A348379 -id:A348379 - 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).

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}]

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

A349059 Number of weakly alternating ordered factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, 1, 3, 3, 8, 1, 8, 1, 8, 3, 3, 1, 18, 2, 3, 4, 8, 1, 11, 1, 16, 3, 3, 3, 22, 1, 3, 3, 18, 1, 11, 1, 8, 8, 3, 1, 38, 2, 8, 3, 8, 1, 18, 3, 18, 3, 3, 1, 32, 1, 3, 8, 28, 3, 11, 1, 8, 3, 11, 1, 56, 1, 3, 8, 8, 3, 11, 1, 38, 8, 3

Offset: 1

Gus Wiseman, Dec 04 2021

nonn

An ordered factorization of n is a finite sequence of positive integers > 1 with product n.

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

			The ordered factorizations for n = 2, 4, 6, 8, 12, 24, 30:
  (2)  (4)    (6)    (8)      (12)     (24)       (30)
       (2*2)  (2*3)  (2*4)    (2*6)    (3*8)      (5*6)
              (3*2)  (4*2)    (3*4)    (4*6)      (6*5)
                     (2*2*2)  (4*3)    (6*4)      (10*3)
                              (6*2)    (8*3)      (15*2)
                              (2*2*3)  (12*2)     (2*15)
                              (2*3*2)  (2*12)     (3*10)
                              (3*2*2)  (2*2*6)    (2*5*3)
                                       (2*4*3)    (3*2*5)
                                       (2*6*2)    (3*5*2)
                                       (3*2*4)    (5*2*3)
                                       (3*4*2)
                                       (4*2*3)
                                       (6*2*2)
                                       (2*2*2*3)
                                       (2*2*3*2)
                                       (2*3*2*2)
                                       (3*2*2*2)

The strong version for compositions is A025047, also A025048, A025049.
The strong case is A348610, complement A348613.
The version for compositions is A349052, complement A349053.
As compositions these are ranked by the complement of A349057.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations, complement A348615.
A335434 counts separable factorizations, complement A333487.
A345164 counts alternating permutations of prime factors, w/ twins A344606.
A345170 counts partitions with an alternating permutation.
A348379 = factorizations w/ alternating permutation, complement A348380.
A348611 counts anti-run ordered factorizations, complement A348616.
A349060 counts weakly alternating partitions, complement A349061.
A349800 = weakly but not strongly alternating compositions, ranked A349799.
Cf. A003242, A122181, A138364, A339846, A339890, A345165, A345167, A345194, A347050, A347438, A347463, A347706.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
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/@facs[n], whkQ[#]||whkQ[-#]&]],{n,100}]

a(2^n) = A349052(n).

A348377 Number of non-alternating compositions of n, excluding twins (x,x).

Original entry on oeis.org

0, 0, 0, 1, 3, 9, 19, 45, 98, 208, 436, 906, 1861, 3803, 7731, 15659, 31628, 63747, 128257, 257722, 517338, 1037652, 2079983, 4167325, 8346203, 16710572, 33449694, 66944254, 133959020, 268028868, 536231902, 1072737537, 2145905284, 4292486690, 8586035992

Offset: 0

Gus Wiseman, Oct 26 2021

nonn

First differs from A348382 at a(6) = 19, A348382(6) = 17. The two non-alternating non-twin compositions of 6 that are not an anti-run are (1,2,3) and (3,2,1).

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(3) = 1 through a(6) = 19 compositions:
  (1,1,1)  (1,1,2)    (1,1,3)      (1,1,4)
           (2,1,1)    (1,2,2)      (1,2,3)
           (1,1,1,1)  (2,2,1)      (2,2,2)
                      (3,1,1)      (3,2,1)
                      (1,1,1,2)    (4,1,1)
                      (1,1,2,1)    (1,1,1,3)
                      (1,2,1,1)    (1,1,2,2)
                      (2,1,1,1)    (1,1,3,1)
                      (1,1,1,1,1)  (1,2,2,1)
                                   (1,3,1,1)
                                   (2,1,1,2)
                                   (2,2,1,1)
                                   (3,1,1,1)
                                   (1,1,1,1,2)
                                   (1,1,1,2,1)
                                   (1,1,2,1,1)
                                   (1,2,1,1,1)
                                   (2,1,1,1,1)
                                   (1,1,1,1,1,1)

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

The version for patterns is A000670(n) - A344605(n).
Non-twin compositions are counted by A051049.
The complement is counted by A344604.
An unordered version is A344654.
The complement is ranked by A345167 \/ A007582.
These compositions are ranked by A345168 \ A007582.
Including twins gives A345192, complement A025047.
The version for factorizations is A347706, or A348380 with twins.
The non-anti-run case is A348382.
A001250 counts alternating permutations.
A011782 counts compositions, strict A032020.
A106356 counts compositions by number of maximal anti-runs.
A114901 counts compositions where each part is adjacent to an equal part.
A261983 counts non-anti-run compositions, complement A003242.
A325535 counts inseparable partitions, ranked by A335448.
A344614 counts compositions avoiding (1,2,3) and (3,2,1) adjacent.
A345165 = partitions with no alternating permutations, ranked by A345171.
A345170 = partitions with an alternating permutation, ranked by A345172.
Cf. A005649, A178470, A238279, A333755, A335126, A344653, A344740, A345166, A345169, A345173, A348379, A348381.

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

For n > 0, a(n) = A345192(n) - 1 if n is even; otherwise A345192(n).

a(26) onwards from Andrew Howroyd, Jan 31 2024

A349797 Number of non-weakly alternating permutations of the multiset of prime factors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0

Offset: 1

Gus Wiseman, Dec 24 2021

nonn

First differs from 2 * A326291 at a(90) = 4, A326291(90) = 3.
The first odd term is a(144) = 7, whose non-weakly alternating permutations are shown in the example below.
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.
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 = 30    60     72      120     144      180
   ---------------------------------------------
    235   2235   22332   22235   222332   22353
    532   2352   23223   22352   223223   23235
          2532   23322   22532   223322   23325
          3225   32232   23225   232232   23523
          5223           23522   233222   23532
          5322           25223   322223   25323
                         25322   322322   32235
                         32252            32253
                         52232            32352
                         53222            32532
                                          33225
                                          35223
                                          35322
                                          52233
                                          52332
                                          53223
                                          53232

Counting all permutations of prime factors gives A008480.
Compositions not of this type are counted by A349052/A129852/A129853.
Compositions of this type are counted by A349053, ranked by A349057.
The complement is counted by A349056.
Partitions of this type are counted by A349061, complement A349060.
The version counting patterns is A350138, complement A349058.
The version counting ordered factorizations is A350139, complement A349059.
The strong case is counted by A350251, complement A345164.
Positions of nonzero terms are A350353.
A001250 counts alternating permutations, complement A348615.
A025047 = alternating compositions, ranked by A345167, complement A345192.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A071321 gives the alternating sum of prime factors, reverse A071322.
A335452 counts anti-run permutations of prime factors, complement A336107.
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.
Cf. A003242, A335433, A335448, A344614, A344652, A344653, A345173, A348613, A349798, A350252, A349800.

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[Flatten[ConstantArray@@@ FactorInteger[n]]], !whkQ[#]&&!whkQ[-#]&]],{n,100}]

a(n) = A008480(n) - A349056(n).

A348609 Numbers with a separable factorization without an alternating permutation.

Original entry on oeis.org

216, 270, 324, 378, 432, 486, 540, 594, 640, 648, 702, 756, 768, 810, 864, 896, 918, 960, 972, 1024, 1026, 1080, 1134, 1152, 1188, 1242, 1280, 1296, 1344, 1350, 1404, 1408, 1458, 1500, 1512, 1536, 1566, 1620, 1664, 1674, 1728, 1750, 1782, 1792, 1836, 1890

Offset: 1

Gus Wiseman, Oct 30 2021

nonn

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts. Alternatively, a multiset is separable if its greatest multiplicity is greater than the sum of the remaining multiplicities plus one.
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 sets.
Note that 216 has separable prime factorization (2*2*2*3*3*3) with an alternating permutation, but the separable factorization (2*3*3*3*4) is has no alternating permutation. See also A345173.

			The terms and their prime factorizations begin:
  216 = 2*2*2*3*3*3
  270 = 2*3*3*3*5
  324 = 2*2*3*3*3*3
  378 = 2*3*3*3*7
  432 = 2*2*2*2*3*3*3
  486 = 2*3*3*3*3*3
  540 = 2*2*3*3*3*5
  594 = 2*3*3*3*11
  640 = 2*2*2*2*2*2*2*5
  648 = 2*2*2*3*3*3*3
  702 = 2*3*3*3*13
  756 = 2*2*3*3*3*7
  768 = 2*2*2*2*2*2*2*2*3
  810 = 2*3*3*3*3*5
  864 = 2*2*2*2*2*3*3*3

Partitions of this type are counted by A345166, ranked by A345173 (a superset).
Compositions of this type are counted by A345195, ranked by A345169.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations, complement A348615.
A025047 counts alternating compositions, complement A345192, ranked by A345167.
A335434 counts separable factorizations, with twins A348383, complement A333487.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A345165 counts partitions w/o an alternating permutation, complement A345170.
A347438 counts factorizations with alternating product 1, additive A119620.
A348379 counts factorizations w/ an alternating permutation, complement A348380.
A348610 counts alternating ordered factorizations, complement A348613.
Cf. A001248, A003242, A038548, A325534, A336103, A344614, A345162, A347050, A347437, A347706, A348377, A348381, A348382.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
sepQ[m_]:=Select[Permutations[m],!MatchQ[#,{_,x_,x_,_}]&]!={};
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Select[Range[1000],Function[n,Select[facs[n],sepQ[#]&&Select[Permutations[#],wigQ]=={}&]!={}]]

A349051 Numbers k such that the k-th composition in standard order is an alternating permutation of {1..k} for some k.

Original entry on oeis.org

0, 1, 5, 6, 38, 41, 44, 50, 553, 562, 582, 593, 610, 652, 664, 708, 788, 808, 16966, 17036, 17048, 17172, 17192, 17449, 17458, 17542, 17676, 17712, 17940, 18000, 18513, 18530, 18593, 18626, 18968, 18992, 19496, 19536, 20625, 20676, 20769, 20868, 21256, 22600

Offset: 1

Gus Wiseman, Nov 08 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 together with the corresponding compositions begins:
        0: ()
        1: (1)
        5: (2,1)
        6: (1,2)
       38: (3,1,2)
       41: (2,3,1)
       44: (2,1,3)
       50: (1,3,2)
      553: (4,2,3,1)
      562: (4,1,3,2)
      582: (3,4,1,2)
      593: (3,2,4,1)
      610: (3,1,4,2)
      652: (2,4,1,3)
      664: (2,3,1,4)
      708: (2,1,4,3)
      788: (1,4,2,3)
      808: (1,3,2,4)
    16966: (5,3,4,1,2)
    17036: (5,2,4,1,3)

Wikipedia, Alternating permutation

These permutations are counted by A001250, complement A348615.
Compositions of this type are counted by A025047, complement A345192.
Subset of A333218, which ranks permutations of initial intervals.
Subset of A345167, which ranks alternating compositions, complement A345168.
A003242 counts Carlitz (anti-run) compositions.
A345163 counts normal partitions with an alternating permutation.
A345164 counts alternating permutations of prime indices.
A345170 counts partitions with an alternating permutation.
Compositions in standard order are the rows of A066099:
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
- GCD and LCM are given by A326674 and A333226.
- Maximal runs and anti-runs are counted by A124767 and A333381.
- Heinz number is given by A333219.
- Runs-resistance is given by A333628.
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Carlitz (anti-run) compositions are ranked by A333489.
Cf. A025048, A025049, A344604, A344615, A344653, A344742, A348377, A348379, A345165.

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[0,1000],Sort[stc[#]]==Range[Length[stc[#]]]&&wigQ[stc[#]]&]

Equals A333218 (permutation) /\ A345167 (alternating).

cp's OEIS Frontend

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

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

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A349059 Number of weakly alternating ordered factorizations of n.

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A348377 Number of non-alternating compositions of n, excluding twins (x,x).

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A349797 Number of non-weakly alternating permutations of the multiset of prime factors of n.