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A032153 Number of ways to partition n elements into pie slices of different sizes. 10
1, 1, 1, 2, 2, 3, 5, 6, 8, 11, 19, 22, 32, 41, 57, 92, 114, 155, 209, 280, 364, 587, 707, 984, 1280, 1737, 2213, 2990, 4390, 5491, 7361, 9650, 12708, 16451, 21567, 27506, 40100, 49201, 65701, 84128, 111278, 140595, 184661, 232356, 300680 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Number of strict necklace compositions of n. A strict necklace composition of n is a finite sequence of distinct positive integers summing to n that is lexicographically minimal among all of its cyclic rotations. In other words, it is a strict composition of n starting with its least part. - Gus Wiseman, May 31 2019

LINKS

Robert Israel, Table of n, a(n) for n = 0..2000

C. G. Bower, Transforms (2)

Index entries for sequences related to Lyndon words

FORMULA

"CGK" (necklace, element, unlabeled) transform of 1, 1, 1, 1, ...

G.f.: Sum_{k >= 1} (k-1)! * x^((k^2+k)/2) / (Product_{j=1..k} 1-x^j). - Vladeta Jovovic, Sep 21 2004

EXAMPLE

From Gus Wiseman, May 31 2019: (Start)

Inequivalent representatives of the a(1) = 1 through a(9) = 11 ways to slice a pie:

  (1)  (2)  (3)   (4)   (5)   (6)    (7)    (8)    (9)

            (12)  (13)  (14)  (15)   (16)   (17)   (18)

                        (23)  (24)   (25)   (26)   (27)

                              (123)  (34)   (35)   (36)

                              (132)  (124)  (125)  (45)

                                     (142)  (134)  (126)

                                            (143)  (135)

                                            (152)  (153)

                                                   (162)

                                                   (234)

                                                   (243)

(End)

MAPLE

N:= 100: # to get a(0)..a(N)

K:= floor(isqrt(1+8*N)/2):

S:= series(1+add((k-1)!*x^((k^2+k)/2)/mul(1-x^j, j=1..k), k=1..K), x, N+1):

seq(coeff(S, x, j), j=0..N); # Robert Israel, Jul 15 2016

MATHEMATICA

max=50; s=Sum[(x^(k(k+1)/2-1)*(k-1)!)/QPochhammer[x, x, k], {k, 1, max}] + O[x]^max; CoefficientList[s, x] (* Jean-Fran├žois Alcover, Jan 19 2016 *)

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

Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n], UnsameQ@@#&], neckQ]], {n, 30}] (* Gus Wiseman, May 31 2019 *)

PROG

(PARI)

N=66;  q='q+O('q^N);

gf=sum(n=1, N, (n-1)!*q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) );

Vec(gf)

/* Joerg Arndt, Oct 20 2012 */

(PARI) seq(n)=[subst(serlaplace(p/y), y, 1) | p <- Vec(y-1+prod(k=1, n, 1 + x^k*y + O(x*x^n)))] \\ Andrew Howroyd, Sep 13 2018

CROSSREFS

Cf. A000079, A008965, A032020, A275972, A325676, A325788.

Sequence in context: A084783 A265853 A129838 * A309223 A116465 A117356

Adjacent sequences:  A032150 A032151 A032152 * A032154 A032155 A032156

KEYWORD

nonn,nice

AUTHOR

Christian G. Bower

EXTENSIONS

a(0)=1 prepended by Andrew Howroyd, Sep 13 2018

STATUS

approved

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Last modified October 23 20:09 EDT 2019. Contains 328373 sequences. (Running on oeis4.)