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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) 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-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 *) 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 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.)