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A348613 Number of non-alternating ordered factorizations of n.

%I A348613 #9 Nov 04 2021 06:26:18
%S A348613 0,0,0,1,0,0,0,1,1,0,0,2,0,0,0,4,0,2,0,2,0,0,0,8,1,0,1,2,0,2,0,9,0,0,
%T A348613 0,11,0,0,0,8,0,2,0,2,2,0,0,25,1,2,0,2,0,8,0,8,0,0,0,16,0,0,2,20,0,2,
%U A348613 0,2,0,2,0,43,0,0,2,2,0,2,0,25,4,0,0,16,0
%N A348613 Number of non-alternating ordered factorizations of n.
%C A348613 An ordered factorization of n is a finite sequence of positive integers > 1 with product n.
%C A348613 A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either.
%H A348613 Wikipedia, <a href="https://en.wikipedia.org/wiki/Alternating_permutation">Alternating permutation</a>
%e A348613 The a(n) ordered factorizations for n = 4, 12, 16, 24, 32, 36:
%e A348613   2*2   2*2*3   4*4       2*2*6     2*2*8       6*6
%e A348613         3*2*2   2*2*4     2*3*4     2*4*4       2*2*9
%e A348613                 4*2*2     4*3*2     4*4*2       2*3*6
%e A348613                 2*2*2*2   6*2*2     8*2*2       3*3*4
%e A348613                           2*2*2*3   2*2*2*4     4*3*3
%e A348613                           2*2*3*2   2*2*4*2     6*3*2
%e A348613                           2*3*2*2   2*4*2*2     9*2*2
%e A348613                           3*2*2*2   4*2*2*2     2*2*3*3
%e A348613                                     2*2*2*2*2   2*3*3*2
%e A348613                                                 3*2*2*3
%e A348613                                                 3*3*2*2
%t A348613 ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
%t A348613 wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
%t A348613 Table[Length[Select[ordfacs[n],!wigQ[#]&]],{n,100}]
%Y A348613 The complementary additive version is A025047, ranked by A345167.
%Y A348613 The additive version is A345192, ranked by A345168, without twins A348377.
%Y A348613 The complement is counted by A348610.
%Y A348613 A001055 counts factorizations, strict A045778, ordered A074206.
%Y A348613 A001250 counts alternating permutations.
%Y A348613 A339846 counts even-length factorizations.
%Y A348613 A339890 counts odd-length factorizations.
%Y A348613 A345165 counts partitions without an alternating permutation, ranked by A345171.
%Y A348613 A345170 counts partitions with an alternating permutation, ranked by A345172.
%Y A348613 A348379 counts factorizations w/ an alternating permutation, with twins A347050.
%Y A348613 A348380 counts factorizations w/o an alternating permutation, w/o twins A347706.
%Y A348613 A348611 counts anti-run ordered factorizations.
%Y A348613 Cf. A038548, A056986, A344614, A344653, A344654, A344740, A347437, A347438, A347463, A348381, A348609.
%K A348613 nonn
%O A348613 1,12
%A A348613 _Gus Wiseman_, Nov 03 2021