A329137 Number of integer partitions of n whose differences are an aperiodic word.

1, 1, 2, 2, 4, 6, 8, 14, 20, 25, 39, 54, 69, 99, 130, 167, 224, 292, 373, 483, 620, 773, 993, 1246, 1554, 1946, 2421, 2987, 3700, 4548, 5575, 6821, 8330, 10101, 12287, 14852, 17935, 21599, 25986, 31132, 37295, 44539, 53112, 63212, 75123, 89055, 105503, 124682

Offset

0,3

Comments

A sequence is aperiodic if its cyclic rotations are all different.

Links

Table of n, a(n) for n=0..47.

Formula

a(n) + A329144(n) = A000041(n).

Example

The a(1) = 1 through a(7) = 14 partitions:
  (1)  (2)    (3)    (4)      (5)        (6)          (7)
       (1,1)  (2,1)  (2,2)    (3,2)      (3,3)        (4,3)
                     (3,1)    (4,1)      (4,2)        (5,2)
                     (2,1,1)  (2,2,1)    (5,1)        (6,1)
                              (3,1,1)    (4,1,1)      (3,2,2)
                              (2,1,1,1)  (2,2,1,1)    (3,3,1)
                                         (3,1,1,1)    (4,2,1)
                                         (2,1,1,1,1)  (5,1,1)
                                                      (2,2,2,1)
                                                      (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 differences:
  ()  ()   ()   ()     ()       ()         ()
      (0)  (1)  (0)    (1)      (0)        (1)
                (2)    (3)      (2)        (3)
                (1,0)  (0,1)    (4)        (5)
                       (2,0)    (3,0)      (0,2)
                       (1,0,0)  (0,1,0)    (1,0)
                                (2,0,0)    (2,1)
                                (1,0,0,0)  (4,0)
                                           (0,0,1)
                                           (1,1,0)
                                           (3,0,0)
                                           (0,1,0,0)
                                           (2,0,0,0)
                                           (1,0,0,0,0)

Mathematica

aperQ[q_]:=Array[RotateRight[q, #1]&, Length[q], 1, UnsameQ];
Table[Length[Select[IntegerPartitions[n], aperQ[Differences[#]]&]], {n, 0, 30}]

Crossrefs

The Heinz numbers of these partitions are given by A329135.

The periodic version is A329144.

The augmented version is A329136.

Aperiodic binary words are A027375.

Aperiodic compositions are A000740.

Numbers whose binary expansion is aperiodic are A328594.

Numbers whose prime signature is aperiodic are A329139.

Cf. A152061, A325356, A329132, A329133, A329134, A329140.

Sequence in context: A291055 A266949 A255710 * A239851 A153958 A216214

Adjacent sequences: A329134 A329135 A329136 * A329138 A329139 A329140

Keyword

nonn

Author

Gus Wiseman, Nov 09 2019

Status

approved