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A348380
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Number of factorizations of n without an alternating permutation. Includes all twins (x*x).
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21
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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, 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, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0
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OFFSET
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1,16
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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 {1..n}.
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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, 144, 192, 384:
(2*2*2*12) (12*12) (3*4*4*4) (4*4*4*6)
(2*2*2*2*6) (2*2*2*18) (2*2*2*24) (2*2*2*48)
(2*2*2*2*2*3) (2*2*2*2*9) (2*2*2*2*12) (2*2*2*2*24)
(2*2*2*2*3*3) (2*2*2*2*2*6) (2*2*2*2*3*8)
(2*2*2*2*3*4) (2*2*2*2*4*6)
(2*2*2*2*2*2*3) (2*2*2*2*2*12)
(2*2*2*2*2*2*6)
(2*2*2*2*2*3*4)
(2*2*2*2*2*2*2*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]]}]];
wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[facs[n], Select[Permutations[#], wigQ]=={}&]], {n, 100}]
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CROSSREFS
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Non-twin partitions of this type are counted by A344654, ranked by A344653.
Twins and partitions not of this type are counted by A344740, ranked by A344742.
Partitions not of this type are counted by A345170, ranked by A345172.
Numbers with a factorization of this type are A348609.
A001250 counts alternating permutations.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
Cf. A038548, A049774, A119620, A289553, A325534, A336107, A344614, A345192, A347437, A347438, A347439, A347442, A347458, A348383, A348611.
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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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