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).

0, 2, 16, 112, 832, 6896, 62368, 619904, 6733312, 79268096, 1010644736, 13833177088, 203128772608, 3175336112128, 52723300200448, 927263962759168, 17221421451378688, 336720980854571008, 6911300635636400128, 148661140496700932096

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

Alois P. Heinz, Table of n, a(n) for n = 1..450

L. Kollár, Optimal sorting of seven element sets, Proceedings of the 12th symposium on Mathematical foundations of computer science 1986, 449-457.

Index entries for sequences related to sorting

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

Cf. A003070, A036604, A117628.

Sequence in context: A370245 A370270 A058121 * A117628 A288971 A288970

Adjacent sequences: A117624 A117625 A117626 * A117628 A117629 A117630

Keyword

nonn

Author

N. J. A. Sloane, Oct 06 2006

Status

approved