A350251 -id:A350251 - cp's OEIS frontend

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

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

Offset: 0

Gus Wiseman, Dec 16 2021

nonn

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

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

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

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

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

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

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

A345194 Number of alternating patterns of length n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 17 2021

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). An alternating pattern is necessarily an anti-run (A005649).
The version with twins (A344605) is identical to this sequence except with a(2) = 3 instead of 2.
From Gus Wiseman, Jan 16 2022: (Start)
Conjecture: Also the number of weakly up/down patterns of length n, where a sequence is weakly up/down if it is alternately weakly increasing and weakly decreasing, starting with an increase. For example, the a(0) = 1 through a(3) = 6 weakly up/down patterns are:
  ()  (1)  (1,1)  (1,1,1)
           (2,1)  (1,1,2)
                  (2,1,1)
                  (2,1,2)
                  (2,1,3)
                  (3,1,2)
(End)

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

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

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

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
allnorm[n_]:=If[n<=0,{{}},Function[s, Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],wigQ]],{n,0,6}]

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

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

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

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

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).

A350252 Number of non-alternating patterns of length n.

Original entry on oeis.org

0, 0, 1, 7, 53, 439, 4121, 43675, 519249, 6867463, 100228877, 1602238783, 27866817297, 524175098299, 10606844137009, 229807953097903, 5308671596791901, 130261745042452855, 3383732450013895721, 92770140175473602755, 2677110186541556215233

Offset: 0

Gus Wiseman, Jan 13 2022

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). An alternating pattern is necessarily an anti-run (A005649).
Conjecture: Also the number of non-weakly up/down (or down/up) patterns of length n. For example:
- The a(3) = 7 non-weakly up/down patterns:
  (121), (122), (123), (132), (221), (231), (321)
- The a(3) = 7 non-weakly down/up patterns:
  (112), (123), (211), (212), (213), (312), (321)
- The a(3) = 7 non-alternating patterns (see example for more):
  (111), (112), (122), (123), (211), (221), (321)

			The a(2) = 1 and a(3) = 7 non-alternating patterns:
  (1,1)  (1,1,1)
         (1,1,2)
         (1,2,2)
         (1,2,3)
         (2,1,1)
         (2,2,1)
         (3,2,1)
The a(4) = 53 non-alternating patterns:
  2112   3124   4123   1112   2134   1234   3112   2113   1123
  2211   3214   4213   1211   2314   1243   3123   2123   1213
  2212   3412   4312   1212   2341   1324   3211   2213   1223
         3421   4321   1221   2413   1342   3212   2311   1231
                       1222   2431   1423   3213   2312   1232
                                     1432   3312   2313   1233
                                            3321   2321   1312
                                                   2331   1321
                                                          1322
                                                          1323
                                                          1332

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

The unordered version is A122746.
The version for compositions is A345192, ranked by A345168, weak A349053.
The complement is counted by A345194, weak A349058.
The version for factorizations is A348613, complement A348610, weak A350139.
The strict case (permutations) is A348615, complement A001250.
The weak version for partitions is A349061, complement A349060.
The weak version for perms of prime indices is A349797, complement A349056.
The weak version is A350138.
The version for perms of prime indices is A350251, complement A345164.
A000670 = patterns (ranked by A333217).
A003242 = anti-run compositions, complement A261983, ranked by A333489.
A005649 = anti-run patterns, complement A069321.
A019536 = necklace patterns.
A025047/A129852/A129853 = alternating compositions, ranked by A345167.
A226316 = patterns avoiding (1,2,3), weakly A052709, complement A335515.
A345163 = normal partitions w/ alternating permutation, complement A345162.
A345170 = partitions w/ alternating permutation, complement A345165.
A349055 = normal multisets w/ alternating permutation, complement A349050.
Cf. A049774, A052955, A096441, A128761, A274230, A335456, A335457, A336103, A344605, A344614.

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[Join@@Permutations/@allnorm[n],!wigQ[#]&]],{n,0,6}]

a(n) = A000670(n) - A345194(n).

Terms a(9) and beyond from Andrew Howroyd, Feb 04 2022

A350137 Nonsquarefree numbers whose prime signature, except possibly the first and last parts, is all even.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 63, 64, 68, 72, 75, 76, 80, 81, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 121, 124, 125, 126, 128, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 169, 171, 172

Offset: 1

Gus Wiseman, Dec 23 2021

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.
Also nonsquarefree numbers whose prime factors, taken in order and with multiplicity, are alternately constant and weakly increasing, starting with either.
Also the Heinz numbers of non-strict integer partitions whose part multiplicities, except possibly the first and last, are all even. These are counted by A349795.

			The terms together with their prime indices begin:
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}

This is the nonsquarefree case of the complement of A349794.
These are the Heinz numbers of the partitions counted by A349795.
A version for compositions is A349799, counted by A349800.
A complementary version is A350140, counted by A349796.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A005117 = squarefree numbers, complement A013929.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A124010 = prime signature, sorted A118914.
A345164 = alternating permutations of prime indices, complement A350251.
A349052/A129852/A129853 = weakly alternating compositions.
A349053 = non-weakly alternating compositions, ranked by A349057.
A349056 = weakly alternating permutations of prime indices.
A349058 = weakly alternating patterns, complement A350138.
A349060 = weakly alternating partitions, complement A349061.
Cf. A000111, A096441, A117298, A335433, A335448, A335452, A344614, A344652, A344653, A345173, A349059, A349797.

Select[Range[100],!SquareFreeQ[#]&&(PrimePowerQ[#]||And@@EvenQ/@Take[Last/@FactorInteger[#],{2,-2}])&]

A350140 Nonsquarefree numbers whose prime signature has at least one odd part other the first or last.

Original entry on oeis.org

60, 84, 120, 132, 140, 150, 156, 168, 204, 220, 228, 240, 260, 264, 270, 276, 280, 294, 300, 308, 312, 315, 336, 340, 348, 364, 372, 378, 380, 408, 420, 440, 444, 456, 460, 476, 480, 490, 492, 495, 516, 520, 528, 532, 540, 552, 560, 564, 572, 580, 585, 588

Offset: 1

Gus Wiseman, Dec 25 2021

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Also Heinz numbers of non-weakly alternating non-strict integer partitions, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. These partitions are counted by A349796. This sequence involves the somewhat degenerate case where no strict increases are allowed.

			The terms together with their Heinz partitions begin (A-E = 10-14):
     60: (3211)      276: (9211)      420: (43211)
     84: (4211)      280: (43111)     440: (53111)
    120: (32111)     294: (4421)      444: (C211)
    132: (5211)      300: (33211)     456: (82111)
    140: (4311)      308: (5411)      460: (9311)
    150: (3321)      312: (62111)     476: (7411)
    156: (6211)      315: (4322)      480: (3211111)
    168: (42111)     336: (421111)    490: (4431)
    204: (7211)      340: (7311)      492: (D211)
    220: (5311)      348: (A211)      495: (5322)
    228: (8211)      364: (6411)      516: (E211)
    240: (321111)    372: (B211)      520: (63111)
    260: (6311)      378: (42221)     528: (521111)
    264: (52111)     380: (8311)      532: (8411)
    270: (32221)     408: (72111)     540: (322211)

Including all nonsquarefree numbers gives A013929, complement A005117.
Subsets include A088860 and A110286.
Signatures of this type are counted by A274230, complement A027383.
The strict instead of non-strict version is A336568, counted by A347548.
A version for compositions allowing strict is A349057, counted by A349053.
Allowing strict partitions gives A349794, counted by A349061.
These partitions are counted by A349796.
The complement in nonsquarefree partitions is A350137, counted by A349795.
A000041 = integer partitions, strict A000009.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A003242 = Carlitz (anti-run) compositions.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A096441 = weakly alternating 0-appended partitions.
A124010 = prime signature, sorted A118914.
A345164 = alternating permutations of prime indices, complement A350251.
A345170 = partitions w/ an alternating permutation, ranked by A345172.
A349052/A129852/A129853 = weakly alternating compositions.
A349056 = weakly alternating permutations of prime indices.
A349058 = weakly alternating patterns, complement A350138.
A349060 = weakly alternating partitions, strong A349801.
A349798 = weakly but not strongly alternating perms of prime indices.
Cf. A000111, A047967, A333213, A335448, A344615, A344653, A345173, A349054, A349059, A349797, A349799.

Select[Range[300],!SquareFreeQ[#]&&PrimeNu[#]>1&& !And@@EvenQ/@Take[Last/@FactorInteger[#],{2,-2}]&]

Complement of A005117 in A349794.

A350250 Numbers k such that the k-th composition in standard order is a non-alternating permutation of an initial interval of positive integers.

Original entry on oeis.org

37, 52, 549, 550, 556, 564, 581, 600, 616, 649, 657, 712, 786, 802, 836, 840, 16933, 16934, 16937, 16940, 16946, 16948, 16965, 16977, 16984, 16994, 17000, 17033, 17041, 17092, 17096, 17170, 17186, 17220, 17224, 17445, 17446, 17452, 17460, 17541, 17569, 17584

Offset: 1

Gus Wiseman, Jan 13 2022

nonn

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

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 and corresponding permutations begin:
     37: (3,2,1)
     52: (1,2,3)
    549: (4,3,2,1)
    550: (4,3,1,2)
    556: (4,2,1,3)
    564: (4,1,2,3)
    581: (3,4,2,1)
    600: (3,2,1,4)
    616: (3,1,2,4)
    649: (2,4,3,1)
    657: (2,3,4,1)
    712: (2,1,3,4)
    786: (1,4,3,2)
    802: (1,3,4,2)
    836: (1,2,4,3)
    840: (1,2,3,4)
  16933: (5,4,3,2,1)

This is the non-alternating case of A333218.
This is the restriction of A345168 to permutations, complement A345167.
These partitions are counted by A348615, complement A001250.
A003242 counts anti-run compositions, patterns A005649.
A025047 counts alternating compositions, directed A025048/A025049.
A345192 counts non-alternating compositions.
A345194 counts alternating patterns, complement A350252.
Statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of maximal anti-runs is A333381.
- Number of distinct parts is A334028.
Classes of standard compositions:
- Weakly decreasing compositions (partitions) are A114994, strict A333256.
- Weakly increasing compositions (multisets) are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Anti-run compositions are A333489, complement A348612.
Cf. A008965, A059893, A164894, A246534, A333217, A344605, A345162, A350251, A345163, A345171, A345172, A348613.

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[#]])&]

A350353 Numbers whose multiset of prime factors has a permutation that is not weakly alternating.

Original entry on oeis.org

30, 36, 42, 60, 66, 70, 72, 78, 84, 90, 100, 102, 105, 108, 110, 114, 120, 126, 130, 132, 138, 140, 144, 150, 154, 156, 165, 168, 170, 174, 180, 182, 186, 190, 195, 196, 198, 200, 204, 210, 216, 220, 222, 225, 228, 230, 231, 234, 238, 240, 246, 252, 255, 258

Offset: 1

Gus Wiseman, Jan 13 2022

nonn

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

			The terms together with a (generally not unique) non-weakly alternating permutation of each multiset of prime indices begin:
   30 : (1,2,3)       100 : (1,3,3,1)
   36 : (1,2,2,1)     102 : (1,2,7)
   42 : (1,2,4)       105 : (2,3,4)
   60 : (1,1,2,3)     108 : (1,2,2,1,2)
   66 : (1,2,5)       110 : (1,3,5)
   70 : (1,3,4)       114 : (1,2,8)
   72 : (1,1,2,2,1)   120 : (1,1,1,2,3)
   78 : (1,2,6)       126 : (1,2,4,2)
   84 : (1,1,2,4)     130 : (1,3,6)
   90 : (1,2,3,2)     132 : (1,1,2,5)

The strong version is A289553, complement A167171.
These are the positions of nonzero terms in A349797.
Below, WA = "weakly alternating":
- WA compositions are counted by A349052/A129852/A129853.
- Non-WA compositions are counted by A349053, ranked by A349057.
- WA permutations of prime factors = A349056, complement A349797.
- WA patterns are counted by A349058, complement A350138.
- WA ordered factorizations are counted by A349059, complement A350139.
- WA partitions are counted by A349060, complement A349061.
A001250 counts alternating permutations, complement A348615.
A008480 counts permutations of prime factors.
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.
A345164 = alternating permutations of prime factors, complement A350251.
Cf. A003242, A335433, A335448, A344652, A344653, A345171, A345172, A345173, A348379, A348613, A349798, A350252, A349800.

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}];
Select[Range[100],Select[Permutations[primeMS[#]],!whkQ[#]&&!whkQ[-#]&]!={}&]

cp's OEIS Frontend

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

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A345194 Number of alternating patterns of length n.

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

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A350252 Number of non-alternating patterns of length n.

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A350137 Nonsquarefree numbers whose prime signature, except possibly the first and last parts, is all even.

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A350140 Nonsquarefree numbers whose prime signature has at least one odd part other the first or last.

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A350250 Numbers k such that the k-th composition in standard order is a non-alternating permutation of an initial interval of positive integers.

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A350353 Numbers whose multiset of prime factors has a permutation that is not weakly alternating.

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