All Questions
Tagged with stirling-numbers enumerative-combinatorics
4 questions
4
votes
1
answer
407
views
Inverse relationship between Stirling numbers of the first and second kind via generating functions
In combinatorics, a well-known result is that the matrix formed by the Stirling numbers of the second kind $\left(S(n,k)\right)_{n,k\geq 0}$ and the matrix of the signed Stirling numbers of the first ...
- 151
4
votes
1
answer
379
views
Counting permutations with a fixed number of descents and an extra condition
I am computing the volumes of certain polytopes and it turns out that knowing a "closed formula" for the following number would help a lot.
Determine the number of permutations $\sigma\in \...
- 1,889
3
votes
2
answers
463
views
Ask for a reference or a proof of a combinatorial identity $\sum_{k=0}^n\binom{2n+1}{2k}\binom {k}{m} =2^{2(n-m)}\frac{2n+1}{2(n-m)+1}\binom{2n-m}{m}$
Could you please recommend a reference to or supply a proof of the following identity \eqref{combin-ID-Maclaurin}, or \eqref{first-equiv-form}, or \eqref{combin-ID-Mac-Equiv}, or \eqref{combin-ID-Mac-...
- 1,101
3
votes
2
answers
403
views
Closed form for product of Stirling numbers of the second kind
What does the following expression evaluate to:
\begin{equation}
\sum\limits_{k=1}^n \dbinom{n}{k} \cdot k! \begin{Bmatrix} n \\ k \end{Bmatrix} \cdot k! \begin{Bmatrix} n \\ k \end{Bmatrix}
\end{...
- 37