cp's OEIS Frontend

A349797 Number of non-weakly alternating permutations of the multiset of prime factors of n.

%I A349797 #12 Jan 07 2022 15:54:38
%S A349797 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,
%T A349797 0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,2,0,0,
%U A349797 0,2,0,4,0,0,0,0,0,2,0,0,0,0,0,6,0,0,0
%N A349797 Number of non-weakly alternating permutations of the multiset of prime factors of n.
%C A349797 First differs from 2 * A326291 at a(90) = 4, A326291(90) = 3.
%C A349797 The first odd term is a(144) = 7, whose non-weakly alternating permutations are shown in the example below.
%C A349797 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 A349797 For n > 1, the multiset of prime factors of n is row n of A027746. The prime indices A112798 can also be used.
%F A349797 a(n) = A008480(n) - A349056(n).
%e A349797 The following are the weakly alternating permutations for selected n.
%e A349797 n = 30    60     72      120     144      180
%e A349797    ---------------------------------------------
%e A349797     235   2235   22332   22235   222332   22353
%e A349797     532   2352   23223   22352   223223   23235
%e A349797           2532   23322   22532   223322   23325
%e A349797           3225   32232   23225   232232   23523
%e A349797           5223           23522   233222   23532
%e A349797           5322           25223   322223   25323
%e A349797                          25322   322322   32235
%e A349797                          32252            32253
%e A349797                          52232            32352
%e A349797                          53222            32532
%e A349797                                           33225
%e A349797                                           35223
%e A349797                                           35322
%e A349797                                           52233
%e A349797                                           52332
%e A349797                                           53223
%e A349797                                           53232
%t A349797 whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
%t A349797 Table[Length[Select[Permutations[Flatten[ConstantArray@@@ FactorInteger[n]]], !whkQ[#]&&!whkQ[-#]&]],{n,100}]
%Y A349797 Counting all permutations of prime factors gives A008480.
%Y A349797 Compositions not of this type are counted by A349052/A129852/A129853.
%Y A349797 Compositions of this type are counted by A349053, ranked by A349057.
%Y A349797 The complement is counted by A349056.
%Y A349797 Partitions of this type are counted by A349061, complement A349060.
%Y A349797 The version counting patterns is A350138, complement A349058.
%Y A349797 The version counting ordered factorizations is A350139, complement A349059.
%Y A349797 The strong case is counted by A350251, complement A345164.
%Y A349797 Positions of nonzero terms are A350353.
%Y A349797 A001250 counts alternating permutations, complement A348615.
%Y A349797 A025047 = alternating compositions, ranked by A345167, complement A345192.
%Y A349797 A056239 adds up prime indices, row sums of A112798, row lengths A001222.
%Y A349797 A071321 gives the alternating sum of prime factors, reverse A071322.
%Y A349797 A335452 counts anti-run permutations of prime factors, complement A336107.
%Y A349797 A345165 counts partitions w/o an alternating permutation, ranked by A345171.
%Y A349797 A345170 counts partitions w/ an alternating permutation, ranked by A345172.
%Y A349797 A348379 counts factorizations with an alternating permutation.
%Y A349797 Cf. A003242, A335433, A335448, A344614, A344652, A344653, A345173, A348613, A349798, A350252, A349800.
%K A349797 nonn
%O A349797 1,30
%A A349797 _Gus Wiseman_, Dec 24 2021