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A329136
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Number of integer partitions of n whose augmented differences are an aperiodic word.
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7
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1, 1, 1, 2, 4, 5, 10, 14, 19, 28, 40, 53, 75, 99, 131, 172, 226, 294, 380, 488, 617, 787, 996, 1250, 1565, 1953, 2425, 3003, 3705, 4559, 5589, 6836, 8329, 10132, 12292, 14871, 17950, 21629, 25988, 31169, 37306, 44569, 53139, 63247, 75133, 89111, 105515, 124737
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OFFSET
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0,4
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COMMENTS
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The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
A sequence is aperiodic if its cyclic rotations are all different.
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LINKS
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FORMULA
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EXAMPLE
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The a(1) = 1 through a(7) = 14 partitions:
(1) (2) (3) (4) (5) (6) (7)
(2,1) (2,2) (4,1) (3,3) (4,3)
(3,1) (2,2,1) (4,2) (5,2)
(2,1,1) (3,1,1) (5,1) (6,1)
(2,1,1,1) (2,2,2) (3,2,2)
(3,2,1) (3,3,1)
(4,1,1) (4,2,1)
(2,2,1,1) (5,1,1)
(3,1,1,1) (2,2,2,1)
(2,1,1,1,1) (3,2,1,1)
(4,1,1,1)
(2,2,1,1,1)
(3,1,1,1,1)
(2,1,1,1,1,1)
With augmented differences:
(1) (2) (3) (4) (5) (6) (7)
(2,1) (1,2) (4,1) (1,3) (2,3)
(3,1) (1,2,1) (3,2) (4,2)
(2,1,1) (3,1,1) (5,1) (6,1)
(2,1,1,1) (1,1,2) (1,3,1)
(2,2,1) (2,1,2)
(4,1,1) (3,2,1)
(1,2,1,1) (5,1,1)
(3,1,1,1) (1,1,2,1)
(2,1,1,1,1) (2,2,1,1)
(4,1,1,1)
(1,2,1,1,1)
(3,1,1,1,1)
(2,1,1,1,1,1)
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MATHEMATICA
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aperQ[q_]:=Array[RotateRight[q, #1]&, Length[q], 1, UnsameQ];
aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], aperQ[aug[#]]&]], {n, 0, 30}]
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CROSSREFS
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The Heinz numbers of these partitions are given by A329133.
The non-augmented version is A329137.
Aperiodic binary words are A027375.
Aperiodic compositions are A000740.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose differences of prime indices are aperiodic are A329135.
Numbers whose prime signature is aperiodic are A329139.
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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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