A344742 -id:A344742 - cp's OEIS frontend

A348379 Number of factorizations of n with an alternating permutation.

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

A347706 Number of factorizations of n that are not a twin (x*x) nor have an alternating permutation.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 28 2021

nonn

First differs from A348381 at a(216) = 4, A348381(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 sets.

			The a(n) factorizations for n = 96, 192, 2160, 576:
  2*2*2*12      3*4*4*4         3*3*3*80       4*4*4*9
  2*2*2*2*6     2*2*2*24        6*6*6*10       2*2*2*72
  2*2*2*2*2*3   2*2*2*2*12      2*2*2*270      2*2*2*2*36
                2*2*2*2*2*6     2*3*3*3*40     2*2*2*2*4*9
                2*2*2*2*3*4     2*2*2*2*135    2*2*2*2*6*6
                2*2*2*2*2*2*3   2*2*2*2*3*45   2*2*2*2*2*18
                                2*2*2*2*5*27   2*2*2*2*3*12
                                2*2*2*2*9*15   2*2*2*2*2*2*9
                                               2*2*2*2*2*3*6
                                               2*2*2*2*2*2*3*3

Wikipedia, Alternating permutation

Positions of nonzero terms are A046099.
Partitions of this type are counted by A344654, ranked by A344653.
Partitions not of this type are counted by A344740, ranked by A344742.
The complement is counted by A347050, without twins A348379.
The version for compositions is A348377.
The version allowing twins is A348380.
The inseparable case is A348381.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations of sets.
A025047 counts alternating or wiggly compositions, ranked by A345167.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A347438 counts factorizations with alternating product 1, additive A119620.
A348610 counts alternating ordered factorizations.
Cf. A049774, A325535, A336103, A344614, A347437, A347439, A347442, A347456, A347463, A348382, A348383.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Function[f,Select[Permutations[f],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]=={}]]],{n,100}]

a(2^n) = A344654(n).

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).

A347050 Number of factorizations of n that are a twin (x*x) or have an alternating permutation.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 9, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 10, 4, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Oct 15 2021

nonn

First differs from A348383 at a(216) = 27, A348383(216) = 28.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
These permutations are ordered factorizations of n with no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z.
The version without twins for n > 0 is a(n) + 1 if n is a perfect square; otherwise a(n).

			The factorizations for n = 4, 12, 24, 30, 36, 48, 60, 64, 72:
  4    12     24     30     36       48       60       64       72
  2*2  2*6    3*8    5*6    4*9      6*8      2*30     8*8      8*9
       3*4    4*6    2*15   6*6      2*24     3*20     2*32     2*36
       2*2*3  2*12   3*10   2*18     3*16     4*15     4*16     3*24
              2*2*6  2*3*5  3*12     4*12     5*12     2*4*8    4*18
              2*3*4         2*2*9    2*3*8    6*10     2*2*16   6*12
                            2*3*6    2*4*6    2*5*6    2*2*4*4  2*4*9
                            3*3*4    3*4*4    3*4*5             2*6*6
                            2*2*3*3  2*2*12   2*2*15            3*3*8
                                     2*2*3*4  2*3*10            3*4*6
                                              2*2*3*5           2*2*18
                                                                2*3*12
                                                                2*2*3*6
                                                                2*3*3*4
                                                                2*2*2*3*3
The a(270) = 19 factorizations:
  (2*3*5*9)   (5*6*9)   (3*90)   (270)
  (3*3*5*6)   (2*3*45)  (5*54)
  (2*3*3*15)  (2*5*27)  (6*45)
              (2*9*15)  (9*30)
              (3*3*30)  (10*27)
              (3*5*18)  (15*18)
              (3*6*15)  (2*135)
              (3*9*10)
Note that (2*3*3*3*5) is separable but has no alternating permutations.

Partitions not of this type are counted by A344654, ranked by A344653.
Partitions of this type are counted by A344740, ranked by A344742.
The complement is counted by A347706, without twins A348380.
The case without twins is A348379.
Dominates A348383, the separable case.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations.
A008480 counts permutations of prime indices, strict A335489.
A025047 counts alternating or wiggly compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
Cf. A038548, A049774, A102726, A119620, A128761, A344614, A347437, A347438, A347458, A348381.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Function[f,Select[Permutations[f],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]!={}]]],{n,100}]

For n > 1, a(n) = A335434(n) + A010052(n).

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

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

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 08 2021

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

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

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

Wikipedia, Alternating permutation

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

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Select[Range[0,1000],Sort[stc[#]]==Range[Length[stc[#]]]&&wigQ[stc[#]]&]

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

A350251 Number of non-alternating permutations of the multiset of prime factors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 4, 1, 0, 1, 2, 0, 2, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 2, 0, 2, 2, 0, 0, 5, 1, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 8, 0, 0, 2, 1, 0, 2, 0, 2, 0, 2, 0, 9, 0, 0, 2, 2, 0, 2, 0, 5, 1, 0, 0, 8, 0, 0, 0

Offset: 1

Gus Wiseman, Jan 08 2022

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).

			The a(n) permutations for selected n:
n = 4    12    24     48      60     72      90     96       120
   ----------------------------------------------------------------
    22   223   2223   22223   2235   22233   2335   222223   22235
         322   2232   22232   2253   22323   2353   222232   22253
               2322   22322   2352   22332   2533   222322   22325
               3222   23222   2532   23223   3235   223222   22352
                      32222   3225   23322   3325   232222   22523
                              3522   32223   3352   322222   22532
                              5223   32232   3532            23225
                              5322   32322   5233            23522
                                     33222   5323            25223
                                             5332            25322
                                                             32225
                                                             32252
                                                             32522
                                                             35222
                                                             52223
                                                             52232
                                                             52322
                                                             53222

The non-anti-run case is A336107, complement A335452.
The complement is counted by A345164, with twins A344606.
Positions of nonzero terms are A345171, counted by A345165.
Positions of zeros are A345172, counted by A345170.
Compositions of this type are counted by A345192, ranked by A345168.
Ordered factorizations of this type counted by A348613, complement A348610.
Compositions weakly of this type are counted by A349053, ranked by A349057.
The weak version is A349797, complement A349056.
The case that is also weakly alternating is A349798, compositions A349800.
Patterns of this type are counted by A350252, complement A345194.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions.
A008480 counts permutations of prime factors (ordered prime factorizations).
A025047/A025048/A025049 count alternating compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798 (row lengths A001222).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344616 gives the alternating sum of prime indices, reverse A316524.
A349052/A129852/A129853 count weakly alternating compositions.
Cf. A071321, A289553, A316523, A344652, A344653, A344742, A345173, A348379, A349061, A349801, A350138, A350139.

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,100}]

a(n) = A008480(n) - A345164(n).

A345193 Heinz numbers of non-twin (x,x) inseparable partitions.

Original entry on oeis.org

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 80, 81, 88, 96, 104, 112, 125, 128, 135, 136, 144, 152, 160, 162, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 272, 288, 296, 297, 304, 320, 324, 328, 336, 343, 344, 351, 352, 368, 375, 376, 384, 400, 405

Offset: 1

Gus Wiseman, Jun 17 2021

nonn

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.

A multiset is separable if it has an anti-run permutation (no adjacent parts equal). This is equivalent to having maximal multiplicity greater than one plus the sum of the remaining multiplicities. For example, the partition (3,2,2,2,1) has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2), so is separable.

			The sequence of terms together with their prime indices begins:
      8: {1,1,1}          112: {1,1,1,1,4}        232: {1,1,1,10}
     16: {1,1,1,1}        125: {3,3,3}            240: {1,1,1,1,2,3}
     24: {1,1,1,2}        128: {1,1,1,1,1,1,1}    243: {2,2,2,2,2}
     27: {2,2,2}          135: {2,2,2,3}          248: {1,1,1,11}
     32: {1,1,1,1,1}      136: {1,1,1,7}          250: {1,3,3,3}
     40: {1,1,1,3}        144: {1,1,1,1,2,2}      256: {1,1,1,1,1,1,1,1}
     48: {1,1,1,1,2}      152: {1,1,1,8}          272: {1,1,1,1,7}
     54: {1,2,2,2}        160: {1,1,1,1,1,3}      288: {1,1,1,1,1,2,2}
     56: {1,1,1,4}        162: {1,2,2,2,2}        296: {1,1,1,12}
     64: {1,1,1,1,1,1}    176: {1,1,1,1,5}        297: {2,2,2,5}
     80: {1,1,1,1,3}      184: {1,1,1,9}          304: {1,1,1,1,8}
     81: {2,2,2,2}        189: {2,2,2,4}          320: {1,1,1,1,1,1,3}
     88: {1,1,1,5}        192: {1,1,1,1,1,1,2}    324: {1,1,2,2,2,2}
     96: {1,1,1,1,1,2}    208: {1,1,1,1,6}        328: {1,1,1,13}
    104: {1,1,1,6}        224: {1,1,1,1,1,4}      336: {1,1,1,1,2,4}

A000041 counts integer partitions.
A001248 lists Heinz numbers of twins (x,x).
A001250 counts wiggly permutations.
A003242 counts anti-run compositions.
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.
A344740 counts twins and partitions w/ wiggly permutation, rank: A344742.
A345164 counts wiggly permutations of prime indices (with twins: A344606).
A345165 counts partitions without a wiggly permutation, ranked by A345171.
A345170 counts partitions with a wiggly permutation, ranked by A345172.
A345192 counts non-wiggly compositions.
Cf. A001222, A071321, A316523, A316524, A335126, A344614, A344616, A344652, A344654, A345169, A345173.

Select[Range[100],!MatchQ[FactorInteger[#],{{,2}}]&&Select[Permutations[Flatten[ConstantArray@@@FactorInteger[#]]],!MatchQ[#,{__,x_,x_,_}]&]=={}&]

Complement of A001248 in A335448.

cp's OEIS Frontend

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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A347706 Number of factorizations of n that are not a twin (x*x) nor have an alternating permutation.

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

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A347050 Number of factorizations of n that are a twin (x*x) or have an alternating permutation.

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

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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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