• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Proving an identity related to Sterling Numbers of the 2nd Kind

  • Thread starter BraedenP
  • Start date
1. The problem statement, all variables and given/known data

I am to prove, by induction, that [tex]S(n,2)=\sum_{m=1}^{n-1}\cdot S(m,1) + \sum_{m=2}^{n-1}\cdot S(m,2)[/tex]

where the S function is the Sterling function (S(n,k) is the number of k-partitions of an n-set)

2. Relevant equations

[tex]S(n,1) = 1[/tex]
[tex]S(n,2) = 2^{n-1}-1[/tex]

3. The attempt at a solution

Creating a base case is easy, and using the first given equation, the first sum in the problem evaluates to simply n-1. So then I'll need to get the second sum into a form that can help me prove this equality by induction.

Since I'm using induction, at some point I'll need to get the right side into a form that I can equate with the second given equation.

How can I start simplifying this to lead me to an answer?
 

Want to reply to this thread?

"Proving an identity related to Sterling Numbers of the 2nd Kind" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top