So let's look at some applications of Saddle-Point Asymptotics to combinatorial
problems. so, one problem that we saw very early on
is that we can write down generating function equations for the number of
permutations that have no long cycles. So in simplest cases, involutions where
all the cycles of link one or two. so immediately from the symbolic method
we get the generating function for involutions.
Is i of z equals e to the z plus e squared over two.
that's a function that's got no singularities and it's immediately
amenable to the saddle point method. so, so we have, e2, a function.
All we need to do is take the derivatives, of that function and set the
first one to zero to find the saddle point.
and then so, that's the function, We take, z plus z squared over 2, then
minus n plus 1 log z, because of the, z to the n plus 1, we put in for Crouch's
formula. saddle-point is where the first
derivative is equal to 0 and so that's turns out to be a quadratic equation in
this example and it's about square root of n minus one half.
Again, we want to work with approximation to the saddle-point to simplify
calculations. and so then, the saddle-point
approximation says that we plug that square root of n to the n minus one half
into the original equation and that immediately gives the saddle-point in
asymptotics. E to the square root of n, square root of
m plus n over 2 minus one fourth. over 2 and the end of the two squared of
pie n. That's a kind of a complicated equation
but that's the asymptotics for this problem.
in, actually, we're interested in factorial time set coefficients, so just
plugging in a Stirling's approximation. gives, that asymptotic expression for
involutions. Fairly straightforward calculation using
saddle point asymptotics. now again, you have to check
susceptibility or give up on the square root to 2 pi N factor.