A348611 -id:A348611 - cp's OEIS frontend

A348610 Number of alternating ordered factorizations of n.

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 12, 1, 3, 3, 6, 1, 11, 1, 7, 3, 3, 3, 15, 1, 3, 3, 12, 1, 11, 1, 6, 6, 3, 1, 23, 1, 6, 3, 6, 1, 12, 3, 12, 3, 3, 1, 28, 1, 3, 6, 12, 3, 11, 1, 6, 3, 11, 1, 33, 1, 3, 6, 6, 3, 11, 1, 23, 4, 3

Offset: 1

Gus Wiseman, Nov 05 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. 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 alternating ordered factorizations of n = 1, 6, 12, 16, 24, 30, 32, 36:
  ()   6     12      16      24      30      32      36
       2*3   2*6     2*8     3*8     5*6     4*8     4*9
       3*2   3*4     8*2     4*6     6*5     8*4     9*4
             4*3     2*4*2   6*4     10*3    16*2    12*3
             6*2             8*3     15*2    2*16    18*2
             2*3*2           12*2    2*15    2*8*2   2*18
                             2*12    3*10    4*2*4   3*12
                             2*4*3   2*5*3           2*6*3
                             2*6*2   3*2*5           2*9*2
                             3*2*4   3*5*2           3*2*6
                             3*4*2   5*2*3           3*4*3
                             4*2*3                   3*6*2
                                                     6*2*3
                                                     2*3*2*3
                                                     3*2*3*2

Wikipedia, Alternating permutation

The additive version (compositions) is A025047 ranked by A345167.
The complementary additive version is A345192, ranked by A345168.
Dominated by A348611 (the anti-run version) at positions A122181.
The complement is counted by A348613.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations, complement A348615.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A347463 counts ordered factorizations with integer alternating product.
A348379 counts factorizations w/ an alternating permutation.
A348380 counts factorizations w/o an alternating permutation.
Cf. A038548, A049774, A056986, A102726, A128761, A344614, A347050, A347464, A347706, A348383.

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}]

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}]

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).

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).

A348381 Number of inseparable factorizations of n that are not a twin (x*x).

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 30 2021

nonn

First differs from A347706 at a(216) = 3, A347706(216) = 4.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
A multiset is inseparable if it has no permutation that is an anti-run, meaning there are always adjacent equal parts. Alternatively, a multiset is inseparable if its maximal multiplicity is at most one plus the sum of its remaining multiplicities.

			The a(n) factorizations for n = 96, 192, 384, 576:
  2*2*2*12      3*4*4*4         4*4*4*6           4*4*4*9
  2*2*2*2*6     2*2*2*24        2*2*2*48          2*2*2*72
  2*2*2*2*2*3   2*2*2*2*12      2*2*2*2*24        2*2*2*2*36
                2*2*2*2*2*6     2*2*2*2*3*8       2*2*2*2*4*9
                2*2*2*2*3*4     2*2*2*2*4*6       2*2*2*2*6*6
                2*2*2*2*2*2*3   2*2*2*2*2*12      2*2*2*2*2*18
                                2*2*2*2*2*2*6     2*2*2*2*3*12
                                2*2*2*2*2*3*4     2*2*2*2*2*2*9
                                2*2*2*2*2*2*2*3   2*2*2*2*2*3*6
                                                  2*2*2*2*2*2*3*3

Positions of nonzero terms are A046099.
Partitions not of this type are counted by A325534 - A000035.
Partitions of this type are counted by A325535 - A000035.
Allowing twins gives A333487.
The case without an alternating permutation is A347706, with twins A348380.
The complement is counted by A348383, without twins A335434.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations of sets.
A008480 counts permutations of prime indices, strict A335489.
A025047 counts alternating or wiggly compositions.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A344654 counts non-twin partitions without an alternating permutation.
A348382 counts non-anti-run compositions that are not a twin.
A348611 counts anti-run ordered factorizations.
Cf. A038548, A336102, A344614, A344653, A344740, A347050, A347437, A347438, A347456, A348379, A348609.

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],!MatchQ[#,{x_,x_}]&&Select[Permutations[#],!MatchQ[#,{_,x_,x_,_}]&]=={}&]],{n,100}]

a(n > 1) = A333487(n) - A010052(n).

a(2^n) = A325535(n) - 1 for odd n, otherwise A325535(n).

A350139 Number of non-weakly alternating ordered factorizations of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 12, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 20, 0, 0, 0, 0, 0, 2, 0, 10, 0, 0, 0, 12, 0

Offset: 1

Gus Wiseman, Dec 24 2021

nonn

The first odd term is a(180) = 69, which has, for example, the non-weakly alternating ordered factorization 2*3*5*3*2.
An ordered factorization of n is a finite sequence of positive integers > 1 with product n. Ordered factorizations are counted by A074206.
We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

			The a(n) ordered factorizations for n = 24, 36, 48, 60:
  (2*3*4)  (2*3*6)    (2*3*8)    (2*5*6)
  (4*3*2)  (6*3*2)    (2*4*6)    (3*4*5)
           (2*3*3*2)  (6*4*2)    (5*4*3)
           (3*2*2*3)  (8*3*2)    (6*5*2)
                      (2*2*3*4)  (10*3*2)
                      (2*3*4*2)  (2*3*10)
                      (2*4*3*2)  (2*2*3*5)
                      (3*2*2*4)  (2*3*5*2)
                      (4*2*2*3)  (2*5*3*2)
                      (4*3*2*2)  (3*2*2*5)
                                 (5*2*2*3)
                                 (5*3*2*2)

Positions of nonzero terms are A122181.
The strong version for compositions is A345192, ranked by A345168.
The strong case is A348613, complement A348610.
The version for compositions is A349053, complement A349052.
As compositions with ones allowed these are ranked by A349057.
The complement is counted by A349059.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations, complement A348615.
A025047 counts weakly alternating compositions, ranked by A345167.
A335434 counts separable factorizations, complement A333487.
A345164 counts alternating perms of prime factors, with twins A344606.
A345170 counts partitions with an alternating permutation.
A348379 counts factorizations w/ alternating perm, complement A348380.
A348611 counts anti-run ordered factorizations, complement A348616.
A349060 counts weakly alternating partitions, complement A349061.
Cf. A003242, A138364, A339846, A339890, A344604, A345194, A347050, A347438, A347463, A347706, A349054.

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) = A349053(n).

A348616 Number of ordered factorizations of n with adjacent equal factors.

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, 6, 1, 0, 1, 2, 0, 0, 0, 9, 0, 0, 0, 9, 0, 0, 0, 6, 0, 0, 0, 2, 2, 0, 0, 19, 1, 2, 0, 2, 0, 6, 0, 6, 0, 0, 0, 8, 0, 0, 2, 18, 0, 0, 0, 2, 0, 0, 0, 31, 0, 0, 2, 2, 0, 0, 0, 19, 4, 0, 0, 8, 0, 0

Offset: 1

Gus Wiseman, Nov 08 2021

nonn

First differs from A348613 at a(24) = 6, A348613(24) = 8.

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

			The a(n) ordered factorizations with at least one pair of adjacent equal factors for n = 12, 24, 36, 60:
   2*2*3    2*2*6      6*6        15*2*2
   3*2*2    6*2*2      2*2*9      2*2*15
            2*2*2*3    3*3*4      2*2*3*5
            2*2*3*2    4*3*3      2*2*5*3
            2*3*2*2    9*2*2      3*2*2*5
            3*2*2*2    2*2*3*3    3*5*2*2
                       2*3*3*2    5*2*2*3
                       3*2*2*3    5*3*2*2
                       3*3*2*2
See also examples in A348611.

Positions of 0's are A005117.
Positions of 4's appear to be A030514.
Positions of 2's appear to be A054753.
Positions of 1's appear to be A168363.
The additive version (compositions) is A261983, complement A003242.
Factorizations with a permutation of this type are counted by A333487.
Factorizations without a permutation of this type are counted by A335434.
The complement is counted by A348611.
As compositions these are ranked by A348612, complement A333489.
Dominated by A348613 (non-alternating ordered factorizations).
A001055 counts factorizations, strict A045778, ordered A074206.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
Cf. A001250, A025047, A122181, A138364, A347050, A347463, A347706, A348379, A348380, A348382, A348610.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
antirunQ[y_]:=Length[y]==Length[Split[y]]
Table[Length[Select[ordfacs[n],!antirunQ[#]&]],{n,100}]

a(n) = A074206(n) - A348611(n).

cp's OEIS Frontend

A348610 Number of alternating ordered factorizations of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A348613 Number of non-alternating ordered factorizations of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A349059 Number of weakly alternating ordered factorizations of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A348381 Number of inseparable factorizations of n that are not a twin (x*x).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A350139 Number of non-weakly alternating ordered factorizations of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A348616 Number of ordered factorizations of n with adjacent equal factors.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula