partitions.
A001250 counts alternating permutations.
A003242 counts anti-run compositions.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts alternating compositions with twins.
A344654 counts non-twin partitions w/o alternating permutation, rank: A344653.
A344740 counts twins and partitions w/ alternating permutation, rank: A344742.
A345166 counts separable partitions w/o alternating permutation, rank: A345173.
A345170 counts partitions with a alternating permutation.
Cf. A001222, A071321, A071322, A316523, A316524, A333489, A335126, A344605, A344614, A344616, A344652, A345163, A345168.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Permutations[Flatten[ConstantArray@@@FactorInteger[n]]],wigQ]],{n,30}]

cp's OEIS Frontend

A025047 Number of alternating compositions, i.e., compositions with alternating increases and decreases, starting with either an increase or a decrease.

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A345192 Number of non-alternating compositions of n.

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A345170 Number of integer partitions of n with an alternating permutation.

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A344654 Number of integer partitions of n of which every permutation has a consecutive monotone triple, i.e., a triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.

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A344653 Every permutation of the prime factors of n has a consecutive monotone triple, i.e., a triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.

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A345165 Number of integer partitions of n without an alternating permutation.

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A345168 Numbers k such that the k-th composition in standard order is not alternating.

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A348615 Number of non-alternating permutations of {1...n}.