A335489 -id:A335489 - cp's OEIS frontend

A032020 Number of compositions (ordered partitions) of n into distinct parts.

1, 1, 1, 3, 3, 5, 11, 13, 19, 27, 57, 65, 101, 133, 193, 351, 435, 617, 851, 1177, 1555, 2751, 3297, 4757, 6293, 8761, 11305, 15603, 24315, 30461, 41867, 55741, 74875, 98043, 130809, 168425, 257405, 315973, 431065, 558327, 751491, 958265, 1277867, 1621273

Offset: 0

Christian G. Bower, Apr 01 1998

Compositions into distinct parts are equivalent to (1,1)-avoiding compositions. - Gus Wiseman, Jun 25 2020

All terms are odd. - Alois P. Heinz, Apr 09 2021

			a(6) = 11 because 6 = 5+1 = 4+2 = 3+2+1 = 3+1+2 = 2+4 = 2+3+1 = 2+1+3 = 1+5 = 1+3+2 = 1+2+3.
From _Gus Wiseman_, Jun 25 2020: (Start)
The a(0) = 1 through a(7) = 13 strict compositions:
  ()  (1)  (2)  (3)    (4)    (5)    (6)      (7)
                (1,2)  (1,3)  (1,4)  (1,5)    (1,6)
                (2,1)  (3,1)  (2,3)  (2,4)    (2,5)
                              (3,2)  (4,2)    (3,4)
                              (4,1)  (5,1)    (4,3)
                                     (1,2,3)  (5,2)
                                     (1,3,2)  (6,1)
                                     (2,1,3)  (1,2,4)
                                     (2,3,1)  (1,4,2)
                                     (3,1,2)  (2,1,4)
                                     (3,2,1)  (2,4,1)
                                              (4,1,2)
                                              (4,2,1)
(End)

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
C. G. Bower, Transforms (2)
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
B. Richmond and A. Knopfmacher, Compositions with distinct parts, Aequationes Mathematicae 49 (1995), pp. 86-97.
B. Richmond and A. Knopfmacher, Compositions with distinct parts, Aequationes Mathematicae 49 (1995), pp. 86-97. (free access)
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Cf. A008289, A032011.
Row sums of A241719.
Column k=1 of A243081, A242447, A261835, A261836, A327244, A327245.
Main diagonal of A261960.
Dominated by A003242 (anti-run compositions).
These compositions are ranked by A233564.
(1,1)-avoiding patterns are counted by A000142.
Numbers with strict prime signature are A130091.
(1,1,1)-avoiding compositions are counted by A232432.
(1,1)-matching compositions are counted by A261982.
Inseparable partitions are counted by A325535.
Patterns matched by compositions are counted by A335456.
Strict permutations of prime indices are counted by A335489.
Cf. A000009, A011782, A102726, A181796, A261962, A333489, A335457.

b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
      -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0))) end:
a:= proc(n) local l; l:=b(n, n): add((i-1)! *l[i], i=1..nops(l)) end:
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 12 2012
# second Maple program:
T:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
      `if`(k=0, `if`(n=0, 1, 0), T(n-k, k) +k*T(n-k, k-1)))
    end:
a:= n-> add(T(n, k), k=0..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 04 2015

f[list_]:=Length[list]!; Table[Total[Map[f, Select[IntegerPartitions[n], Sort[#] == Union[#] &]]], {n, 0,30}]
T[n_, k_] := T[n, k] = If[k<0 || n<0, 0, If[k==0, If[n==0, 1, 0], T[n-k, k] + k*T[n-k, k-1]]]; a[n_] := Sum[T[n, k], {k, 0, Floor[(Sqrt[8*n + 1] - 1) / 2]}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Sep 22 2015, after Alois P. Heinz *)

N=66;  q='q+O('q^N);
gf=sum(n=0,N, n!*q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
Vec(gf)
/* Joerg Arndt, Oct 20 2012 */

Q(N) = { \\ A008289
  my(q = vector(N)); q[1] = [1, 0, 0, 0];
  for (n = 2, N,
    my(m = (sqrtint(8*n+1) - 1)\2);
    q[n] = vector((1 + (m>>2)) << 2); q[n][1] = 1;
    for (k = 2, m, q[n][k] = q[n-k][k] + q[n-k][k-1]));
  return(q);
};
seq(N) = concat(1, apply(q -> sum(k = 1, #q, q[k] * k!), Q(N)));
seq(43) \\ Gheorghe Coserea, Sep 09 2018

"AGK" (ordered, elements, unlabeled) transform of 1, 1, 1, 1, ...
G.f.: Sum_{k>=0} k! * x^((k^2+k)/2) / Product_{j=1..k} (1-x^j). - David W. Wilson May 04 2000
a(n) = Sum_{m=1..n} A008289(n,m)*m!. - Geoffrey Critzer, Sep 07 2012

A335448 Numbers whose prime indices are inseparable.

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, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336, 343, 344, 351, 352

Offset: 1

Gus Wiseman, Jun 21 2020

nonn

First differs from A212164 in lacking 72.
First differs from A293243 in lacking 72.
No terms are squarefree.
Also Heinz numbers of inseparable partitions (A325535). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
These are also numbers that can be written as a product of prime numbers, each different from the last but not necessarily different from those prior to the last.
A multiset is inseparable iff its maximal multiplicity is greater than one plus the sum of its remaining multiplicities.

			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}

Complement of A335433.
Separations are counted by A003242 and A335452 and ranked by A333489.
Permutations of prime indices are counted by A008480.
Inseparable partitions are counted by A325535.
Strict permutations of prime indices are counted by A335489.
Cf. A000670, A000961, A005117, A056239, A112798, A181796, A261962, A333221, A335451.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Permutations[primeMS[#]],!MatchQ[#,{_,x_,x_,_}]&]=={}&]

A335433 Numbers whose multiset of prime indices is separable.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83

Offset: 1

Gus Wiseman, Jul 02 2020

nonn

First differs from A212167 in having 72.
Includes all squarefree numbers A005117.
A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Also Heinz numbers of separable partitions (A325534). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Also numbers that cannot be written as a product of prime numbers, each different from the last but not necessarily different from those prior to the last.

			The sequence of terms together with their prime indices begins:
      1: {}          20: {1,1,3}       39: {2,6}
      2: {1}         21: {2,4}         41: {13}
      3: {2}         22: {1,5}         42: {1,2,4}
      5: {3}         23: {9}           43: {14}
      6: {1,2}       26: {1,6}         44: {1,1,5}
      7: {4}         28: {1,1,4}       45: {2,2,3}
     10: {1,3}       29: {10}          46: {1,9}
     11: {5}         30: {1,2,3}       47: {15}
     12: {1,1,2}     31: {11}          50: {1,3,3}
     13: {6}         33: {2,5}         51: {2,7}
     14: {1,4}       34: {1,7}         52: {1,1,6}
     15: {2,3}       35: {3,4}         53: {16}
     17: {7}         36: {1,1,2,2}     55: {3,5}
     18: {1,2,2}     37: {12}          57: {2,8}
     19: {8}         38: {1,8}         58: {1,10}

The version for a multiset with prescribed multiplicities is A335127.
Separable factorizations are counted by A335434.
The complement is A335448.
Separations are counted by A003242 and A335452 and ranked by A333489.
Permutations of prime indices are counted by A008480.
Inseparable partitions are counted by A325535.
Strict permutations of prime indices are counted by A335489.
Cf. A000961, A005117, A056239, A112798, A114938, A181796, A292884, A335407, A335451, A335516.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Permutations[primeMS[#]],!MatchQ[#,{_,x_,x_,_}]&]!={}&]

A335452 Number of separations (Carlitz compositions or anti-runs) of the prime indices 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, 6, 1, 0, 2, 2, 2, 2, 1, 2, 2, 0, 1, 6, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 6, 1, 2, 1, 0, 2, 6, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 2, 6, 1, 0, 0, 2, 1, 6, 2, 2, 2

Offset: 1

Gus Wiseman, Jun 21 2020

nonn

The first term that is not a factorial number is a(180) = 12.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A separation (or Carlitz composition) of a multiset is a permutation with no adjacent equal parts.
a(n) depends only on the prime signature of n. - Andrew Howroyd, Feb 03 2021

			The a(n) separations for n = 2, 6, 30, 180:
  (1)  (12)  (123)  (12123)
       (21)  (132)  (12132)
             (213)  (12312)
             (231)  (12321)
             (312)  (13212)
             (321)  (21213)
                    (21231)
                    (21312)
                    (21321)
                    (23121)
                    (31212)
                    (32121)

Andrew Howroyd, Table of n, a(n) for n = 1..4096
Wikipedia, Permutation pattern

Separations are counted by A003242 and ranked by A333489.
Patterns are counted by A000670 and ranked by A333217.
Permutations of prime indices are counted by A008480.
Inseparable partitions are counted by A325535 and ranked by A335448.
Cf. A000961, A005117, A056239, A112798, A181796, A261962, A333221, A335451, A335454, A335465, A335489.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],!MatchQ[#,{_,x_,x_,_}]&]],{n,100}]

F(i, j, r, t) = {sum(k=max(0, i-j), min(min(i,t), (i-j+t)\2), binomial(i, k)*binomial(r-i+1, t+i-j-2*k)*binomial(t-1, k-i+j))}
count(sig)={my(s=vecsum(sig), r=0, v=[1]); for(p=1, #sig, my(t=sig[p]); v=vector(s-r-t+1, j, sum(i=1, #v, v[i]*F(i-1, j-1, r, t))); r += t); v[1]}
a(n)={count(factor(n)[,2])} \\ Andrew Howroyd, Feb 03 2021

A344606 Number of alternating permutations of the prime factors of n, counting multiplicity, including twins (x,x).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 1, 2, 0, 1, 1, 4, 1, 0, 2, 2, 2, 2, 1, 2, 2, 0, 1, 4, 1, 1, 1, 2, 1, 0, 1, 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, May 28 2021

nonn

Differs from A335448 in having a(x^2) = 0 and a(270) = 0.
These are permutations of the prime factors of n, counting multiplicity, with no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z.
The version without twins (x,x) is A345164, which is identical to this sequence except when n is the square of a prime.

			The permutations for n = 2, 6, 30, 180, 210, 300, 420, 720, 840:
  2   23   253   23253   2537   25253   23275   2323252   232527
      32   325   32325   2735   25352   25273   2325232   232725
           352   32523   3275   32525   25372   2523232   252327
           523   35232   3527   35252   27253             252723
                 52323   3725   52325   27352             272325
                         5273   52523   32527             272523
                         5372           32725             325272
                         5723           35272             327252
                         7253           37252             523272
                         7352           52327             527232
                                        52723             723252
                                        57232             725232
                                        72325
                                        72523
For example, there are no alternating permutations of the prime factors of 270 because the only anti-runs are {3,2,3,5,3} and {3,5,3,2,3}, neither of which is alternating, so a(270) = 0.

The version for permutations is A001250.
The extension to anti-run permutations is A335452.
The version for compositions is A344604.
The version for patterns is A344605.
Positions of zeros are A344653 (counted by A344654).
Not including twins (x,x) gives A345164.
A008480 counts permutations of prime indices (strict: A335489, rank: A333221).
A056239 adds up prime indices,  row sums of A112798.
A071321 and A071322 are signed sums of prime factors.
A316523 is a signed sum of prime multiplicities.
A316524 and A344616 are signed sums of prime indices.
A325534 counts separable partitions (ranked by A335433).
A325535 counts inseparable partitions (ranked by A335448).
A344740 counts partitions with an alternating permutation or twin (x,x).
Cf. A000041, A000607, A000961, A001222, A003242, A026424, A028260, A049774, A103919, A181796, A343938, A344652.

Table[Length[Select[Permutations[Flatten[ConstantArray@@@FactorInteger[n]]],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]],{n,100}]

A335126 A multiset whose multiplicities are the prime indices of n is inseparable.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 28, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 71, 73, 74, 76, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 101, 102, 103, 104, 106

Offset: 1

Gus Wiseman, Jul 01 2020

nonn

A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			The sequence of terms together with the corresponding multisets begins:
   3: {1,1}
   5: {1,1,1}
   7: {1,1,1,1}
  10: {1,1,1,2}
  11: {1,1,1,1,1}
  13: {1,1,1,1,1,1}
  14: {1,1,1,1,2}
  17: {1,1,1,1,1,1,1}
  19: {1,1,1,1,1,1,1,1}
  21: {1,1,1,1,2,2}
  22: {1,1,1,1,1,2}
  23: {1,1,1,1,1,1,1,1,1}
  26: {1,1,1,1,1,1,2}
  28: {1,1,1,1,2,3}
  29: {1,1,1,1,1,1,1,1,1,1}

The complement is A335127.
Anti-run compositions are A003242.
Anti-runs are ranked by A333489.
Separable partitions are A325534.
Inseparable partitions are A325535.
Separable factorizations are A335434.
Inseparable factorizations are A333487.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Patterns contiguously matched by compositions are A335457.
Cf. A025487, A056239, A106351, A112798, A114938, A181819, A181821, A278990, A292884, A335407, A335489, A335516, A335838.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Select[Permutations[nrmptn[#]],!MatchQ[#,{_,x_,x_,_}]&]=={}&]

A335127 A multiset whose multiplicities are the prime indices of n is separable.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 77, 80, 81, 84, 90, 96, 98, 99, 100, 105, 108, 110, 112, 120, 121, 125, 126, 128, 132, 135, 140, 143, 144, 147, 150, 154, 160, 162, 165

Offset: 1

Gus Wiseman, Jul 02 2020

nonn

A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			The sequence together with the corresponding multisets begins:
   1: {}
   2: {1}
   4: {1,2}
   6: {1,1,2}
   8: {1,2,3}
   9: {1,1,2,2}
  12: {1,1,2,3}
  15: {1,1,1,2,2}
  16: {1,2,3,4}
  18: {1,1,2,2,3}
  20: {1,1,1,2,3}
  24: {1,1,2,3,4}
  25: {1,1,1,2,2,2}
  27: {1,1,2,2,3,3}
  30: {1,1,1,2,2,3}

The complement is A335126.
Anti-run compositions are A003242.
Anti-runs are ranked by A333489.
Separable partitions are A325534.
Inseparable partitions are A325535.
Separable factorizations are A335434.
Inseparable factorizations are A333487.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Patterns contiguously matched by compositions are A335457.
Cf. A056239, A106351, A112798, A114938, A292884, A335489, A335516, A335838.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Select[Permutations[nrmptn[#]],!MatchQ[#,{_,x_,x_,_}]&]!={}&]

A335125 Number of anti-run permutations of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 0, 6, 2, 0, 0, 6, 0, 0, 1, 24, 0, 12, 0, 2, 0, 0, 0, 36, 2, 0, 30, 0, 0, 10, 0, 120, 0, 0, 1, 84, 0, 0, 0, 24, 0, 3, 0, 0, 38, 0, 0, 240, 2, 18, 0, 0, 0, 246, 0, 6, 0, 0, 0, 96, 0, 0, 24, 720, 0, 0, 0, 0, 0, 14, 0, 660, 0, 0, 74, 0, 1, 0, 0

Offset: 1

Gus Wiseman, Jul 01 2020

nonn

A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

An anti-run is a sequence with no adjacent equal parts.

			The a(n) permutations for n = 2, 4, 42, 8, 30, 18:
  (1)  (12)  (1212131)  (123)  (121213)  (12123)
       (21)  (1213121)  (132)  (121231)  (12132)
             (1312121)  (213)  (121312)  (12312)
                        (231)  (121321)  (12321)
                        (312)  (123121)  (13212)
                        (321)  (131212)  (21213)
                               (132121)  (21231)
                               (212131)  (21312)
                               (213121)  (21321)
                               (312121)  (23121)
                                         (31212)
                                         (32121)

Positions of zeros are A335126.
Positions of nonzeros are A335127.
The version for the prime indices themselves is A335452.
Anti-run compositions are A003242.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Separable factorizations are A335434.
Inseparable partitions are ranked by A335448.
Patterns contiguously matched by compositions are A335457.
Strict permutations of prime indices are A335489.
Cf. A019472, A056239, A106351, A112798, A114938, A278990, A292884, A325535, A335407, A335463, A335516, A335838.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Length[Select[Permutations[nrmptn[n]],!MatchQ[#,{_,x_,x_,_}]&]],{n,100}]

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).

A335451 Number of permutations of the prime indices of n with all equal parts contiguous and none appearing more than twice.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 2, 1, 2, 2, 0, 1, 2, 1, 2, 2, 2, 1, 0, 1, 2, 0, 2, 1, 6, 1, 0, 2, 2, 2, 2, 1, 2, 2, 0, 1, 6, 1, 2, 2, 2, 1, 0, 1, 2, 2, 2, 1, 0, 2, 0, 2, 2, 1, 6, 1, 2, 2, 0, 2, 6, 1, 2, 2, 6, 1, 0, 1, 2, 2, 2, 2, 6, 1, 0, 0, 2, 1, 6, 2, 2, 2

Offset: 1

Gus Wiseman, Jun 21 2020

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(90) = 6 permutations are (1,2,2,3), (1,3,2,2), (2,2,1,3), (2,2,3,1), (3,1,2,2), (3,2,2,1).

Wikipedia, Permutation pattern

Separations are counted by A003242 and A335452 and ranked by A333489.
Permutations of prime indices are counted by A008480.
Unsorted prime signature is A124010. Sorted prime signature is A118914.
Permutations of prime indices with equal parts contiguous are A333175.
STC-numbers of permutations of prime indices are A333221.
(1,2,1) and (2,1,2)-avoiding permutations of prime indices are A333175.
Numbers whose prime indices are inseparable are A335448.
(1,2,1) or (2,1,2)-matching permutations of prime indices are A335460.
(1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
Strict permutations of prime indices are counted by A335489.
Cf. A056239, A056986, A112798, A181796, A335446, A335463.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],!MatchQ[#,{_,x_,,x_,_}]&]],{n,100}]

a(n) = A001221(n)! if n is cubefree, otherwise 0.

cp's OEIS Frontend

A032020 Number of compositions (ordered partitions) of n into distinct parts.

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A335448 Numbers whose prime indices are inseparable.

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A335433 Numbers whose multiset of prime indices is separable.

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A335452 Number of separations (Carlitz compositions or anti-runs) of the prime indices of n.

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A344606 Number of alternating permutations of the prime factors of n, counting multiplicity, including twins (x,x).

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A335126 A multiset whose multiplicities are the prime indices of n is inseparable.

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A335127 A multiset whose multiplicities are the prime indices of n is separable.

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A335125 Number of anti-run permutations of a multiset whose multiplicities are the prime indices of n.

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