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%I #10 Dec 10 2021 11:12:22
%S 1,1,1,1,1,2,1,1,1,2,1,3,1,2,2,1,1,3,1,3,2,2,1,4,1,2,1,3,1,4,1,1,2,2,
%T 2,4,1,2,2,4,1,4,1,3,3,2,1,5,1,3,2,3,1,4,2,4,2,2,1,6,1,2,3,1,2,4,1,3,
%U 2,4,1,6,1,2,3,3,2,4,1,5,1,2,1,6,2,2,2
%N Number of weakly alternating permutations of the multiset of prime factors of n.
%C We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. Then a sequence is alternating in the sense of A025047 iff it is a weakly alternating anti-run.
%C A prime index of n is a number m such that prime(m) divides n. For n > 1, the multiset of prime factors of n is row n of A027746. The prime indices A112798 can also be used.
%e The following are the weakly alternating permutations for selected n:
%e n = 2 6 12 24 48 60 90 120 180
%e ----------------------------------------------------------
%e 2 23 223 2223 22223 2253 2335 22253 22335
%e 32 232 2232 22232 2325 2533 22325 22533
%e 322 2322 22322 2523 3253 22523 23253
%e 3222 23222 3252 3325 23252 23352
%e 32222 3522 3352 25232 25233
%e 5232 3523 32225 25332
%e 5233 32522 32325
%e 5332 35222 32523
%e 52223 33252
%e 52322 33522
%e 35232
%e 52323
%e 53322
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
%t Table[Length[Select[Permutations[primeMS[n]],whkQ[#]||whkQ[-#]&]],{n,100}]
%Y Counting all permutations of prime factors gives A008480.
%Y The variation counting anti-run permutations is A335452.
%Y The strong case is A345164, with twins A344606.
%Y Compositions of this type are counted by A349052, also A129852 and A129853.
%Y Compositions not of this type are counted by A349053, ranked by A349057.
%Y The version for patterns is A349058, strong A345194.
%Y The version for ordered factorizations is A349059, strong A348610.
%Y Partitions of this type are counted by A349060, complement A349061.
%Y The complement is counted by A349797.
%Y The non-alternating case is A349798.
%Y A001250 counts alternating permutations, complement A348615.
%Y A003242 counts Carlitz (anti-run) compositions.
%Y A025047 counts alternating or wiggly compositions, ranked by A345167.
%Y A056239 adds up prime indices, row sums of A112798, row lengths A001222.
%Y A071321 gives the alternating sum of prime factors, reverse A071322.
%Y A344616 gives the alternating sum of prime indices, reverse A316524.
%Y A345165 counts partitions w/o an alternating permutation, ranked by A345171.
%Y A345170 counts partitions w/ an alternating permutation, ranked by A345172.
%Y A348379 counts factorizations with an alternating permutation.
%Y A349800 counts weakly but not strongly alternating compositions.
%Y Cf. A028234, A051119, A096441, A335433, A335448, A344614, A344652, A344653, A345173, A345192.
%K nonn
%O 1,6
%A _Gus Wiseman_, Dec 02 2021
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