A345168 -id:A345168 - cp's OEIS frontend

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

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

Offset: 0

Gus Wiseman, Nov 03 2021

nonn

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

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

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

Wikipedia, Alternating permutation

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

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

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

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

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

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 13 2021

nonn

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

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

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

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

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

A344740 Number of integer partitions of n with a permutation that has no consecutive monotone triple, i.e., no triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 10, 15, 19, 26, 36, 49, 64, 85, 111, 147, 191, 245, 315, 405, 515, 652, 823, 1036, 1295, 1617, 2011, 2493, 3076, 3788, 4650, 5696, 6952, 8464, 10280, 12461, 15059, 18163, 21858, 26255, 31463, 37642, 44933, 53555, 63704, 75654, 89683, 106163, 125445, 148021

Offset: 0

Gus Wiseman, Jun 12 2021

nonn

These partitions are characterized by either being a twin (x,x) or having a wiggly permutation. A sequence is wiggly if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no wiggly permutations, even though it has anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

			The a(1) = 1 through a(8) = 15 partitions:
  (1)  (2)    (3)    (4)      (5)      (6)        (7)          (8)
       (1,1)  (2,1)  (2,2)    (3,2)    (3,3)      (4,3)        (4,4)
                     (3,1)    (4,1)    (4,2)      (5,2)        (5,3)
                     (2,1,1)  (2,2,1)  (5,1)      (6,1)        (6,2)
                              (3,1,1)  (3,2,1)    (3,2,2)      (7,1)
                                       (4,1,1)    (3,3,1)      (3,3,2)
                                       (2,2,1,1)  (4,2,1)      (4,2,2)
                                                  (5,1,1)      (4,3,1)
                                                  (3,2,1,1)    (5,2,1)
                                                  (2,2,1,1,1)  (6,1,1)
                                                               (3,2,2,1)
                                                               (3,3,1,1)
                                                               (4,2,1,1)
                                                               (2,2,2,1,1)
                                                               (3,2,1,1,1)
For example, the partition (3,2,2,1) has the two wiggly permutations (2,3,1,2) and (2,1,3,2), so is counted under a(8).

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

The complement is counted by A344654.
The Heinz numbers of these partitions are A344742, complement A344653.
The normal case starts 1, 1, 1, then becomes A345163, complement A345162.
Not counting twins (x,x) gives A345170, ranked by A345172.
A001250 counts wiggly permutations.
A003242 counts anti-run compositions.
A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts wiggly compositions with twins.
A344605 counts wiggly patterns with twins.
A344606 counts wiggly permutations of prime indices with twins.
A344614 counts compositions with no consecutive strictly monotone triple.
A345164 counts wiggly permutations of prime indices.
A345165 counts partitions without a wiggly permutation, ranked by A345171.
A345192 counts non-wiggly compositions.
Cf. A000041, A000070, A102726, A103919, A333489, A344607, A344612, A344615, A345166, A345168, A345169.

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

a(n) = A345170(n) for n odd; a(n) = A345170(n) + 1 for n even.

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

a(33) onwards from Joseph Likar, Sep 05 2023

A345171 Numbers whose multiset of prime factors has no alternating permutation.

Original entry on oeis.org

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 80, 81, 88, 96, 104, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 270, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336, 343, 344, 351

Offset: 1

Gus Wiseman, Jun 13 2021

nonn

First differs from A335448 in having 270.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
Also Heinz numbers of integer partitions without a wiggly permutation, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   16: {1,1,1,1}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
   32: {1,1,1,1,1}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   49: {4,4}
   54: {1,2,2,2}
   56: {1,1,1,4}
   64: {1,1,1,1,1,1}
   80: {1,1,1,1,3}
   81: {2,2,2,2}
   88: {1,1,1,5}
   96: {1,1,1,1,1,2}

Removing squares of primes A001248 gives A344653, counted by A344654.
A superset of A335448, which is counted by A325535.
Positions of 0's in A345164.
The partitions with these Heinz numbers are counted by A345165.
The complement is A345172, counted by A345170.
The separable case is A345173, counted by A345166.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions, complement A261983.
A025047 counts alternating or wiggly compositions, directed A025048, A025049.
A325534 counts separable partitions, ranked by A335433.
A344606 counts alternating permutations of prime indices with twins.
A344742 ranks twins and partitions with an alternating permutation.
A345192 counts non-alternating compositions.
Cf. A001222, A071321, A071322, A316523, A316524, A335126, A344604, A344616, A344652, A344740, A345163, A345168, A345193, A345195, A348380, A348609.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Select[Range[100],Select[Permutations[Flatten[ ConstantArray@@@FactorInteger[#]]],wigQ]=={}&]

A348612 Numbers k such that the k-th composition in standard order is not an anti-run, i.e., has adjacent equal parts.

Original entry on oeis.org

3, 7, 10, 11, 14, 15, 19, 21, 23, 26, 27, 28, 29, 30, 31, 35, 36, 39, 42, 43, 46, 47, 51, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 71, 73, 74, 75, 78, 79, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 99, 100, 103, 106, 107, 110, 111, 112, 113, 114, 115, 116

Offset: 1

Gus Wiseman, Nov 03 2021

nonn

First differs from A345168 in lacking 37, corresponding to the composition (3,2,1).

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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.

			The terms and corresponding standard compositions begin:
     3: (1,1)          35: (4,1,1)        61: (1,1,1,2,1)
     7: (1,1,1)        36: (3,3)          62: (1,1,1,1,2)
    10: (2,2)          39: (3,1,1,1)      63: (1,1,1,1,1,1)
    11: (2,1,1)        42: (2,2,2)        67: (5,1,1)
    14: (1,1,2)        43: (2,2,1,1)      71: (4,1,1,1)
    15: (1,1,1,1)      46: (2,1,1,2)      73: (3,3,1)
    19: (3,1,1)        47: (2,1,1,1,1)    74: (3,2,2)
    21: (2,2,1)        51: (1,3,1,1)      75: (3,2,1,1)
    23: (2,1,1,1)      53: (1,2,2,1)      78: (3,1,1,2)
    26: (1,2,2)        55: (1,2,1,1,1)    79: (3,1,1,1,1)
    27: (1,2,1,1)      56: (1,1,4)        83: (2,3,1,1)
    28: (1,1,3)        57: (1,1,3,1)      84: (2,2,3)
    29: (1,1,2,1)      58: (1,1,2,2)      85: (2,2,2,1)
    30: (1,1,1,2)      59: (1,1,2,1,1)    86: (2,2,1,2)
    31: (1,1,1,1,1)    60: (1,1,1,3)      87: (2,2,1,1,1)

Constant run compositions are counted by A000005, ranked by A272919.
Counting these compositions by sum and length gives A131044.
These compositions are counted by A261983.
The complement is A333489, counted by A003242.
The non-alternating case is A345168, complement A345167.
A011782 counts compositions, strict A032020.
A238279 counts compositions by sum and number of maximal runs.
A274174 counts compositions with equal parts contiguous.
A336107 counts non-anti-run permutations of prime factors.
A345195 counts non-alternating anti-runs, ranked by A345169.
For compositions in standard order (rows of A066099):
- Length is A000120.
- Sum is A070939
- Maximal runs are counted by A124767.
- Strict compositions are ranked by A233564.
- Maximal anti-runs are counted by A333381.
- Runs-resistance is A333628.
Cf. A029931, A048793, A106356, A114901, A167606, A178470, A228351, A244164, A262046, A335452, A335464.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[100],MatchQ[stc[#],{_,x_,x_,_}]&]

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

Original entry on oeis.org

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

A349052 Number of weakly alternating compositions of n.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 28, 52, 91, 161, 280, 491, 850, 1483, 2573, 4469, 7757, 13472, 23378, 40586, 70438, 122267, 212210, 368336, 639296, 1109620, 1925916, 3342755, 5801880, 10070133, 17478330, 30336518, 52653939, 91389518, 158621355, 275313226, 477850887, 829388075

Offset: 0

Gus Wiseman, Nov 29 2021

nonn

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

			The a(5) = 16 compositions:
  (1,1,1,1,1)  (1,1,1,2)  (1,1,3)  (1,4)  (5)
               (1,1,2,1)  (1,2,2)  (2,3)
               (1,2,1,1)  (1,3,1)  (3,2)
               (2,1,1,1)  (2,1,2)  (4,1)
                          (2,2,1)
                          (3,1,1)
The a(6) = 28 compositions:
  (111111)  (11112)  (1113)  (114)  (15)  (6)
            (11121)  (1122)  (132)  (24)
            (11211)  (1131)  (141)  (33)
            (12111)  (1212)  (213)  (42)
            (21111)  (1311)  (222)  (51)
                     (2121)  (231)
                     (2211)  (312)
                     (3111)  (411)

Andrew Howroyd, Table of n, a(n) for n = 0..1000 (Terms 0..55 from Martin Ehrenstein)

The strong case is A025047, ranked by A345167.
The directed versions are A129852 and A129853, strong A025048 and A025049.
The complement is counted by A349053, strong A345192.
The version for permutations of prime indices is A349056, strong A345164.
The complement is ranked by A349057, strong A345168.
The version for patterns is A349058, strong A345194.
The multiplicative version is A349059, strong A348610.
An unordered version (partitions) is A349060, complement A349061.
The non-alternating case is A349800, ranked by A349799.
A001250 counts alternating permutations, complement A348615.
A001700 counts compositions of 2n with alternating sum 0.
A003242 counts Carlitz (anti-run) compositions.
A011782 counts compositions.
A106356 counts compositions by number of maximal anti-runs.
A344604 counts alternating compositions with twins.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A349054 counts strict alternating compositions.
Cf. A000041, A008965, A102726, A114901, A128761, A261983, A333213, A333755, A344614, A344615, A345165, A345195.

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[-#]&]],{n,0,10}]

C(n,f)={my(M=matrix(n,n,j,k,k>=j), s=M[,n]); for(b=1, n, f=!f; M=matrix(n,n,j,k, if(k1,M[j-k,k-1]) ))); for(k=2, n, M[,k]+=M[,k-1]); s+=M[,n]); s~}
seq(n) = concat([1], C(n,0) + C(n,1) - vector(n,j,numdiv(j))) \\ Andrew Howroyd, Jan 31 2024

a(21)-a(37) from Martin Ehrenstein, Jan 08 2022

A345163 Number of integer partitions of n with an alternating permutation covering an initial interval of positive integers.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 6, 7, 8, 11, 12, 16, 20, 23, 27, 34, 41, 48, 57, 68, 80, 94, 110, 130, 153, 175, 203, 239, 275, 317, 365, 420, 483, 553, 632, 720, 825, 938, 1064, 1211, 1370, 1550, 1755, 1982, 2235, 2517, 2830, 3182, 3576, 4006, 4487, 5027, 5619, 6275, 7007, 7812

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 the anti-run permutations (2,3,2,3,2,1,2), (2,3,2,1,2,3,2), and (2,1,2,3,2,3,2).

A partition with k parts is alternating if and only every part has a multiplicity no greater than k/2, except either the smallest or largest part may have a multiplicity of (k+1)/2 when k is odd. - Andrew Howroyd, Jan 31 2024

			The a(3) = 1 through a(12) = 7 partitions:
  21  211  221  321   3211   3221   3321    4321     33221    33321
                2211  22111  22211  32211   33211    43211    43221
                             32111  222111  322111   322211   332211
                                            2221111  332111   432111
                                                     2222111  3222111
                                                     3221111  3321111
                                                              22221111
For example, the partition (3,3,2,1,1,1,1) has the alternating permutations (1,3,1,3,1,2,1), (1,3,1,2,1,3,1), and (1,2,1,3,1,3,1), so is counted under a(12).

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

Not requiring an alternating permutation gives A000670, ranked by A333217.
The complement in covering partitions is counted by A345162.
Not requiring normality gives A345170, ranked by A345172.
A000041 counts integer partitions.
A001250 counts alternating permutations.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344605 counts alternating patterns with twins.
A345164 counts alternating permutations of prime indices.
A345165 counts partitions without a alternating permutation, ranked by A345171.
A349051 ranks alternating compositions.
Cf. A000070, A103919, A335126, A344604, A344607, A344615, A344653, A344654, A344740, A345166, A345167, A345168.

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}]

\\ See also A345162 for a faster program.
ok(k,p)={my(S=Set(p)); foreach(S, t, my(c=k+#p-2*(1+#select(x->x==t, p))); if(c<0, return(c==-1 && (t==1||t==k)))); 1}
a(n)={sum(k=1, (sqrtint(8*n+1)-1)\2, s=0; forpart(p=n-binomial(k+1,2), s+=ok(k,Vec(p)), k); s)} \\ Andrew Howroyd, Jan 31 2024

The Heinz numbers of these partitions are A333217 /\ A345172.

a(n) = A000009(n) - A345162(n). - Andrew Howroyd, Jan 31 2024

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

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

A349057 Numbers k such that the k-th composition in standard order is not weakly alternating.

Original entry on oeis.org

37, 46, 52, 53, 69, 75, 78, 92, 93, 101, 104, 105, 107, 110, 116, 117, 133, 137, 139, 142, 150, 151, 156, 157, 165, 174, 180, 181, 184, 185, 186, 187, 190, 197, 200, 201, 203, 206, 208, 209, 210, 211, 214, 215, 220, 221, 229, 232, 233, 235, 238, 244, 245, 261

Offset: 1

Gus Wiseman, Dec 04 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly 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 compositions begin:
   37: (3,2,1)
   46: (2,1,1,2)
   52: (1,2,3)
   53: (1,2,2,1)
   69: (4,2,1)
   75: (3,2,1,1)
   78: (3,1,1,2)
   92: (2,1,1,3)
   93: (2,1,1,2,1)
  101: (1,3,2,1)
  104: (1,2,4)
  105: (1,2,3,1)
  107: (1,2,2,1,1)
  110: (1,2,1,1,2)
  116: (1,1,2,3)
  117: (1,1,2,2,1)

The strong case is A345168, complement A345167, counted by A345192.
The strong anti-run case is A345169, counted by A345195.
Including all non-anti-runs gives A348612, complement A333489.
These compositions are counted by A349053, complement A349052.
The directed cases are counted by A129852 (incr.) and A129853 (decr.).
The complement for patterns is A349058, strong A345194.
The complement for ordered factorizations is A349059, strong A348610.
Partitions of this type are counted by A349061, complement A349060.
Partitions of this type are ranked by A349794.
Non-strict partitions of this type are counted by A349796.
Permutations of prime indices of this type are counted by A349797.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions, complement A261983.
A011782 counts compositions.
A025047 counts alternating/wiggly compositions, directed A025048, A025049.
A345164 counts alternating permutations of prime indices, weak A349056.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A349054 counts strict alternating compositions.
Cf. A001700, A096441, A128761, A344615, A344654, A345173, A348613, A349051, A349794, A349795, A349799.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
whkQ[y_]:=And@@Table[If[EvenQ[m], y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Select[Range[0,100],!whkQ[stc[#]]&&!whkQ[-stc[#]]&]

cp's OEIS Frontend

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

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

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A344740 Number of integer partitions of n with a permutation that has no consecutive monotone triple, i.e., no triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.

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A345171 Numbers whose multiset of prime factors has no alternating permutation.

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A348612 Numbers k such that the k-th composition in standard order is not an anti-run, i.e., has adjacent equal parts.

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

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A349052 Number of weakly alternating compositions of n.

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A345163 Number of integer partitions of n with an alternating permutation covering an initial interval of positive integers.

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