A335516 - OEIS

%I #10 Jun 27 2020 09:07:43

%S 1,2,2,3,2,3,2,4,3,3,2,5,2,3,3,5,2,5,2,5,3,3,2,7,3,3,4,5,2,4,2,6,3,3,

%T 3,7,2,3,3,7,2,4,2,5,5,3,2,9,3,5,3,5,2,7,3,7,3,3,2,7,2,3,5,7,3,4,2,5,

%U 3,4,2,10,2,3,5,5,3,4,2,9,5,3,2,7,3,3,3

%N Number of normal patterns contiguously matched by the prime indices of n in increasing or decreasing order, counting multiplicity.

%C First differs from A181796 at a(180) = 9, A181796(180) = 10.

%C First differs from A335549 at a(90) = 7, A335549(90) = 8.

%C 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.

%C We define a (normal) 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 contiguously match a pattern P if there is a contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) contiguously matches (1,1,2) and (2,1,1) but not (2,1,2), (1,2,1), (1,2,2), or (2,2,1).

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

%H Gus Wiseman, <a href="/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>

%e The a(n) patterns for n = 2, 30, 12, 60, 120, 540, 1500:

%e   ()   ()     ()     ()      ()       ()        ()

%e   (1)  (1)    (1)    (1)     (1)      (1)       (1)

%e        (12)   (11)   (11)    (11)     (11)      (11)

%e        (123)  (12)   (12)    (12)     (12)      (12)

%e               (112)  (112)   (111)    (111)     (111)

%e                      (123)   (112)    (112)     (112)

%e                      (1123)  (123)    (122)     (122)

%e                              (1112)   (1112)    (123)

%e                              (1123)   (1122)    (1123)

%e                              (11123)  (1222)    (1222)

%e                                       (11222)   (1233)

%e                                       (12223)   (11233)

%e                                       (112223)  (12333)

%e                                                 (112333)

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];

%t Table[Length[Union[mstype/@ReplaceList[primeMS[n],{___,s___,___}:>{s}]]],{n,100}]

%Y The version for standard compositions is A335458.

%Y The not necessarily contiguous version is A335549.

%Y Patterns are counted by A000670 and ranked by A333217.

%Y A number's prime indices are given in the rows of A112798.

%Y Contiguous subsequences of standard compositions are A124771.

%Y Contiguous sub-partitions of prime indices are counted by A335519.

%Y Minimal avoided patterns of prime indices are counted by A335550.

%Y Patterns contiguously matched by partitions are counted by A335838.

%Y Cf. A000005, A056239, A056986, A108917, A124770, A181796, A269134, A333224, A334299, A335457, A335837.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 26 2020