Landau's function

Landau's function "g"("n") is defined for every natural number "n" to be the largest order of an element of the symmetric group "S"_"n". Equivalently, "g"("n") is the largest least common multiple of any partition of "n".

For instance, 5 = 2 + 3 and lcm(2,3) = 6. No other partition of 5 yields a bigger lcm, so "g"(5) = 6. An element of order 6 in the group "S"₅ can be written in cycle notation as (1 2) (3 4 5).

The integer sequence "g"(0) = 1, "g"(1) = 1, "g"(2) = 2, "g"(3) = 3, "g"(4) = 4, "g"(5) = 6, "g"(6) = 6, "g"(7) = 12, "g"(8) = 15, ... is [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000793 A000793] .

The sequence is named after Edmund Landau, who proved in 1902 (reference [1] below) that:$lim\_\{n\; oinfty\}frac\{ln(g(n))\}\{sqrt\{n\; ln(n)\; =\; 1$(where ln denotes the natural logarithm).

The statement that :

References

# E. Landau, "Über die Maximalordnung der Permutationen gegebenen Grades [On the maximal order of permutations of given degree] ", Arch. Math. Phys. Ser. 3, vol. 5, 1903, pp. 92-103.
# W. Miller, "The maximum order of an element of a finite symmetric group" , American Mathematical Monthly, vol. 94, 1987, pp. 497-506.
# J.-L. Nicolas, "On Landau's function g(n)", in "The Mathematics of Paul Erdős", vol. 1, Springer Verlag, 1997, pp. 228-240.

External links

On-Line Encyclopedia of Integer Sequences: Sequence [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000793 A000793,] Landau's function on the natural numbers.