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A335549 Number of normal patterns matched by the multiset of prime indices of n in weakly increasing order.

%I A335549 #9 Jun 30 2020 09:56:11
%S A335549 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 A335549 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 A335549 3,4,2,10,2,3,5,5,3,4,2,9,5,3,2,7,3,3,3
%N A335549 Number of normal patterns matched by the multiset of prime indices of n in weakly increasing order.
%C A335549 First differs from A181796 at a(90) = 8 A181796(90) = 7.
%C A335549 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 A335549 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 A335549 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H A335549 Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%e A335549 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 A335549 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A335549 mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
%t A335549 Table[Length[Union[mstype/@Subsets[primeMS[n]]]],{n,100}]
%Y A335549 The version for standard compositions instead of prime indices is A335454.
%Y A335549 Permutations of prime indices are counted by A008480.
%Y A335549 Permutations are counted by A000142 and ranked by A333218.
%Y A335549 Patterns are counted by A000670 and ranked by A333217.
%Y A335549 Subset-sums are counted by A304792 and ranked by A299701.
%Y A335549 Patterns matched by compositions of n are counted by A335456(n).
%Y A335549 Minimal patterns avoided by a standard composition are counted by A335465.
%Y A335549 Cf. A056239, A108917, A112798, A333221, A335452, A335457, A335458.
%K A335549 nonn
%O A335549 1,2
%A A335549 _Gus Wiseman_, Jun 21 2020