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.