A325590 Number of necklace compositions of n with circular differences all equal to 1 or -1.

0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 3, 3, 4, 4, 5, 7, 6, 7, 10, 10, 11, 15, 16, 18, 23, 25, 32, 38, 43, 53, 64, 73, 89, 108, 131, 153, 188, 223, 272, 329, 395, 475, 583, 697, 848, 1027, 1247, 1506, 1837, 2223, 2708, 3282, 3993, 4848, 5913, 7175, 8745, 10640

Offset

1,9

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.

The circular differences of a sequence c of length k are c_{i + 1} - c_i for i < k and c_1 - c_i for i = k. For example, the circular differences of (1,2,1,3) are (1,-1,2,-2).

Up to rotation, a(n) is the number of ways to arrange positive integers summing to n in a circle such that adjacent parts differ by 1 or -1.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Example

The first 16 terms count the following compositions:
   3: (12)
   5: (23)
   6: (1212)
   7: (34)
   8: (1232)
   9: (45)
   9: (121212)
  10: (2323)
  11: (56)
  11: (121232)
  12: (2343)
  12: (12121212)
  13: (67)
  13: (123232)
  14: (3434)
  14: (12121232)
  15: (78)
  15: (123432)
  15: (232323)
  15: (1212121212)
  16: (3454)
  16: (12321232)
  16: (12123232)
The a(21) = 7 necklace compositions:
  (10,11)
  (2,3,4,5,4,3)
  (3,4,3,4,3,4)
  (1,2,1,2,1,2,3,4,3,2)
  (1,2,3,2,1,2,3,2,3,2)
  (1,2,1,2,3,2,3,2,3,2)
  (1,2,1,2,1,2,1,2,1,2,1,2,1,2)

Mathematica

neckQ[q_]:=Array[OrderedQ[{q, RotateRight[q, #]}]&, Length[q]-1, 1, And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], neckQ[#]&&(SameQ[1, ##]&@@Abs[Differences[Append[#, First[#]]]])&]], {n, 15}]

Prog

(PARI)
step(R, n, s)={matrix(n, n, i, j, if(i>j, if(j>s, R[i-j, j-s]) + if(j+s<=n, R[i-j, j+s])) )}
a(n)={sum(k=1, n, my(R=matrix(n, n, i, j, i==j&&abs(i-k)==1), t=0, m=1); while(R, R=step(R, n, 1); m++; t+=sumdiv(n, d, R[d, k]*d*eulerphi(n/d))/m ); t/n)} \\ Andrew Howroyd, Aug 23 2019

Crossrefs

Cf. A000079, A000740, A008965, A034297, A173258, A325556, A325588, A325589, A325591.

Sequence in context: A140426 A146879 A231577 * A277210 A304777 A058762

Adjacent sequences: A325587 A325588 A325589 * A325591 A325592 A325593

Keyword

nonn

Author

Gus Wiseman, May 12 2019

Extensions

a(26)-a(40) from Lars Blomberg, Jun 11 2019

Terms a(41) and beyond from Andrew Howroyd, Aug 23 2019

Status

approved