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A347706
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Number of factorizations of n that are not a twin (x*x) nor have an alternating permutation.
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24
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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, 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, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0
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OFFSET
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1,32
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COMMENTS
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A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
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.
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LINKS
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FORMULA
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EXAMPLE
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The a(n) factorizations for n = 96, 192, 2160, 576:
2*2*2*12 3*4*4*4 3*3*3*80 4*4*4*9
2*2*2*2*6 2*2*2*24 6*6*6*10 2*2*2*72
2*2*2*2*2*3 2*2*2*2*12 2*2*2*270 2*2*2*2*36
2*2*2*2*2*6 2*3*3*3*40 2*2*2*2*4*9
2*2*2*2*3*4 2*2*2*2*135 2*2*2*2*6*6
2*2*2*2*2*2*3 2*2*2*2*3*45 2*2*2*2*2*18
2*2*2*2*5*27 2*2*2*2*3*12
2*2*2*2*9*15 2*2*2*2*2*2*9
2*2*2*2*2*3*6
2*2*2*2*2*2*3*3
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MATHEMATICA
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facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], Function[f, Select[Permutations[f], !MatchQ[#, {___, x_, y_, z_, ___}/; x<=y<=z||x>=y>=z]&]=={}]]], {n, 100}]
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CROSSREFS
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Positions of nonzero terms are A046099.
Partitions not of this type are counted by A344740, ranked by A344742.
The version for compositions is A348377.
The version allowing twins is A348380.
A001250 counts alternating permutations of sets.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A347438 counts factorizations with alternating product 1, additive A119620.
A348610 counts alternating ordered factorizations.
Cf. A049774, A325535, A336103, A344614, A347437, A347439, A347442, A347456, A347463, A348382, A348383.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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