A335521 Number of (1,2,3)-avoiding permutations of the prime indices of n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 5, 1, 1, 2, 2, 2, 6, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 1, 2, 5, 1, 3, 2, 5, 1, 10, 1, 2, 3, 3, 2, 5, 1, 5, 1, 2, 1, 9, 2, 2, 2

Offset

1,6

Comments

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.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

Links

Table of n, a(n) for n=1..87.

Wikipedia, Permutation pattern

Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Formula

For n > 0, a(n) + A335520(n) = A008480(n).

Example

The a(n) permutations for n = 1, 6, 12, 24, 30, 36, 60, 72, 120:
  ()  (12)  (112)  (1112)  (132)  (1122)  (1132)  (11122)  (11132)
      (21)  (121)  (1121)  (213)  (1212)  (1312)  (11212)  (11312)
            (211)  (1211)  (231)  (1221)  (1321)  (11221)  (11321)
                   (2111)  (312)  (2112)  (2113)  (12112)  (13112)
                           (321)  (2121)  (2131)  (12121)  (13121)
                                  (2211)  (2311)  (12211)  (13211)
                                          (3112)  (21112)  (21113)
                                          (3121)  (21121)  (21131)
                                          (3211)  (21211)  (21311)
                                                  (22111)  (23111)
                                                           (31112)
                                                           (31121)
                                                           (31211)
                                                           (32111)

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Select[Permutations[primeMS[n]], !MatchQ[#, {___, x_, ___, y_, ___, z_, ___}/; x<y<z]&]], {n, 100}]

Crossrefs

These compositions are counted by A102726.

Patterns avoiding this pattern are counted by A226316.

The complement A335520 is the matching version.

Permutations of prime indices are counted by A008480.

Patterns are counted by A000670 and ranked by A333217.

Anti-run permutations of prime indices are counted by A335452.

Cf. A056239, A056986, A112798, A238279, A281188, A333221, A333755, A335456, A335460, A335462, A335463.

Sequence in context: A081707 A368414 A303707 * A323087 A321747 A008480

Adjacent sequences: A335518 A335519 A335520 * A335522 A335523 A335524

Keyword

nonn

Author

Gus Wiseman, Jun 19 2020

Status

approved