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

Author

Gus Wiseman, Nov 09 2019

Keywords

• nonn

Comments

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

Examples

			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)
		

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.

Programs

Mathematica

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

Formula

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