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A349800
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Number of integer compositions of n that are weakly alternating and have at least two adjacent equal parts.
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16
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0, 0, 1, 1, 4, 9, 16, 33, 62, 113, 205, 373, 664, 1190, 2113, 3744, 6618, 11683, 20564, 36164, 63489, 111343, 195042, 341357, 596892, 1042976, 1821179, 3178145, 5543173, 9663545, 16839321, 29332231, 51075576, 88908912, 154722756, 269186074, 468221264
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OFFSET
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0,5
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COMMENTS
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We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.
This sequence counts compositions that are weakly but not strongly alternating; also weakly alternating non-anti-run compositions.
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LINKS
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FORMULA
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EXAMPLE
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The a(2) = 1 through a(6) = 16 compositions:
(1,1) (1,1,1) (2,2) (1,1,3) (3,3)
(1,1,2) (1,2,2) (1,1,4)
(2,1,1) (2,2,1) (2,2,2)
(1,1,1,1) (3,1,1) (4,1,1)
(1,1,1,2) (1,1,1,3)
(1,1,2,1) (1,1,2,2)
(1,2,1,1) (1,1,3,1)
(2,1,1,1) (1,3,1,1)
(1,1,1,1,1) (2,2,1,1)
(3,1,1,1)
(1,1,1,1,2)
(1,1,1,2,1)
(1,1,2,1,1)
(1,2,1,1,1)
(2,1,1,1,1)
(1,1,1,1,1,1)
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MATHEMATICA
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wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]==Length[y] &&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
whkQ[y_]:=And@@Table[If[EvenQ[m], y[[m]]<=y[[m+1]], y[[m]]>=y[[m+1]]], {m, 1, Length[y]-1}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], (whkQ[#]||whkQ[-#])&&!wigQ[#]&]], {n, 0, 10}]
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CROSSREFS
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The version counting permutations of prime indices is A349798.
These compositions are ranked by A349799.
A345165 = partitions without an alternating permutation, ranked by A345171.
A345170 = partitions with an alternating permutation, ranked by A345172.
A348377 = non-alternating non-twin compositions.
Weakly alternating:
- A349057 = complement of standard composition numbers (too dense).
Cf. A008965, A011782, A027383, A096441, A274230, A333213, A344614, A344615, A348382, A348613, A349796, A350140.
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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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