Archive of posts with tag “combinatorics”

• May 9, 2023

  The distribution when indistinguishable balls are put into boxes

  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 $n,k\to\infty$ with $k/n$ fixed, the distribution tends to be a geometric distribution.

  • Categories: math
  • Tags: probability, combinatorics, ruby
• Nov 9, 2022

  A polynomial whose sum of coefficients is a factorial

  The function $\left(1-z\right)^{n+1}\sum_{k=1}^\infty k^nz^k$ 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.

  • Categories: math
  • Tags: combinatorics, number sequence, from zhihu
• Nov 6, 2022

  Solving linear homogeneous ODE with constant coefficients

  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…

  • Categories: math
  • Tags: calculus, linear algebra, combinatorics, ode, long paper, from zhihu