A345170 -id:A345170 - cp's OEIS frontend

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

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

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}]

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).

A345162 Number of integer partitions of n with no alternating permutation covering an initial interval of positive integers.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 3, 3, 5, 6, 6, 8, 10, 11, 15, 16, 18, 23, 27, 30, 35, 41, 47, 54, 62, 71, 82, 92, 103, 121, 137, 151, 173, 195, 220, 248, 277, 311, 350, 393, 435, 488, 546, 605, 678, 754, 835, 928, 1029, 1141, 1267, 1400, 1544, 1712, 1891, 2081, 2298, 2533, 2785, 3068

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 (3,3,2,2,2,2,1) has no alternating permutations, even though it has 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).

Sequences covering an initial interval (patterns) are counted by A000670 and ranked by A333217.

			The a(2) = 1 through a(10) = 6 partitions:
  11  111  1111  2111   21111   2221     221111    22221      32221
                 11111  111111  211111   2111111   321111     222211
                                1111111  11111111  2211111    3211111
                                                   21111111   22111111
                                                   111111111  211111111
                                                              1111111111

Andrew Howroyd, Table of n, a(n) for n = 0..500

The complement in covering partitions is counted by A345163.
Not requiring normality gives A345165, ranked by A345171.
The separable case is A345166.
A000041 counts integer partitions.
A000670 counts patterns, ranked by A333217.
A001250 counts alternating permutations.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A025047 counts alternating or wiggly compositions, directed A025048/A025049.
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.
A345164 counts alternating permutations of prime indices.
A345170 counts partitions with a alternating permutation, ranked by A345172.
Cf. A000070, A103919, A335126, A344614, A344615, A344653, A344654, A344740, A344742, A345167, A345168, A345192, A348609.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[IntegerPartitions[n],normQ[#]&&Select[Permutations[#],wigQ[#]&]=={}&]],{n,0,15}]

P(n,m)={Vec(1/prod(k=1, m, 1-y*x^k, 1+O(x*x^n)))}
a(n) = {(n >= 2) + sum(k=2, (sqrtint(8*n+1)-1)\2, my(r=n-binomial(k+1,2), v=P(r, k)); sum(i=1, min(k,2*r\k), sum(j=k-1, (2*r-(k-1)*(i-1))\(i+1), my(p=(j+k+(i==1||i==k))\2); if(p*i<=r, polcoef(v[r-p*i+1],j-p)) )))} \\ Andrew Howroyd, Jan 31 2024

a(n) = A000009(n) - A345163(n). - Andrew Howroyd, Jan 31 2024

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

A349054 Number of alternating strict compositions of n. Number of alternating (up/down or down/up) permutations of strict integer partitions of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 9, 11, 15, 21, 35, 41, 59, 75, 103, 155, 193, 255, 339, 443, 569, 841, 1019, 1365, 1743, 2295, 2879, 3785, 5151, 6417, 8301, 10625, 13567, 17229, 21937, 27509, 37145, 45425, 58345, 73071, 93409, 115797, 147391, 182151, 229553, 297061, 365625

Offset: 0

Gus Wiseman, Dec 21 2021

nonn

A strict composition of n is a finite sequence of distinct positive integers summing to n.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either.
The case starting with an increase (or decrease, it doesn't matter in the enumeration) is counted by A129838.

			The a(1) = 1 through a(7) = 11 compositions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)
            (1,2)  (1,3)  (1,4)  (1,5)    (1,6)
            (2,1)  (3,1)  (2,3)  (2,4)    (2,5)
                          (3,2)  (4,2)    (3,4)
                          (4,1)  (5,1)    (4,3)
                                 (1,3,2)  (5,2)
                                 (2,1,3)  (6,1)
                                 (2,3,1)  (1,4,2)
                                 (3,1,2)  (2,1,4)
                                          (2,4,1)
                                          (4,1,2)

Wikipedia, Alternating permutation

Ranking sequences are put in parentheses below.
This is the strict case of A025047/A025048/A025049 (A345167).
This is the alternating case of A032020 (A233564).
The unordered case (partitions) is A065033.
The directed case is A129838.
A001250 = alternating permutations (A349051), complement A348615 (A350250).
A003242 = Carlitz (anti-run) compositions, complement A261983.
A011782 = compositions, unordered A000041.
A345165 = partitions without an alternating permutation (A345171).
A345170 = partitions with an alternating permutation (A345172).
A345192 = non-alternating compositions (A345168).
A345195 = non-alternating anti-run compositions (A345169).
A349800 = weakly but not strongly alternating compositions (A349799).
A349052 = weakly alternating compositions, complement A349053 (A349057).
Cf. A000111, A008965, A015723, A114901, A128761, A129852, A129853, A218074, A333213, A344614, A345164, A345194, A349060, A349795.

g:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(g(o-1+j, u-j), j=1..u))
    end:
b:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
      `if`(k=0, `if`(n=0, 2, 0), b(n-k, k)+b(n-k, k-1)))
    end:
a:= n-> add(b(n, k)*g(k, 0), k=0..floor((sqrt(8*n+1)-1)/2))-1:
seq(a(n), n=0..46);  # Alois P. Heinz, Dec 22 2021

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

a(n) = 2 * A129838(n) - 1.

G.f.: Sum_{n>0} A001250(n)*x^(n*(n+1)/2)/Product_{k=1..n}(1-x^k).

A349058 Number of weakly alternating patterns of length n.

Original entry on oeis.org

1, 1, 3, 11, 43, 203, 1123, 7235, 53171, 439595, 4037371, 40787579, 449500595, 5366500163, 68997666867, 950475759899, 13966170378907, 218043973366091, 3604426485899203, 62894287709616755, 1155219405655975763, 22279674547003283003, 450151092568978825707

Offset: 0

Gus Wiseman, Dec 04 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.

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

			The a(1) = 1 through a(3) = 11 patterns:
  (1)  (1,1)  (1,1,1)
       (1,2)  (1,1,2)
       (2,1)  (1,2,1)
              (1,2,2)
              (1,3,2)
              (2,1,1)
              (2,1,2)
              (2,1,3)
              (2,2,1)
              (2,3,1)
              (3,1,2)

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

The strict case is A001250, complement A348615.
The strong case of compositions is A025047, ranked by A345167.
The unordered version is A052955.
The strong case is A345194, with twins A344605. Also the directed case.
The version for compositions is A349052, complement A349053.
The version for permutations of prime indices: A349056, complement A349797.
The version for compositions is ranked by A349057.
The version for ordered factorizations is A349059, strong A348610.
The version for partitions is A349060, complement A349061.
A003242 counts Carlitz (anti-run) compositions.
A005649 counts anti-run patterns.
A344604 counts alternating compositions with twins.
A345163 counts normal partitions with an alternating permutation.
A345170 counts partitions w/ an alternating permutation, complement A345165.
A345192 counts non-alternating compositions, ranked by A345168.
A349055 counts multisets w/ an alternating permutation, complement A349050.
Cf. A049774, A096441, A129852, A129853, A336103, A344614, A344615, A344740, A345164, A348613, A349794.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s, y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
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/@allnorm[n],whkQ[#]||whkQ[-#]&]],{n,0,6}]

R(n,k)={my(v=vector(k,i,1), u=vector(n)); for(r=1, n, if(r%2==0, my(s=v[k]); forstep(i=k, 2, -1, v[i] = s - v[i-1]); v[1] = s); for(i=2, k, v[i] += v[i-1]); u[r]=v[k]); u}
seq(n)= {concat([1], -vector(n,i,1) + 2*sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ) )} \\ Andrew Howroyd, Jan 13 2024

a(9)-a(18) from Alois P. Heinz, Dec 10 2021

a(19) onwards from Andrew Howroyd, Jan 13 2024

A349800 Number of integer compositions of n that are weakly alternating and have at least two adjacent equal parts.

Original entry on oeis.org

0, 0, 1, 1, 4, 9, 16, 33, 62, 113, 205, 373, 664, 1190, 2113, 3744, 6618, 11683, 20564, 36164, 63489, 111343, 195042, 341357, 596892, 1042976, 1821179, 3178145, 5543173, 9663545, 16839321, 29332231, 51075576, 88908912, 154722756, 269186074, 468221264

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.

This sequence counts compositions that are weakly but not strongly alternating; also weakly alternating non-anti-run compositions.

			The a(2) = 1 through a(6) = 16 compositions:
  (1,1)  (1,1,1)  (2,2)      (1,1,3)      (3,3)
                  (1,1,2)    (1,2,2)      (1,1,4)
                  (2,1,1)    (2,2,1)      (2,2,2)
                  (1,1,1,1)  (3,1,1)      (4,1,1)
                             (1,1,1,2)    (1,1,1,3)
                             (1,1,2,1)    (1,1,2,2)
                             (1,2,1,1)    (1,1,3,1)
                             (2,1,1,1)    (1,3,1,1)
                             (1,1,1,1,1)  (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

This is the weakly alternating case of A345192, ranked by A345168.
The case of partitions is A349795, ranked by A350137.
The version counting permutations of prime indices is A349798.
These compositions are ranked by A349799.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A003242 = Carlitz (anti-run) compositions, ranked by A333489.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A261983 = non-anti-run compositions, ranked by A348612.
A345165 = partitions without an alternating permutation, ranked by A345171.
A345170 = partitions with an alternating permutation, ranked by A345172.
A345173 = non-alternating anti-run partitions, ranked by A345166.
A345195 = non-alternating anti-run compositions, ranked by A345169.
A348377 = non-alternating non-twin compositions.
A349801 = non-alternating partitions, ranked by A289553.
Weakly alternating:
- A349052 = compositions, directed A129852/A129853, complement A349053.
- A349056 = permutations of prime indices, complement A349797.
- A349057 = complement of standard composition numbers (too dense).
- A349058 = patterns, complement A350138.
- A349059 = ordered factorizations, complement A350139.
- A349060 = partitions, complement A349061.
Cf. A008965, A011782, A027383, A096441, A274230, A333213, A344614, A344615, A348382, A348613, A349796, A350140.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y] &&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],(whkQ[#]||whkQ[-#])&&!wigQ[#]&]],{n,0,10}]

a(n) = A349052(n) - A025047(n). - Andrew Howroyd, Jan 31 2024

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

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

cp's OEIS Frontend

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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A344652 Number of permutations of the prime indices of n with no adjacent triples (..., x, y, z, ...) such that x <= y <= z.

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A349059 Number of weakly alternating ordered factorizations of n.

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A345162 Number of integer partitions of n with no alternating permutation covering an initial interval of positive integers.

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A349054 Number of alternating strict compositions of n. Number of alternating (up/down or down/up) permutations of strict integer partitions of n.

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A349058 Number of weakly alternating patterns of length n.