Mar 3, 2025
Indistinguishability plays an important role in enumerative problems in combinatorics. This article explains the concept and significance of particle indistinguishability in statistical mechanics.
May 9, 2023
Suppose there are n distinguishable boxes and k indistinguishable balls. Now, we randomly put the balls into the boxes. For each of the boxes, what is the probability that it contains m balls? This is a simple combanitorics problem that can be solved by the stars and bars method. It turns out that in the limit n,k→∞ with k/n fixed, the distribution tends to be a geometric distribution.
Nov 9, 2022
The function (1−z)n+1∑k=1∞knzk is a polynomial of degree n w.r.t. z, and the sum of its coefficients is n!. This turns out to be properties of Eulerian numbers.
Nov 6, 2022
By using power series, we can prove that the problem of solving linear homogeneous ODE with constant coefficients can be reduced to the problem of solving a polynomial with those coefficients. This article illustrates this point in detail, but it uses a very awful notation…