

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
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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



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):


MATHEMATICA

q[n_] := Log[2, n!] // Ceiling; a[n_] := (q[n]+1)*n!  2^q[n]; Array[a, 20] (* JeanFrançois 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



KEYWORD

nonn


AUTHOR



STATUS

approved



