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A325588 Number of necklace compositions of n with equal circular differences up to sign. 4
1, 2, 3, 4, 4, 7, 5, 9, 8, 10, 8, 17, 9, 14, 15, 22, 12, 23, 14, 31, 23, 25, 19, 48, 25, 35, 36, 56, 33, 59, 43, 86, 64, 74, 76, 136, 95, 127, 138, 219, 178, 245, 249, 372, 370, 445, 506, 747, 730, 907, 1069, 1431, 1544, 1927, 2268, 2981, 3332, 4074, 4896, 6320 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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).

LINKS

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

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

EXAMPLE

The a(1) = 1 through a(8) = 9 compositions:

  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)

       (11)  (12)   (13)    (14)     (15)      (16)       (17)

             (111)  (22)    (23)     (24)      (25)       (26)

                    (1111)  (11111)  (33)      (34)       (35)

                                     (222)     (1111111)  (44)

                                     (1212)               (1232)

                                     (111111)             (1313)

                                                          (2222)

                                                          (11111111)

MATHEMATICA

neckQ[q_]:=Array[OrderedQ[{q, RotateRight[q, #]}]&, Length[q]-1, 1, And];

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], neckQ[#]&&SameQ@@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])) )}

w(n, s)={sum(k=1, n, my(R=matrix(n, n, i, j, i==j&&abs(i-k)==s), t=0, m=1); while(R, R=step(R, n, s); m++; t+=sumdiv(n, d, R[d, k]*d*eulerphi(n/d))/m ); t/n)}

a(n) = {numdiv(max(1, n)) + sum(s=1, n-1, w(n, s))} \\ Andrew Howroyd, Aug 24 2019

CROSSREFS

Cf. A000079, A000740, A008965, A049988, A175342, A325549, A325556, A325558, A325590.

Sequence in context: A347700 A049988 A079247 * A244903 A342337 A167932

Adjacent sequences:  A325585 A325586 A325587 * A325589 A325590 A325591

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 11 2019

EXTENSIONS

Terms a(26) and beyond from Andrew Howroyd, Aug 24 2019

STATUS

approved

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Last modified October 25 03:56 EDT 2021. Contains 348237 sequences. (Running on oeis4.)