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A347706 Number of factorizations of n that are not a twin (x*x) nor have an alternating permutation.

%I A347706 #14 Nov 03 2021 10:48:31
%S A347706 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,2,0,0,
%T A347706 0,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,4,0,0,0,0,
%U A347706 0,0,0,1,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0
%N A347706 Number of factorizations of n that are not a twin (x*x) nor have an alternating permutation.
%C A347706 First differs from A348381 at a(216) = 4, A348381(216) = 3.
%C A347706 A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
%C A347706 A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). Alternating permutations of multisets are a generalization of alternating or up-down permutations of sets.
%H A347706 Wikipedia, <a href="https://en.wikipedia.org/wiki/Alternating_permutation">Alternating permutation</a>
%F A347706 a(2^n) = A344654(n).
%e A347706 The a(n) factorizations for n = 96, 192, 2160, 576:
%e A347706   2*2*2*12      3*4*4*4         3*3*3*80       4*4*4*9
%e A347706   2*2*2*2*6     2*2*2*24        6*6*6*10       2*2*2*72
%e A347706   2*2*2*2*2*3   2*2*2*2*12      2*2*2*270      2*2*2*2*36
%e A347706                 2*2*2*2*2*6     2*3*3*3*40     2*2*2*2*4*9
%e A347706                 2*2*2*2*3*4     2*2*2*2*135    2*2*2*2*6*6
%e A347706                 2*2*2*2*2*2*3   2*2*2*2*3*45   2*2*2*2*2*18
%e A347706                                 2*2*2*2*5*27   2*2*2*2*3*12
%e A347706                                 2*2*2*2*9*15   2*2*2*2*2*2*9
%e A347706                                                2*2*2*2*2*3*6
%e A347706                                                2*2*2*2*2*2*3*3
%t A347706 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A347706 Table[Length[Select[facs[n],Function[f,Select[Permutations[f],!MatchQ[#,{___,x_,y_,z_,___}/;x<=y<=z||x>=y>=z]&]=={}]]],{n,100}]
%Y A347706 Positions of nonzero terms are A046099.
%Y A347706 Partitions of this type are counted by A344654, ranked by A344653.
%Y A347706 Partitions not of this type are counted by A344740, ranked by A344742.
%Y A347706 The complement is counted by A347050, without twins A348379.
%Y A347706 The version for compositions is A348377.
%Y A347706 The version allowing twins is A348380.
%Y A347706 The inseparable case is A348381.
%Y A347706 A001055 counts factorizations, strict A045778, ordered A074206.
%Y A347706 A001250 counts alternating permutations of sets.
%Y A347706 A025047 counts alternating or wiggly compositions, ranked by A345167.
%Y A347706 A339846 counts even-length factorizations.
%Y A347706 A339890 counts odd-length factorizations.
%Y A347706 A347438 counts factorizations with alternating product 1, additive A119620.
%Y A347706 A348610 counts alternating ordered factorizations.
%Y A347706 Cf. A049774, A325535, A336103, A344614, A347437, A347439, A347442, A347456, A347463, A348382, A348383.
%K A347706 nonn
%O A347706 1,32
%A A347706 _Gus Wiseman_, Oct 28 2021