OFFSET
1,6
COMMENTS
Depends only on unsorted prime signature (A124010), but not 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. 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
EXAMPLE
The a(n) permutations for n = 2, 10, 36, 54, 324, 30, 1458, 90:
(1) (13) (1122) (1222) (112222) (123) (1222222) (1223)
(31) (2112) (2122) (211222) (132) (2122222) (1322)
(2211) (2212) (221122) (213) (2212222) (2123)
(2221) (222112) (231) (2221222) (2213)
(222211) (312) (2222122) (2231)
(321) (2222212) (3122)
(2222221) (3212)
(3221)
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
The matching version is A335446.
Patterns are counted by A000670.
(1,2,1)-avoiding patterns are counted by A001710.
Permutations of prime indices are counted by A008480.
(1,2,1) and (2,1,2)-avoiding permutations of prime indices are counted by A333175.
STC-numbers of permutations of prime indices are A333221.
(1,2,1) and (2,1,2)-avoiding permutations of prime indices are A335448.
Patterns matched by standard compositions are counted by A335454.
(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.
Dimensions of downsets of standard compositions are A335465.
(1,2,1)-avoiding compositions are ranked by A335467.
(1,2,1)-avoiding compositions are counted by A335471.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 14 2020
STATUS
approved