cp's OEIS Frontend

A349056 Number of weakly alternating permutations of the multiset of prime factors of n.

%I A349056 #10 Dec 10 2021 11:12:22
%S A349056 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 A349056 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 A349056 2,4,1,6,1,2,3,3,2,4,1,5,1,2,1,6,2,2,2
%N A349056 Number of weakly alternating permutations of the multiset of prime factors of n.
%C A349056 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 A349056 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 A349056 The following are the weakly alternating permutations for selected n:
%e A349056 n = 2   6    12    24     48      60     90     120     180
%e A349056    ----------------------------------------------------------
%e A349056     2   23   223   2223   22223   2253   2335   22253   22335
%e A349056         32   232   2232   22232   2325   2533   22325   22533
%e A349056              322   2322   22322   2523   3253   22523   23253
%e A349056                    3222   23222   3252   3325   23252   23352
%e A349056                           32222   3522   3352   25232   25233
%e A349056                                   5232   3523   32225   25332
%e A349056                                          5233   32522   32325
%e A349056                                          5332   35222   32523
%e A349056                                                 52223   33252
%e A349056                                                 52322   33522
%e A349056                                                         35232
%e A349056                                                         52323
%e A349056                                                         53322
%t A349056 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A349056 whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
%t A349056 Table[Length[Select[Permutations[primeMS[n]],whkQ[#]||whkQ[-#]&]],{n,100}]
%Y A349056 Counting all permutations of prime factors gives A008480.
%Y A349056 The variation counting anti-run permutations is A335452.
%Y A349056 The strong case is A345164, with twins A344606.
%Y A349056 Compositions of this type are counted by A349052, also A129852 and A129853.
%Y A349056 Compositions not of this type are counted by A349053, ranked by A349057.
%Y A349056 The version for patterns is A349058, strong A345194.
%Y A349056 The version for ordered factorizations is A349059, strong A348610.
%Y A349056 Partitions of this type are counted by A349060, complement A349061.
%Y A349056 The complement is counted by A349797.
%Y A349056 The non-alternating case is A349798.
%Y A349056 A001250 counts alternating permutations, complement A348615.
%Y A349056 A003242 counts Carlitz (anti-run) compositions.
%Y A349056 A025047 counts alternating or wiggly compositions, ranked by A345167.
%Y A349056 A056239 adds up prime indices, row sums of A112798, row lengths A001222.
%Y A349056 A071321 gives the alternating sum of prime factors, reverse A071322.
%Y A349056 A344616 gives the alternating sum of prime indices, reverse A316524.
%Y A349056 A345165 counts partitions w/o an alternating permutation, ranked by A345171.
%Y A349056 A345170 counts partitions w/ an alternating permutation, ranked by A345172.
%Y A349056 A348379 counts factorizations with an alternating permutation.
%Y A349056 A349800 counts weakly but not strongly alternating compositions.
%Y A349056 Cf. A028234, A051119, A096441, A335433, A335448, A344614, A344652, A344653, A345173, A345192.
%K A349056 nonn
%O A349056 1,6
%A A349056 _Gus Wiseman_, Dec 02 2021