A345167 -id:A345167 - cp's OEIS frontend

A348380 Number of factorizations of n without an alternating permutation. Includes all twins (x*x).

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Oct 28 2021

nonn

First differs from A333487 at a(216) = 4, A333487(216) = 3.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
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). Alternating permutations of multisets are a generalization of alternating or up-down permutations of {1..n}.

			The a(n) factorizations for n = 96, 144, 192, 384:
  (2*2*2*12)     (12*12)        (3*4*4*4)        (4*4*4*6)
  (2*2*2*2*6)    (2*2*2*18)     (2*2*2*24)       (2*2*2*48)
  (2*2*2*2*2*3)  (2*2*2*2*9)    (2*2*2*2*12)     (2*2*2*2*24)
                 (2*2*2*2*3*3)  (2*2*2*2*2*6)    (2*2*2*2*3*8)
                                (2*2*2*2*3*4)    (2*2*2*2*4*6)
                                (2*2*2*2*2*2*3)  (2*2*2*2*2*12)
                                                 (2*2*2*2*2*2*6)
                                                 (2*2*2*2*2*3*4)
                                                 (2*2*2*2*2*2*2*3)

Wikipedia, Alternating permutation

The inseparable case is A333487, complement A335434, without twins A348381.
Non-twin partitions of this type are counted by A344654, ranked by A344653.
Twins and partitions not of this type are counted by A344740, ranked by A344742.
Partitions of this type are counted by A345165, ranked by A345171.
Partitions not of this type are counted by A345170, ranked by A345172.
The case without twins is A347706.
The complement is counted by A348379, with twins A347050.
Numbers with a factorization of this type are A348609.
An ordered version is A348613, complement A348610.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations.
A025047 counts alternating or wiggly compositions, ranked by A345167.
A325535 counts inseparable partitions, ranked by A335448.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
Cf. A038548, A049774, A119620, A289553, A325534, A336107, A344614, A345192, A347437, A347438, A347439, A347442, A347458, A348383, A348611.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[facs[n],Select[Permutations[#],wigQ]=={}&]],{n,100}]

a(2^n) = A345165(n).

A374253 Numbers k such that the k-th composition in standard order matches the patterns (1,2,1) or (2,1,2).

Original entry on oeis.org

13, 22, 25, 27, 29, 45, 46, 49, 51, 53, 54, 55, 57, 59, 61, 76, 77, 82, 86, 89, 90, 91, 93, 94, 97, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 115, 117, 118, 119, 121, 123, 125, 141, 148, 150, 153, 155, 156, 157, 162, 165, 166, 173, 174, 177, 178

Offset: 1

Gus Wiseman, Jul 13 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)
  27: (1,2,1,1)
  29: (1,1,2,1)
  45: (2,1,2,1)
  46: (2,1,1,2)
  49: (1,4,1)
  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)
  61: (1,1,1,2,1)
  76: (3,1,3)
  77: (3,1,2,1)
  82: (2,3,2)
  86: (2,2,1,2)
  89: (2,1,3,1)

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

Permutations of prime indices of this type are counted by A335460.
Compositions of this type are counted by A335548.
The complement is A374249, counted by A274174.
The anti-run case is A374254.
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],!UnsameQ@@First/@Split[stc[#]]&]

Equals A335466 \/ A335468.

A357184 Numbers k such that the k-th composition in standard order has the same length as its alternating sum.

Original entry on oeis.org

0, 1, 9, 19, 22, 28, 34, 69, 74, 84, 104, 132, 135, 141, 153, 177, 225, 265, 271, 274, 283, 286, 292, 307, 310, 316, 328, 355, 358, 364, 376, 400, 451, 454, 460, 472, 496, 520, 523, 526, 533, 538, 553, 562, 593, 610, 673, 706, 833, 898, 1041, 1047, 1053, 1058

Offset: 1

Gus Wiseman, Sep 28 2022

nonn

A composition of n is a finite sequence of positive integers summing to n. 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 alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The sequence together with the corresponding compositions begins:
    0: ()
    1: (1)
    9: (3,1)
   19: (3,1,1)
   22: (2,1,2)
   28: (1,1,3)
   34: (4,2)
   69: (4,2,1)
   74: (3,2,2)
   84: (2,2,3)
  104: (1,2,4)
  132: (5,3)
  135: (5,1,1,1)
  141: (4,1,2,1)
  153: (3,1,3,1)
  177: (2,1,4,1)
  225: (1,1,5,1)

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

See link for sequences related to standard compositions.
For product equal to sum we have A335404, counted by A335405.
For sum equal to twice alternating sum we have A348614, counted by A262977.
These compositions are counted by A357182.
For absolute value we have A357184, counted by A357183.
The case of partitions is counted by A357189.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A025047 counts alternating compositions, ranked by A345167.
A032020 counts strict compositions, ranked by A233564.
A124754 gives alternating sums of standard compositions.
A238279 counts compositions by sum and number of maximal runs.
A357136 counts compositions by alternating sum.
Cf. A000120, A070939, A114220, A114901, A178470, A242882, A262046, A301987.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],Length[stc[#]]==ats[stc[#]]&]

A347050 Number of factorizations of n that are a twin (x*x) or have an alternating permutation.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 9, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 10, 4, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Oct 15 2021

nonn

First differs from A348383 at a(216) = 27, A348383(216) = 28.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
These permutations are ordered factorizations of n with no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z.
The version without twins for n > 0 is a(n) + 1 if n is a perfect square; otherwise a(n).

			The factorizations for n = 4, 12, 24, 30, 36, 48, 60, 64, 72:
  4    12     24     30     36       48       60       64       72
  2*2  2*6    3*8    5*6    4*9      6*8      2*30     8*8      8*9
       3*4    4*6    2*15   6*6      2*24     3*20     2*32     2*36
       2*2*3  2*12   3*10   2*18     3*16     4*15     4*16     3*24
              2*2*6  2*3*5  3*12     4*12     5*12     2*4*8    4*18
              2*3*4         2*2*9    2*3*8    6*10     2*2*16   6*12
                            2*3*6    2*4*6    2*5*6    2*2*4*4  2*4*9
                            3*3*4    3*4*4    3*4*5             2*6*6
                            2*2*3*3  2*2*12   2*2*15            3*3*8
                                     2*2*3*4  2*3*10            3*4*6
                                              2*2*3*5           2*2*18
                                                                2*3*12
                                                                2*2*3*6
                                                                2*3*3*4
                                                                2*2*2*3*3
The a(270) = 19 factorizations:
  (2*3*5*9)   (5*6*9)   (3*90)   (270)
  (3*3*5*6)   (2*3*45)  (5*54)
  (2*3*3*15)  (2*5*27)  (6*45)
              (2*9*15)  (9*30)
              (3*3*30)  (10*27)
              (3*5*18)  (15*18)
              (3*6*15)  (2*135)
              (3*9*10)
Note that (2*3*3*3*5) is separable but has no alternating permutations.

Partitions not of this type are counted by A344654, ranked by A344653.
Partitions of this type are counted by A344740, ranked by A344742.
The complement is counted by A347706, without twins A348380.
The case without twins is A348379.
Dominates A348383, the separable case.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations.
A008480 counts permutations of prime indices, strict A335489.
A025047 counts alternating or wiggly compositions, ranked by A345167.
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, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
Cf. A038548, A049774, A102726, A119620, A128761, A344614, A347437, A347438, A347458, A348381.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Function[f,Select[Permutations[f],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]!={}]]],{n,100}]

For n > 1, a(n) = A335434(n) + A010052(n).

A349050 Number of multisets of size n that have no alternating permutations and cover an initial interval of positive integers.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 8, 12, 20, 32, 48, 80, 112, 192, 256, 448, 576, 1024, 1280, 2304, 2816, 5120, 6144, 11264, 13312, 24576, 28672, 53248, 61440, 114688, 131072, 245760, 278528, 524288, 589824, 1114112, 1245184, 2359296, 2621440, 4980736, 5505024

Offset: 0

Gus Wiseman, Dec 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 does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). Alternating permutations of multisets are a generalization of alternating or up-down permutations of {1..n}.

			The multiset {1,2,2,2,2,3,3} has no alternating permutations, even though it does have the three anti-run permutations (2,1,2,3,2,3,2), (2,3,2,1,2,3,2), (2,3,2,3,2,1,2), so is counted under a(7).
The a(2) = 1 through a(7) = 12 multisets:
  {11}  {111}  {1111}  {11111}  {111111}  {1111111}
               {1112}  {11112}  {111112}  {1111112}
               {1222}  {12222}  {111122}  {1111122}
                       {12223}  {111123}  {1111123}
                                {112222}  {1122222}
                                {122222}  {1122223}
                                {122223}  {1222222}
                                {123333}  {1222223}
                                          {1222233}
                                          {1222234}
                                          {1233333}
                                          {1233334}
As compositions:
  (2)  (3)  (4)    (5)      (6)      (7)
            (1,3)  (1,4)    (1,5)    (1,6)
            (3,1)  (4,1)    (2,4)    (2,5)
                   (1,3,1)  (4,2)    (5,2)
                            (5,1)    (6,1)
                            (1,1,4)  (1,1,5)
                            (1,4,1)  (1,4,2)
                            (4,1,1)  (1,5,1)
                                     (2,4,1)
                                     (5,1,1)
                                     (1,1,4,1)
                                     (1,4,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Frankie Mennicucci, On the Number of Strong Dominating Sets in a Graph, Master's Thesis, Montclair State Univ. (2024). See p. 32.
Wikipedia, Alternating permutation

The case of weakly decreasing multiplicities is A025065.
The inseparable case is A336102.
A separable instead of alternating version is A336103.
The version for partitions is A345165.
The version for factorizations is A348380, complement A348379.
The complement (still covering an initial interval) is counted by A349055.
A000670 counts sequences covering an initial interval, anti-run A005649.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions, ranked by A333489.
A025047 = alternating compositions, ranked by A345167, also A025048/A025049.
A049774 counts permutations avoiding the consecutive pattern (1,2,3).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A344654 counts partitions w/o an alternating permutation, ranked by A344653.
Cf. A019472, A052841, A096441, A106351, A106356, A335125, A335126, A344614, A347050, A347706, A348377, A349054.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[allnorm[n],Select[Permutations[#],wigQ]=={}&]],{n,0,7}]

a(n) = if(n==0, 0, if(n%2==0, (n+2)*2^(n/2-3), (n-1)*2^((n-1)/2-2))) \\ Andrew Howroyd, Jan 13 2024

a(n) = A011782(n) - A349055(n).

a(n) = (n+2)*2^(n/2-3) for even n > 0; a(n) = (n-1)*2^((n-5)/2) for odd n. - Andrew Howroyd, Jan 13 2024

Terms a(10) and beyond from Andrew Howroyd, Jan 13 2024

A349056 Number of weakly alternating permutations of the multiset of prime factors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 4, 1, 1, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 3, 1, 2, 4, 1, 3, 2, 4, 1, 6, 1, 2, 3, 3, 2, 4, 1, 5, 1, 2, 1, 6, 2, 2, 2

Offset: 1

Gus Wiseman, Dec 02 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 alternating in the sense of A025047 iff it is a weakly alternating anti-run.

A prime index of n is a number m such that prime(m) divides n. For n > 1, the multiset of prime factors of n is row n of A027746. The prime indices A112798 can also be used.

			The following are the weakly alternating permutations for selected n:
n = 2   6    12    24     48      60     90     120     180
   ----------------------------------------------------------
    2   23   223   2223   22223   2253   2335   22253   22335
        32   232   2232   22232   2325   2533   22325   22533
             322   2322   22322   2523   3253   22523   23253
                   3222   23222   3252   3325   23252   23352
                          32222   3522   3352   25232   25233
                                  5232   3523   32225   25332
                                         5233   32522   32325
                                         5332   35222   32523
                                                52223   33252
                                                52322   33522
                                                        35232
                                                        52323
                                                        53322

Counting all permutations of prime factors gives A008480.
The variation counting anti-run permutations is A335452.
The strong case is A345164, with twins A344606.
Compositions of this type are counted by A349052, also A129852 and A129853.
Compositions not of this type are counted by A349053, ranked by A349057.
The version for patterns is A349058, strong A345194.
The version for ordered factorizations is A349059, strong A348610.
Partitions of this type are counted by A349060, complement A349061.
The complement is counted by A349797.
The non-alternating case is A349798.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions.
A025047 counts alternating or wiggly compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A071321 gives the alternating sum of prime factors, reverse A071322.
A344616 gives the alternating sum of prime indices, reverse A316524.
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.
A349800 counts weakly but not strongly alternating compositions.
Cf. A028234, A051119, A096441, A335433, A335448, A344614, A344652, A344653, A345173, A345192.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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[Permutations[primeMS[n]],whkQ[#]||whkQ[-#]&]],{n,100}]

A345195 Number of non-alternating anti-run compositions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 4, 10, 23, 49, 96, 192, 368, 692, 1299, 2403, 4400, 8029, 14556, 26253, 47206, 84574, 151066, 269244, 478826, 849921, 1506309, 2665829, 4711971, 8319763, 14675786, 25865400, 45552678, 80171353, 141015313, 247905305, 435614270, 765132824

Offset: 0

Gus Wiseman, Jun 17 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 does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

An anti-run (separation or Carlitz composition) is a sequence with no adjacent equal parts.

			The a(9) = 23 anti-runs:
  (1,2,6)  (1,2,4,2)  (1,2,1,2,3)
  (1,3,5)  (1,2,5,1)  (1,2,3,1,2)
  (2,3,4)  (1,3,4,1)  (1,2,3,2,1)
  (4,3,2)  (1,4,3,1)  (1,3,2,1,2)
  (5,3,1)  (1,5,2,1)  (2,1,2,3,1)
  (6,2,1)  (2,1,2,4)  (2,1,3,2,1)
           (2,4,2,1)  (3,2,1,2,1)
           (3,1,2,3)
           (3,2,1,3)
           (4,2,1,2)

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

Non-anti-run compositions are counted by A261983.
A version counting partitions is A345166, ranked by A345173.
These compositions are ranked by A345169.
Non-alternating compositions are counted by A345192, ranked by A345168.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A025047 counts alternating or wiggly compositions, ranked by A345167.
A032020 counts strict compositions.
A106356 counts compositions by number of maximal anti-runs.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A345164 counts alternating permutations of prime indices, w/ twins A344606.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A345194 counts alternating patterns (with twins: A344605).
Cf. A005649, A008965, A114901, A178470, A333755, A344604, A344614, A344654, A344740, A345162, A345163, A348380, A348612, A348613.

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

a(n) = A003242(n) - A025047(n).

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

A349055 Number of multisets of size n that have an alternating permutation and cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 1, 3, 5, 12, 24, 52, 108, 224, 464, 944, 1936, 3904, 7936, 15936, 32192, 64512, 129792, 259840, 521472, 1043456, 2091008, 4183040, 8375296, 16752640, 33525760, 67055616, 134156288, 268320768, 536739840, 1073496064, 2147205120, 4294443008, 8589344768

Offset: 0

Gus Wiseman, Dec 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 does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). Alternating permutations of multisets are a generalization of alternating or up-down permutations of {1..n}.

The multisets that have an alternating permutation are those which have no part with multiplicity greater than floor(n/2) except for odd n when either the smallest or largest part can have multiplicity ceiling(n/2). - Andrew Howroyd, Jan 13 2024

			The multiset {1,2,2,3} has alternating permutations (2,1,3,2), (2,3,1,2), so is counted under a(4).
The a(1) = 1 through a(5) = 12 multisets:
  {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,2}
              {1,2,2}  {1,1,2,3}  {1,1,1,2,3}
              {1,2,3}  {1,2,2,3}  {1,1,2,2,2}
                       {1,2,3,3}  {1,1,2,2,3}
                       {1,2,3,4}  {1,1,2,3,3}
                                  {1,1,2,3,4}
                                  {1,2,2,3,3}
                                  {1,2,2,3,4}
                                  {1,2,3,3,3}
                                  {1,2,3,3,4}
                                  {1,2,3,4,4}
                                  {1,2,3,4,5}
As compositions:
  (1)  (1,1)  (1,2)    (2,2)      (2,3)
              (2,1)    (1,1,2)    (3,2)
              (1,1,1)  (1,2,1)    (1,1,3)
                       (2,1,1)    (1,2,2)
                       (1,1,1,1)  (2,1,2)
                                  (2,2,1)
                                  (3,1,1)
                                  (1,1,1,2)
                                  (1,1,2,1)
                                  (1,2,1,1)
                                  (2,1,1,1)
                                  (1,1,1,1,1)

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

The strong inseparable case is A025065.
A separable instead of alternating version is A336103, complement A336102.
The case of weakly decreasing multiplicities is A336106.
The version for non-twin partitions is A344654, ranked by A344653.
The complement for non-twin partitions is A344740, ranked by A344742.
The complement for partitions is A345165, ranked by A345171.
The version for partitions is A345170, ranked by A345172.
The version for factorizations is A348379, complement A348380.
The complement (still covering an initial interval) is counted by A349050.
A000670 counts sequences covering an initial interval, anti-run A005649.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions, ranked by A333489.
A025047 = alternating compositions, ranked by A345167, also A025048/A025049.
A049774 counts permutations avoiding the consecutive pattern (1,2,3).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
Cf. A019472, A052841, A096441, A106351, A106356, A335125, A335127, A344614, A347050, A347706, A348610, A348613, A349054.

allnorm[n_]:=If[n<=0,{{}},Function[s, Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[allnorm[n], Select[Permutations[#],wigQ]!={}&]],{n,0,7}]

a(n) = if(n==0, 1, 2^(n-1) - if(n%2==0, (n+2)*2^(n/2-3), (n-1)*2^((n-5)/2))) \\ Andrew Howroyd, Jan 13 2024

a(n) = A011782(n) - A349050(n).

a(n) = 2^(n-1) - (n+2)*2^(n/2-3) for even n > 0; a(n) = 2^(n-1) - (n-1)*2^((n-5)/2) for odd n. - Andrew Howroyd, Jan 13 2024

Terms a(10) and beyond from Andrew Howroyd, Jan 13 2024

A357183 Number of integer compositions with the same length as the absolute value of their alternating sum.

Original entry on oeis.org

1, 1, 0, 0, 2, 3, 2, 5, 12, 22, 26, 58, 100, 203, 282, 616, 962, 2045, 2982, 6518, 9858, 21416, 31680, 69623, 104158, 228930, 339978, 751430, 1119668, 2478787, 3684082, 8182469, 12171900, 27082870, 40247978, 89748642, 133394708, 297933185, 442628598, 990210110

Offset: 0

Gus Wiseman, Sep 28 2022

nonn

A composition of n is a finite sequence of positive integers summing to n.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The a(1) = 1 through a(8) = 12 compositions:
  (1)  (13)  (113)  (24)  (124)  (35)
       (31)  (212)  (42)  (151)  (53)
             (311)        (223)  (1115)
                          (322)  (1151)
                          (421)  (1214)
                                 (1313)
                                 (1412)
                                 (1511)
                                 (2141)
                                 (3131)
                                 (4121)
                                 (5111)

For product instead of length we have A114220.
For sum equal to twice alternating sum we have A262977, ranked by A348614.
For product equal to sum we have A335405, ranked by A335404.
This is the absolute value version of A357182.
These compositions are ranked by A357185.
The case of partitions is A357189.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A025047 counts alternating compositions, ranked by A345167.
A124754 gives alternating sums of standard compositions.
A238279 counts compositions by sum and number of maximal runs.
A261983 counts non-anti-run compositions.
A357136 counts compositions by alternating sum.
Cf. A000120, A032020, A070939, A106356, A114901, A131044, A178470, A233564, A242882, A262046, A301987.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[#]==Abs[ats[#]]&]],{n,0,15}]

a(21)-a(39) from Alois P. Heinz, Sep 29 2022

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}]

cp's OEIS Frontend

A348380 Number of factorizations of n without an alternating permutation. Includes all twins (x*x).

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A374253 Numbers k such that the k-th composition in standard order matches the patterns (1,2,1) or (2,1,2).

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A357184 Numbers k such that the k-th composition in standard order has the same length as its alternating sum.

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A347050 Number of factorizations of n that are a twin (x*x) or have an alternating permutation.

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A349050 Number of multisets of size n that have no alternating permutations and cover an initial interval of positive integers.

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A349056 Number of weakly alternating permutations of the multiset of prime factors of n.

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A345195 Number of non-alternating anti-run compositions of n.

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A349055 Number of multisets of size n that have an alternating permutation and cover an initial interval of positive integers.

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