\magnification=1200
\hsize=4in
\nopagenumbers
\noindent
%
%
{\bf Lane Clark}
%
%
\medskip
\noindent
%
%
{\bf An Asymptotic Expansion for the Number of  Permutations with a Certain Number of Inversions}
%
%
\vskip 5mm
\noindent
%
%
%
%
Let $b(n,k)$ denote the number of permutations of $\{1,\ldots,n\}$
with precisely $k$ inversions.
We represent $b(n,k)$ as a real
trigonometric integral and then use the method
of Laplace to give a complete
asymptotic expansion of the integral.
Among the consequences, we have a complete asymptotic expansion
for $b(n,k)/n!$ for a range of $k$ including
the maximum of the $b(n,k)/n!$.



\bye

.