A348609 -id:A348609 - cp's OEIS frontend

A345165 Number of integer partitions of n without an alternating permutation.

0, 0, 1, 1, 2, 2, 5, 5, 8, 11, 17, 20, 29, 37, 51, 65, 85, 106, 141, 175, 223, 277, 351, 432, 540, 663, 820, 999, 1226, 1489, 1817, 2192, 2654, 3191, 3847, 4603, 5517, 6578, 7853, 9327, 11084, 13120, 15533, 18328, 21621, 25430, 29905, 35071, 41111, 48080, 56206, 65554, 76420, 88918

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 (3,2,2,2,1) has no alternating permutations, even though it has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

			The a(2) = 1 through a(9) = 11 partitions:
  (11)  (111)  (22)    (2111)   (33)      (2221)     (44)        (333)
               (1111)  (11111)  (222)     (4111)     (2222)      (3222)
                                (3111)    (31111)    (5111)      (6111)
                                (21111)   (211111)   (41111)     (22221)
                                (111111)  (1111111)  (221111)    (51111)
                                                     (311111)    (321111)
                                                     (2111111)   (411111)
                                                     (11111111)  (2211111)
                                                                 (3111111)
                                                                 (21111111)
                                                                 (111111111)

Joseph Likar, Table of n, a(n) for n = 0..1000
Joseph Likar, Java Implementation using QBinomials

Excluding twins (x,x) gives A344654, complement A344740.
The normal case is A345162, complement A345163.
The complement is counted by A345170, ranked by A345172.
The Heinz numbers of these partitions are A345171.
The version for factorizations is A348380, complement A348379.
A version for ordered factorizations is A348613, complement A348610.
A000041 counts integer partitions.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A025047 counts alternating or wiggly compositions.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts alternating compositions with twins.
A345164 counts alternating permutations of prime indices, w/ twins A344606.
A345192 counts non-alternating compositions, without twins A348377.
Cf. A000070, A025048, A025049, A103919, A335126, A344605, A344607, A344615, A344653, A345166, A345167, A345168, A345169, A347706, A348609.

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

a(26) onwards by Joseph Likar, Aug 21 2023

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

Original entry on oeis.org

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).

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]=={}&]

A348379 Number of factorizations of n with an alternating permutation.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 6, 1, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 1, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 10, 3, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Oct 28 2021

nonn

First differs from A335434 at a(216) = 27, A335434(216) = 28. Also differs from A335434 at a(270) = 19, A335434(270) = 20.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
All of the counted factorizations are separable (A335434).
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(270) = 19 factorizations:
  (2*3*3*15)  (2*3*45)  (2*135)  (270)
  (2*3*5*9)   (2*5*27)  (3*90)
  (3*3*5*6)   (2*9*15)  (5*54)
              (3*3*30)  (6*45)
              (3*5*18)  (9*30)
              (3*6*15)  (10*27)
              (3*9*10)  (15*18)
              (5*6*9)

Wikipedia, Alternating permutation

Partitions not of this type are counted by A345165, ranked by A345171.
Partitions of this type are counted by A345170, ranked by A345172.
Twins and partitions of this type are counted by A344740, ranked by A344742.
The case with twins is A347050.
The complement is counted by A348380, without twins A347706.
The ordered version is A348610.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations.
A025047 counts alternating or wiggly compositions, ranked by A345167.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
Cf. A038548, A056986, A325534, A335452, A347437, A347438, A347439, A347442, A347456, A347463, A348381, A348383, A348609.

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) = A345170(n).

A345166 Number of separable integer partitions of n without an alternating permutation.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 5, 6, 7, 10, 14, 18, 21, 27, 35, 42, 54, 65, 78, 95, 117, 140, 170, 202, 239, 286, 343, 401, 476, 562, 660, 775, 910, 1056, 1241, 1444, 1678, 1948, 2267, 2615, 3031, 3502, 4036, 4647, 5356, 6143, 7068, 8101, 9274, 10613, 12151, 13856

Offset: 0

Gus Wiseman, Jun 13 2021

nonn

A partition is separable if it has an anti-run permutation (no adjacent parts equal).
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).
The partitions counted by this sequence are those with 2m-1 parts with m being the multiplicity of a part which is neither the smallest or largest part. For example, 4322221 is such a partition since the multiplicity of 2 is 4, the total number of parts is 7, and 2 is neither the smallest or largest part. - Andrew Howroyd, Jan 15 2024

			The a(10) = 1 through a(16) = 6 partitions:
    32221  42221  52221  62221    43331    43332    53332
                         3222211  72221    53331    63331
                                  4222211  82221    92221
                                           3322221  4322221
                                           5222211  6222211
                                                    322222111

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

Allowing alternating permutations gives A325534, ranked by A335433.
Not requiring separability gives A345165, ranked by A345171.
Permutations of this type are ranked by A345169.
The Heinz numbers of these partitions are A345173.
Numbers with a factorization of this type are A348609.
A000041 counts integer partitions.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A325535 counts inseparable partitions, ranked by A335448.
A344654 counts non-twin partitions w/o alt permutation, rank A344653.
A345162 counts normal partitions w/o alt permutation, complement A345163.
A345170 counts partitions w/ alt permutation, ranked by A345172.
Cf. A000070, A103919, A335126, A344604, A344607, A344615, A344740, A344742, A345164, A345166, A345168, A345192, A348379.

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

The Heinz numbers of these partitions are A345173 = A345171 /\ A335433.

a(n) = A325534(n) - A345170(n). - Andrew Howroyd, Jan 15 2024

a(26) onwards from Andrew Howroyd, Jan 15 2024

A348613 Number of non-alternating ordered factorizations of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 8, 1, 0, 1, 2, 0, 2, 0, 9, 0, 0, 0, 11, 0, 0, 0, 8, 0, 2, 0, 2, 2, 0, 0, 25, 1, 2, 0, 2, 0, 8, 0, 8, 0, 0, 0, 16, 0, 0, 2, 20, 0, 2, 0, 2, 0, 2, 0, 43, 0, 0, 2, 2, 0, 2, 0, 25, 4, 0, 0, 16, 0

Offset: 1

Gus Wiseman, Nov 03 2021

nonn

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

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

			The a(n) ordered factorizations for n = 4, 12, 16, 24, 32, 36:
  2*2   2*2*3   4*4       2*2*6     2*2*8       6*6
        3*2*2   2*2*4     2*3*4     2*4*4       2*2*9
                4*2*2     4*3*2     4*4*2       2*3*6
                2*2*2*2   6*2*2     8*2*2       3*3*4
                          2*2*2*3   2*2*2*4     4*3*3
                          2*2*3*2   2*2*4*2     6*3*2
                          2*3*2*2   2*4*2*2     9*2*2
                          3*2*2*2   4*2*2*2     2*2*3*3
                                    2*2*2*2*2   2*3*3*2
                                                3*2*2*3
                                                3*3*2*2

Wikipedia, Alternating permutation

The complementary additive version is A025047, ranked by A345167.
The additive version is A345192, ranked by A345168, without twins A348377.
The complement is counted by A348610.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A345165 counts partitions without an alternating permutation, ranked by A345171.
A345170 counts partitions with an alternating permutation, ranked by A345172.
A348379 counts factorizations w/ an alternating permutation, with twins A347050.
A348380 counts factorizations w/o an alternating permutation, w/o twins A347706.
A348611 counts anti-run ordered factorizations.
Cf. A038548, A056986, A344614, A344653, A344654, A344740, A347437, A347438, A347463, A348381, A348609.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/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[ordfacs[n],!wigQ[#]&]],{n,100}]

A345173 Numbers whose multiset of prime factors is separable but has no alternating permutation.

Original entry on oeis.org

270, 378, 594, 702, 918, 1026, 1242, 1566, 1620, 1674, 1750, 1998, 2214, 2268, 2322, 2538, 2625, 2750, 2862, 3186, 3250, 3294, 3564, 3618, 3834, 3942, 4050, 4125, 4212, 4250, 4266, 4482, 4750, 4806, 4875, 5238, 5454, 5508, 5562, 5670, 5750, 5778, 5886, 6102

Offset: 1

Gus Wiseman, Jun 13 2021

nonn

A multiset is separable if it has an anti-run permutation (no adjacent parts equal).
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).
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
   270: {1,2,2,2,3}
   378: {1,2,2,2,4}
   594: {1,2,2,2,5}
   702: {1,2,2,2,6}
   918: {1,2,2,2,7}
  1026: {1,2,2,2,8}
  1242: {1,2,2,2,9}
  1566: {1,2,2,2,10}
  1620: {1,1,2,2,2,2,3}
  1674: {1,2,2,2,11}
  1750: {1,3,3,3,4}
  1998: {1,2,2,2,12}
  2214: {1,2,2,2,13}
  2268: {1,1,2,2,2,2,4}
  2322: {1,2,2,2,14}

The partitions with these Heinz numbers are counted by A345166.
Permutations of this type are ranked by A345169.
Numbers with a factorization of this type are counted by A348609.
A000041 counts integer partitions.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A025047 counts alternating compositions, ascend A025048, descend A025049.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices with twins.
A344740 counts twins and partitions with an alternating permutation.
A345164 counts alternating permutations of prime factors.
A345165 counts partitions without an alternating permutation.
A345170 counts partitions with an alternating permutation.
A345192 counts non-alternating compositions, without twins A348377.
A348379 counts factorizations with an alternating permutation.
Cf. A001222, A071321, A316524, A335126, A344614, A344616, A344652, A344653, A345163, A345168, A345193, A347706, A348380, A348613.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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_,_}];
Select[Range[1000],Select[Permutations[primeMS[#]],wigQ]=={}&&!Select[Permutations[primeMS[#]],sepQ]=={}&]

Equals A345171 /\ A335433.

A345169 Numbers k such that the k-th composition in standard order is a non-alternating anti-run.

Original entry on oeis.org

37, 52, 69, 101, 104, 105, 133, 137, 150, 165, 180, 197, 200, 208, 209, 210, 261, 265, 274, 278, 300, 301, 308, 325, 328, 357, 360, 361, 389, 393, 400, 401, 406, 416, 417, 418, 421, 422, 436, 517, 521, 529, 530, 534, 549, 550, 556, 557, 564, 581, 600, 601, 613

Offset: 1

Gus Wiseman, Jun 15 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).
An anti-run (separation or Carlitz composition) is a sequence with no adjacent equal parts.

			The sequence of terms together with their binary indices begins:
     37: (3,2,1)      210: (1,2,3,2)      400: (1,3,5)
     52: (1,2,3)      261: (6,2,1)        401: (1,3,4,1)
     69: (4,2,1)      265: (5,3,1)        406: (1,3,2,1,2)
    101: (1,3,2,1)    274: (4,3,2)        416: (1,2,6)
    104: (1,2,4)      278: (4,2,1,2)      417: (1,2,5,1)
    105: (1,2,3,1)    300: (3,2,1,3)      418: (1,2,4,2)
    133: (5,2,1)      301: (3,2,1,2,1)    421: (1,2,3,2,1)
    137: (4,3,1)      308: (3,1,2,3)      422: (1,2,3,1,2)
    150: (3,2,1,2)    325: (2,4,2,1)      436: (1,2,1,2,3)
    165: (2,3,2,1)    328: (2,3,4)        517: (7,2,1)
    180: (2,1,2,3)    357: (2,1,3,2,1)    521: (6,3,1)
    197: (1,4,2,1)    360: (2,1,2,4)      529: (5,4,1)
    200: (1,3,4)      361: (2,1,2,3,1)    530: (5,3,2)
    208: (1,2,5)      389: (1,5,2,1)      534: (5,2,1,2)
    209: (1,2,4,1)    393: (1,4,3,1)      549: (4,3,2,1)

Wikipedia, Alternating permutation

A version counting partitions is A345166, ranked by A345173.
These compositions are counted by A345195.
A001250 counts alternating permutations, complement A348615.
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.
A345164 counts alternating permutations of prime indices.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A345192 counts non-alternating compositions.
A345194 counts alternating patterns (with twins: A344605).
Statistics of standard compositions:
- Length is A000120.
- Constant runs are A124767.
- Heinz number is A333219.
- Anti-runs are A333381.
- Runs-resistance is A333628.
- Number of distinct parts is A334028.
- Non-anti-runs are A348612.
Classes of standard compositions:
- Weakly decreasing compositions (partitions) are A114994.
- Weakly increasing compositions (multisets) are A225620.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Strictly increasing compositions (sets) are A333255.
- Strictly decreasing compositions (strict partitions) are A333256.
- Anti-runs are A333489.
- Alternating compositions are A345167.
- Non-Alternating compositions are A345168.
Cf. A001222, A008965, A238279, A344614, A344615, A344652, A344653, A344654, A345162, A345163, A345193, A348609, 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];
sepQ[y_]:=!MatchQ[y,{_,x_,x_,_}];
Select[Range[0,1000],sepQ[stc[#]]&&!wigQ[stc[#]]&]

Intersection of A345168 (non-alternating) and A333489 (anti-run).

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

Original entry on oeis.org

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).

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

cp's OEIS Frontend

A345165 Number of integer partitions of n without an alternating permutation.

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

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

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A348379 Number of factorizations of n with an alternating permutation.

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A345166 Number of separable integer partitions of n without an alternating permutation.

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A348613 Number of non-alternating ordered factorizations of n.

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A345173 Numbers whose multiset of prime factors is separable but has no alternating permutation.

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A345169 Numbers k such that the k-th composition in standard order is a non-alternating anti-run.

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