A335460 Number of (1,2,1) or (2,1,2)-matching permutations of the prime indices of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 6, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 8, 0, 0, 1, 1, 0, 0, 0, 3, 0, 0, 0, 6, 0, 0, 0

Offset

1,24

Comments

Depends only on sorted prime signature (A118914).

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.

Example

The a(n) compositions for n = 12, 24, 48, 36, 60, 72:
  (121)  (1121)  (11121)  (1212)  (1213)  (11212)
         (1211)  (11211)  (1221)  (1231)  (11221)
                 (12111)  (2112)  (1312)  (12112)
                          (2121)  (1321)  (12121)
                                  (2131)  (12211)
                                  (3121)  (21112)
                                          (21121)
                                          (21211)

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_, ___, x_, ___}/; x!=y]&]], {n, 100}]

Crossrefs

Positions of zeros are A303554.

The (1,2,1)-matching part is A335446.

The (2,1,2)-matching part is A335453.

Replacing "or" with "and" gives A335462.

Permutations of prime indices are counted by A008480.

Unsorted prime signature is A124010. Sorted prime signature is A118914.

STC-numbers of permutations of prime indices are A333221.

(1,2,1) and (2,1,2)-avoiding permutations of prime indices are A333175.

Patterns matched by standard compositions are counted by A335454.

(1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.

Dimensions of downsets of standard compositions are A335465.

Cf. A056239, A056986, A112798, A158005, A181796, A335451, A335452, A335463.

Sequence in context: A076948 A255309 A335446 * A367077 A368073 A086071

Adjacent sequences: A335457 A335458 A335459 * A335461 A335462 A335463

Keyword

nonn

Author

Gus Wiseman, Jun 20 2020

Status

approved