Where are "Special Numbers" mentioned in Concrete Maths used?

Question

I was glancing through the contents of Concrete Maths online. I had at least heard most of the functions and tricks mentioned but there is a whole section on Special Numbers. These numbers include Stirling Numbers, Eulerian Numbers, Harmonic Numbers so on. Now I have never encountered any of these weird numbers. How do they aid in computational problems? Where are they generally used?

Answer 1

Harmonic Numbers appear almost everywhere! Musical Harmonies, analysis of Quicksort... Stirling Numbers (first and second kind) arise in a variety of combinatorics and partitioning problems. Eulerian Numbers also occur several places, most notably in permutations and coefficients of polylogarithm functions.

Answer 2

A lot of the numbers you mentioned are used in the analysis of algorithms. You may not have these numbers in your code, but you'll need them if you want to estimate how long it will take for your code to run. You might see them in your code too. Some of these numbers are related to combinatorics, counting how many ways something can happen.

Sometimes it's not enough to know how many possibilities there are because you need to enumerate over the possibilities. Volume 4 of Knuth's TAOCP, in progress, gives the algorithms you need.

Here's an example of using Fibonacci numbers as part of a numerical integration problem.

Harmonic numbers are a discrete analog of logarithms and so they come up in difference equations just like logs come up in differential equations. Here's an example of physical applications of harmonic means, related to harmonic numbers. See the book Gamma for many examples of harmonic numbers in action, especially the chapter "It's a harmonic world."