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A348610
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Number of alternating ordered factorizations of n.
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25
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1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 12, 1, 3, 3, 6, 1, 11, 1, 7, 3, 3, 3, 15, 1, 3, 3, 12, 1, 11, 1, 6, 6, 3, 1, 23, 1, 6, 3, 6, 1, 12, 3, 12, 3, 3, 1, 28, 1, 3, 6, 12, 3, 11, 1, 6, 3, 11, 1, 33, 1, 3, 6, 6, 3, 11, 1, 23, 4, 3
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OFFSET
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1,6
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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. 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).
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LINKS
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Table of n, a(n) for n=1..82.
Wikipedia, Alternating permutation
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EXAMPLE
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The alternating ordered factorizations of n = 1, 6, 12, 16, 24, 30, 32, 36:
() 6 12 16 24 30 32 36
2*3 2*6 2*8 3*8 5*6 4*8 4*9
3*2 3*4 8*2 4*6 6*5 8*4 9*4
4*3 2*4*2 6*4 10*3 16*2 12*3
6*2 8*3 15*2 2*16 18*2
2*3*2 12*2 2*15 2*8*2 2*18
2*12 3*10 4*2*4 3*12
2*4*3 2*5*3 2*6*3
2*6*2 3*2*5 2*9*2
3*2*4 3*5*2 3*2*6
3*4*2 5*2*3 3*4*3
4*2*3 3*6*2
6*2*3
2*3*2*3
3*2*3*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 additive version (compositions) is A025047 ranked by A345167.
The complementary additive version is A345192, ranked by A345168.
Dominated by A348611 (the anti-run version) at positions A122181.
The complement is counted by A348613.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations, complement A348615.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A347463 counts ordered factorizations with integer alternating product.
A348379 counts factorizations w/ an alternating permutation.
A348380 counts factorizations w/o an alternating permutation.
Cf. A038548, A049774, A056986, A102726, A128761, A344614, A347050, A347464, A347706, A348383.
Sequence in context: A294787 A294788 A254578 * A296120 A050345 A348611
Adjacent sequences: A348607 A348608 A348609 * A348611 A348612 A348613
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KEYWORD
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nonn
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AUTHOR
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Gus Wiseman, Nov 05 2021
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STATUS
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approved
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