A032153 Number of ways to partition n elements into pie slices of different sizes.

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

Offset: 0

Author

Christian G. Bower

Keywords

• nonn
• nice

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

Examples

			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)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 2001 terms from Robert Israel)
• C. G. Bower, Transforms (2)
• Index entries for sequences related to Lyndon words

Crossrefs

Cf. A000079, A008289, A008965, A032020, A275972, A325676, A325788.

Programs

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
# second Maple program:
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 `if`(n=0, 1, b(n$2, -1)):
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 12 2020

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-François 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 *)

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

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

a(n) = Sum_{k=1..floor((sqrt(8*n+1)-1)/2)} (k-1)! * A008289(n,k) for n > 0. - Alois P. Heinz, Aug 07 2020

Extensions

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