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approved

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Gus Wiseman, <a href="https:/A102726/docs.googlea102726.com/document/d/e/2PACX-1vQiRtjvNcvMLAqzpp4R2HmiWiiFE3qLundk8xemwExxqIURDW4WPlsJ1S3VhB1X7kTVrvYkupa2ZXlW/pubtxt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>

References found in the link are not all repeated here.

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

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

(1,2,1)-avoiding compositions are ranked by A335467.

(1,2,1)-avoiding compositions are counted by A335471.

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

approved

editing

proposed

approved

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References found in the link are not all repeated here.

References found in the link are not repeated below.

(1,2,1) and (2,1,2)-avoiding permutations of prime indices are counted by A335448.

(1,2,1) or (2,1,2)-matching permutations of prime indices are counted by A335460.

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

allocated for Gus WisemanNumber of (1,2,1)-avoiding permutations of the prime indices of n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 6, 1, 2, 2, 2, 1, 2, 1, 3, 2, 2, 1, 4, 2, 2, 2, 2, 1, 6, 1, 2, 2, 1, 2, 6, 1, 2, 2, 6, 1, 3, 1, 2, 3, 2, 2, 6, 1, 2, 1, 2, 1, 6, 2, 2, 2

1,6

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).

Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>

Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vQiRtjvNcvMLAqzpp4R2HmiWiiFE3qLundk8xemwExxqIURDW4WPlsJ1S3VhB1X7kTVrvYkupa2ZXlW/pub">Sequences counting and ranking compositions by the patterns they match or avoid.</a>

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)

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}]

The matching version is A335446.

References found in the link are not repeated below.

Patterns are counted by A000670.

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 counted by A335448.

Patterns matched by standard compositions are counted by A335454.

(1,2,1) or (2,1,2)-matching permutations of prime indices are counted by A335460.

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

Dimensions of downsets of standard compositions are A335465.

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

allocated

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Gus Wiseman, Jun 14 2020

approved

editing

allocated for Gus Wiseman

allocated

approved