A325681 Number of necklace compositions of n such that every restriction to a circular subinterval has a different sum.

1, 1, 2, 2, 3, 3, 6, 6, 11, 9, 16, 16, 27, 23, 46, 42, 73, 63, 112, 102

Offset

1,3

Comments

A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.

A circular subinterval is a sequence of consecutive indices where the first and last indices are also considered consecutive.

Links

Table of n, a(n) for n=1..20.

Example

The a(1) = 1 through a(10) = 9 necklace compositions (A = 10):
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)
            (12)  (13)  (14)  (15)  (16)   (17)   (18)   (19)
                        (23)  (24)  (25)   (26)   (27)   (28)
                                    (34)   (35)   (36)   (37)
                                    (124)  (125)  (45)   (46)
                                    (142)  (152)  (126)  (127)
                                                  (135)  (136)
                                                  (153)  (163)
                                                  (162)  (172)
                                                  (234)
                                                  (243)

Mathematica

neckQ[q_]:=Array[OrderedQ[{q, RotateRight[q, #]}]&, Length[q]-1, 1, And];
suball[q_]:=Join[Take[q, #]&/@Select[Tuples[Range[Length[q]], 2], OrderedQ], Drop[q, #]&/@Select[Tuples[Range[2, Length[q]-1], 2], OrderedQ]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], neckQ[#]&&UnsameQ@@Total/@suball[#]&]], {n, 15}]

Crossrefs

Cf. A000079, A000740, A008965, A108917, A143823, A169942, A276024.

Cf. A325676, A325677, A325678, A325679, A325682, A325683, A325687.

Sequence in context: A292225 A238786 A238547 * A116450 A054172 A236971

Adjacent sequences: A325678 A325679 A325680 * A325682 A325683 A325684

Keyword

nonn,more

Author

Gus Wiseman, May 13 2019

Status

approved