@pieter @bones There is also Flajolet and Sedgewick's “Analytical Combinatorics”, a much bigger work, which in many aspects was much clearer to me. It is completely accessible online (https://ac.cs.princeton.edu/home).

For the real beginner, I would still recommend “Concrete Mathematics” (https://en.wikipedia.org/wiki/Concrete_Mathematics). That's at least where I learned it first.

#combinatorics #GeneratingFunctions

This was my first time attempting to solve a problem using multivariate generating functions. Most of the process was quite miraculous, but there were still a couple of tricky parts. Check it out!
https://thalestriangles.blogspot.com/2023/03/an-average-number-of-1s-and-2s.html
#math #blog #GeneratingFunctions

We know that the Fibonacci (or Virahanka) numbers can be used to count how many ways to express each integer as a sum of 1s and 2s:
1=1 (1 way)
2=1+1=2 (2 ways)
3=1+1+1=1+2=2+1 (3 ways)
4=1+1+1+1=1+1+2=1+2+1=2+1+1=2+2 (5 ways)
5=1+1+1+1+1=1+1+1+2=1+1+2+1=1+2+1+1=1+2+2=2+1+1+1=2+1+2=2+2+1 (8 ways)
and so on. But what if we want to count the number of terms used in all these sums, or (equivalently) the average number of terms, or (again, equivalently) the average ratio of 1s to 2s? That’s the problem addressed in this post.
https://thalestriangles.blogspot.com/2023/03/an-average-number-of-1s-and-2s.html
#math #Fibonacci #GeneratingFunctions

I wonder whether Pólya substitution (https://en.wikipedia.org/wiki/P%C3%B3lya_enumeration_theorem#Full,_weighted_version) has an application in ordinary analysis.

The Pólya substitution of a function \(f(x,y,z)\) into a function \(Z(t_1,t_2,\dots,t_n)\) results in the function\[Z[f](x,y,z)=Z(f(x,y,z),f(x^2,y^2,z^2),\dots,f(x^n,y^n,z^n)).\]This operation arises in the context of enumerative combinatorics, but I would like to know whether there is something similar in ordinary multi-variable analysis.

#combinatorics #analysis #generatingfunctions