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A117627 Let f(n) = minimum of average number of comparisons needed for any sorting method for n elements and let g(n) = n!*f(n). Sequence gives a lower bound on g(n). 5
0, 2, 16, 112, 832, 6896, 62368, 619904, 6733312, 79268096, 1010644736, 13833177088, 203128772608, 3175336112128, 52723300200448, 927263962759168, 17221421451378688, 336720980854571008, 6911300635636400128, 148661140496700932096 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Sorting methods have been constructed such that the lower bound of f(n) is achieved for n=1, 2, 3, 4, 5, 6, 9 and 10. Y. Césari was the first to show that f(7) is not obtainable. He also constructed optimal solutions for n=9 and 10. L. Kollár showed that the minimum number of comparisons needed for n=7 is 62416. - Dmitry Kamenetsky, Jun 11 2015
REFERENCES
Y. Césari, Questionnaire codage et tris, PhD Thesis, University of Paris, 1968.
D. E. Knuth, TAOCP, Vol. 3, Section 5.3.1.
LINKS
L. Kollár, Optimal sorting of seven element sets, Proceedings of the 12th symposium on Mathematical foundations of computer science 1986, 449-457.
FORMULA
Knuth gives an explicit formula.
a(n) = (q(n)+1)*n! - 2^q(n) with q(n) = A003070(n).
MAPLE
q:= n-> ceil(log[2](n!)):
a:= n-> (q(n)+1)*n! - 2^q(n):
seq(a(n), n=1..30); # Alois P. Heinz, Jun 11 2015
MATHEMATICA
q[n_] := Log[2, n!] // Ceiling; a[n_] := (q[n]+1)*n! - 2^q[n]; Array[a, 20] (* Jean-François Alcover, Feb 13 2016 *)
PROG
(PARI) a(n) = { my(N=n!, q = ceil(log(N)/log(2))); return ((q+1)*N - 2^q); } \\ Michel Marcus, Apr 21 2013
CROSSREFS
Sequence in context: A370245 A370270 A058121 * A117628 A288971 A288970
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 06 2006
STATUS
approved

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Last modified May 30 14:25 EDT 2024. Contains 372968 sequences. (Running on oeis4.)