A335462 Number of (1,2,1) and (2,1,2)-matching permutations of the prime indices of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Jun 20 2020

Keywords

• nonn

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).

Examples

			The a(n) permutations for n = 36, 72, 270, 144, 300:
  (1,2,1,2)  (1,1,2,1,2)  (2,1,2,3,2)  (1,1,1,2,1,2)  (1,2,3,1,3)
  (2,1,2,1)  (1,2,1,1,2)  (2,1,3,2,2)  (1,1,2,1,1,2)  (1,3,1,2,3)
             (1,2,1,2,1)  (2,2,1,3,2)  (1,1,2,1,2,1)  (1,3,1,3,2)
             (2,1,1,2,1)  (2,2,3,1,2)  (1,2,1,1,1,2)  (1,3,2,1,3)
             (2,1,2,1,1)  (2,3,1,2,2)  (1,2,1,1,2,1)  (1,3,2,3,1)
                          (2,3,2,1,2)  (1,2,1,2,1,1)  (2,1,3,1,3)
                                       (2,1,1,1,2,1)  (2,3,1,3,1)
                                       (2,1,1,2,1,1)  (3,1,2,1,3)
                                       (2,1,2,1,1,1)  (3,1,2,3,1)
                                                      (3,1,3,1,2)
                                                      (3,1,3,2,1)
                                                      (3,2,1,3,1)
		

Links

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

Crossrefs

The avoiding version is A333175.

Replacing "and" with "or" gives A335460.

Positions of nonzero terms are A335463.

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.

Patterns matched by standard compositions are counted by A335454.

Dimensions of downsets of standard compositions are A335465.

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

Programs

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_,x_,_,y_,_,x_,_}/;x>y]&]],{n,100}]