OFFSET
0,4
COMMENTS
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.
EXAMPLE
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)
MATHEMATICA
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}]
CROSSREFS
The Heinz numbers of these partitions are given by A329133.
The periodic version is A329143.
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 09 2019
STATUS
approved