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Results tagged with enumerative-combinatorics
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user 48831
5
votes
Accepted
Result attribution for eigenvalues of a matrix of Pascal-type
I don't know a reference. One way to show the eigenvalues starts from the observation (which can be proved using
generating functions)
that $\sum_{i=0}^n {i\choose k} A_{i,j}={2n+1 \choose n-k} {j+k …
- 3,255
8
votes
Accepted
The number of ways to merge a permutation with itself
By @Max Alexeyev's solution above $N_{2k-1}^{\sigma}=tr(M_{k}(P_{\sigma}M_{k}P_{\sigma}^{-1}))$.
The eigenvalues and eigenvectors of $M_k$ are given here: Result attribution for eigenvalues of a matri …
- 3,255
2
votes
Asking for a proof for a sum of products of binomials: an "interesting" identity?
A generating function proof.
As $\arcsin(z)=\sum_{k\geq 0} \frac{1}{2k+1} {2k \choose k} \frac{z^{2k+1}}{4^k}$ and $\frac{1}{\sqrt{1-z^2}}=\sum_{k\geq 0}{2k \choose k}\frac{z^{2k}}{4^k}$
we have that
…
- 3,255