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A349050
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Number of multisets of size n that have no alternating permutations and cover an initial interval of positive integers.
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19
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0, 0, 1, 1, 3, 4, 8, 12, 20, 32, 48, 80, 112, 192, 256, 448, 576, 1024, 1280, 2304, 2816, 5120, 6144, 11264, 13312, 24576, 28672, 53248, 61440, 114688, 131072, 245760, 278528, 524288, 589824, 1114112, 1245184, 2359296, 2621440, 4980736, 5505024
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OFFSET
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0,5
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COMMENTS
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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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a(n) = (n+2)*2^(n/2-3) for even n > 0; a(n) = (n-1)*2^((n-5)/2) for odd n. - Andrew Howroyd, Jan 13 2024
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EXAMPLE
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The multiset {1,2,2,2,2,3,3} has no alternating permutations, even though it does have the three anti-run permutations (2,1,2,3,2,3,2), (2,3,2,1,2,3,2), (2,3,2,3,2,1,2), so is counted under a(7).
The a(2) = 1 through a(7) = 12 multisets:
{11} {111} {1111} {11111} {111111} {1111111}
{1112} {11112} {111112} {1111112}
{1222} {12222} {111122} {1111122}
{12223} {111123} {1111123}
{112222} {1122222}
{122222} {1122223}
{122223} {1222222}
{123333} {1222223}
{1222233}
{1222234}
{1233333}
{1233334}
As compositions:
(2) (3) (4) (5) (6) (7)
(1,3) (1,4) (1,5) (1,6)
(3,1) (4,1) (2,4) (2,5)
(1,3,1) (4,2) (5,2)
(5,1) (6,1)
(1,1,4) (1,1,5)
(1,4,1) (1,4,2)
(4,1,1) (1,5,1)
(2,4,1)
(5,1,1)
(1,1,4,1)
(1,4,1,1)
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MATHEMATICA
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allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[allnorm[n], Select[Permutations[#], wigQ]=={}&]], {n, 0, 7}]
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PROG
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(PARI) a(n) = if(n==0, 0, if(n%2==0, (n+2)*2^(n/2-3), (n-1)*2^((n-1)/2-2))) \\ Andrew Howroyd, Jan 13 2024
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CROSSREFS
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The case of weakly decreasing multiplicities is A025065.
A separable instead of alternating version is A336103.
The version for partitions is A345165.
The complement (still covering an initial interval) is counted by A349055.
A000670 counts sequences covering an initial interval, anti-run A005649.
A049774 counts permutations avoiding the consecutive pattern (1,2,3).
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A344654 counts partitions w/o an alternating permutation, ranked by A344653.
Cf. A019472, A052841, A096441, A106351, A106356, A335125, A335126, A344614, A347050, A347706, A348377, A349054.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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