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A348613
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Number of non-alternating ordered factorizations of n.
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24
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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, 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, 0, 2, 0, 2, 0, 43, 0, 0, 2, 2, 0, 2, 0, 25, 4, 0, 0, 16, 0
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OFFSET
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1,12
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COMMENTS
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An ordered factorization of n is a finite sequence of positive integers > 1 with product n.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either.
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LINKS
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EXAMPLE
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The a(n) ordered factorizations for n = 4, 12, 16, 24, 32, 36:
2*2 2*2*3 4*4 2*2*6 2*2*8 6*6
3*2*2 2*2*4 2*3*4 2*4*4 2*2*9
4*2*2 4*3*2 4*4*2 2*3*6
2*2*2*2 6*2*2 8*2*2 3*3*4
2*2*2*3 2*2*2*4 4*3*3
2*2*3*2 2*2*4*2 6*3*2
2*3*2*2 2*4*2*2 9*2*2
3*2*2*2 4*2*2*2 2*2*3*3
2*2*2*2*2 2*3*3*2
3*2*2*3
3*3*2*2
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MATHEMATICA
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ordfacs[n_]:=If[n<=1, {{}}, Join@@Table[Prepend[#, d]&/@ordfacs[n/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[ordfacs[n], !wigQ[#]&]], {n, 100}]
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CROSSREFS
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The complement is counted by A348610.
A001250 counts alternating permutations.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A345165 counts partitions without an alternating permutation, ranked by A345171.
A345170 counts partitions with an alternating permutation, ranked by A345172.
A348379 counts factorizations w/ an alternating permutation, with twins A347050.
A348380 counts factorizations w/o an alternating permutation, w/o twins A347706.
A348611 counts anti-run ordered factorizations.
Cf. A038548, A056986, A344614, A344653, A344654, A344740, A347437, A347438, A347463, A348381, A348609.
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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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