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A335450 Number of (2,1,2)-avoiding permutations of the prime indices of n. 6
1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 6, 1, 1, 2, 2, 2, 3, 1, 2, 2, 4, 1, 6, 1, 3, 3, 2, 1, 5, 1, 2, 2, 3, 1, 2, 2, 4, 2, 2, 1, 12, 1, 2, 3, 1, 2, 6, 1, 3, 2, 6, 1, 4, 1, 2, 2, 3, 2, 6, 1, 5, 1, 2, 1, 12, 2, 2 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..86.

Wikipedia, Permutation pattern

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

EXAMPLE

The permutations for n = 2, 6, 12, 24, 30, 48, 60, 90:

  (1)  (12)  (112)  (1112)  (123)  (11112)  (1123)  (1223)

       (21)  (211)  (2111)  (132)  (21111)  (1132)  (1322)

                            (213)           (2113)  (2123)

                            (231)           (2311)  (2213)

                            (312)           (3112)  (2231)

                            (321)           (3211)  (3122)

                                                    (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

Positions of ones are A000961.

Replacing (2,1,2) with (1,2,1) gives A335449.

The matching version is A335453.

Patterns are counted by A000670.

(2,1,2)-avoiding patterns are counted by A001710.

Permutations of prime indices are counted by A008480.

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

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

(2,1,2)-avoiding compositions are ranked by A335469.

(2,1,2)-avoiding compositions are counted by A335473.

(2,2,1)-avoiding compositions are ranked by A335524.

(1,2,2)-avoiding compositions are ranked by A335525.

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

Sequence in context: A330757 A322373 A332288 * A324191 A238946 A326516

Adjacent sequences:  A335447 A335448 A335449 * A335451 A335452 A335453

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 14 2020

STATUS

approved

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Last modified September 18 22:34 EDT 2021. Contains 347548 sequences. (Running on oeis4.)