A329137 - OEIS

%I #6 Nov 09 2019 16:26:27

%S 1,1,2,2,4,6,8,14,20,25,39,54,69,99,130,167,224,292,373,483,620,773,

%T 993,1246,1554,1946,2421,2987,3700,4548,5575,6821,8330,10101,12287,

%U 14852,17935,21599,25986,31132,37295,44539,53112,63212,75123,89055,105503,124682

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

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

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

%e The a(1) = 1 through a(7) = 14 partitions:

%e   (1)  (2)    (3)    (4)      (5)        (6)          (7)

%e        (1,1)  (2,1)  (2,2)    (3,2)      (3,3)        (4,3)

%e                      (3,1)    (4,1)      (4,2)        (5,2)

%e                      (2,1,1)  (2,2,1)    (5,1)        (6,1)

%e                               (3,1,1)    (4,1,1)      (3,2,2)

%e                               (2,1,1,1)  (2,2,1,1)    (3,3,1)

%e                                          (3,1,1,1)    (4,2,1)

%e                                          (2,1,1,1,1)  (5,1,1)

%e                                                       (2,2,2,1)

%e                                                       (3,2,1,1)

%e                                                       (4,1,1,1)

%e                                                       (2,2,1,1,1)

%e                                                       (3,1,1,1,1)

%e                                                       (2,1,1,1,1,1)

%e With differences:

%e   ()  ()   ()   ()     ()       ()         ()

%e       (0)  (1)  (0)    (1)      (0)        (1)

%e                 (2)    (3)      (2)        (3)

%e                 (1,0)  (0,1)    (4)        (5)

%e                        (2,0)    (3,0)      (0,2)

%e                        (1,0,0)  (0,1,0)    (1,0)

%e                                 (2,0,0)    (2,1)

%e                                 (1,0,0,0)  (4,0)

%e                                            (0,0,1)

%e                                            (1,1,0)

%e                                            (3,0,0)

%e                                            (0,1,0,0)

%e                                            (2,0,0,0)

%e                                            (1,0,0,0,0)

%t aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];

%t Table[Length[Select[IntegerPartitions[n],aperQ[Differences[#]]&]],{n,0,30}]

%Y The Heinz numbers of these partitions are given by A329135.

%Y The periodic version is A329144.

%Y The augmented version is A329136.

%Y Aperiodic binary words are A027375.

%Y Aperiodic compositions are A000740.

%Y Numbers whose binary expansion is aperiodic are A328594.

%Y Numbers whose prime signature is aperiodic are A329139.

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Nov 09 2019