A335549 - OEIS

%I #9 Jun 30 2020 09:56:11

%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 matched by the multiset of prime indices of n in weakly increasing order.

%C First differs from A181796 at a(90) = 8 A181796(90) = 7.

%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 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).

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

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

%e The Heinz number of (1,2,2,3) is 90 and it matches 8 patterns: (), (1), (11), (12), (112), (122), (123), (1223); so a(90) = 8.

%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/@Subsets[primeMS[n]]]],{n,100}]

%Y The version for standard compositions instead of prime indices is A335454.

%Y Permutations of prime indices are counted by A008480.

%Y Permutations are counted by A000142 and ranked by A333218.

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

%Y Subset-sums are counted by A304792 and ranked by A299701.

%Y Patterns matched by compositions of n are counted by A335456(n).

%Y Minimal patterns avoided by a standard composition are counted by A335465.

%Y Cf. A056239, A108917, A112798, A333221, A335452, A335457, A335458.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 21 2020