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A348380 Number of factorizations of n without an alternating permutation. Includes all twins (x*x).

%I A348380 #8 Nov 03 2021 10:48:57
%S A348380 0,0,0,1,0,0,0,1,1,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,1,0,1,0,0,0,0,2,0,0,
%T A348380 0,1,0,0,0,1,0,0,0,0,0,0,0,2,1,0,0,0,0,1,0,1,0,0,0,0,0,0,0,5,0,0,0,0,
%U A348380 0,0,0,1,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0
%N A348380 Number of factorizations of n without an alternating permutation. Includes all twins (x*x).
%C A348380 First differs from A333487 at a(216) = 4, A333487(216) = 3.
%C A348380 A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
%C A348380 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 {1..n}.
%H A348380 Wikipedia, <a href="https://en.wikipedia.org/wiki/Alternating_permutation">Alternating permutation</a>
%F A348380 a(2^n) = A345165(n).
%e A348380 The a(n) factorizations for n = 96, 144, 192, 384:
%e A348380   (2*2*2*12)     (12*12)        (3*4*4*4)        (4*4*4*6)
%e A348380   (2*2*2*2*6)    (2*2*2*18)     (2*2*2*24)       (2*2*2*48)
%e A348380   (2*2*2*2*2*3)  (2*2*2*2*9)    (2*2*2*2*12)     (2*2*2*2*24)
%e A348380                  (2*2*2*2*3*3)  (2*2*2*2*2*6)    (2*2*2*2*3*8)
%e A348380                                 (2*2*2*2*3*4)    (2*2*2*2*4*6)
%e A348380                                 (2*2*2*2*2*2*3)  (2*2*2*2*2*12)
%e A348380                                                  (2*2*2*2*2*2*6)
%e A348380                                                  (2*2*2*2*2*3*4)
%e A348380                                                  (2*2*2*2*2*2*2*3)
%t A348380 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A348380 wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
%t A348380 Table[Length[Select[facs[n],Select[Permutations[#],wigQ]=={}&]],{n,100}]
%Y A348380 The inseparable case is A333487, complement A335434, without twins A348381.
%Y A348380 Non-twin partitions of this type are counted by A344654, ranked by A344653.
%Y A348380 Twins and partitions not of this type are counted by A344740, ranked by A344742.
%Y A348380 Partitions of this type are counted by A345165, ranked by A345171.
%Y A348380 Partitions not of this type are counted by A345170, ranked by A345172.
%Y A348380 The case without twins is A347706.
%Y A348380 The complement is counted by A348379, with twins A347050.
%Y A348380 Numbers with a factorization of this type are A348609.
%Y A348380 An ordered version is A348613, complement A348610.
%Y A348380 A001055 counts factorizations, strict A045778, ordered A074206.
%Y A348380 A001250 counts alternating permutations.
%Y A348380 A025047 counts alternating or wiggly compositions, ranked by A345167.
%Y A348380 A325535 counts inseparable partitions, ranked by A335448.
%Y A348380 A339846 counts even-length factorizations.
%Y A348380 A339890 counts odd-length factorizations.
%Y A348380 Cf. A038548, A049774, A119620, A289553, A325534, A336107, A344614, A345192, A347437, A347438, A347439, A347442, A347458, A348383, A348611.
%K A348380 nonn
%O A348380 1,16
%A A348380 _Gus Wiseman_, Oct 28 2021