All Questions
Tagged with combinatorial-identities linear-algebra
5 questions
11
votes
1
answer
579
views
Catalan determinants in search of a proof: Part II
This problem involves the Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$.
I can prove the below equality by computing each of the two sides, directly. That means, there is an algebraic proof.
...
8
votes
1
answer
811
views
(0,1)-matrix congruence: is it known?
[[UPDATE: This work has now been published at SIAM J Discrete Math.: Formulae for the Alon–Tarsi Conjecture.]]
By equating two formulae (one congruence by Glynn (1) (which has just appeared) and one ...
6
votes
1
answer
535
views
A surprising identity: $\det[\cos\pi\frac{jk}n]_{1\le j,k\le n}=(-1)^{\lfloor\frac{n+1}2\rfloor}(n/2)^{(n-1)/2}$
On the basis of my computation, here I pose my following conjecture involving the cosine function.
Conjecture. For any positive integer $n$, we have the identity
$$\frac1{2n}\det\left[\cos\pi\frac{jk}...
3
votes
0
answers
172
views
Equality of determinants: a direct justification request
Let $I_k$ denote the enumeration of involutions among permutations in $\mathfrak{S}_k$. Here is yet another cute finding for which I ask a:
Question. Is there a direct proof (or interpretation or ...
2
votes
2
answers
258
views
Equal-valued determinants in search of a proof: Part III
Encouraged by David's proof for my earlier MO question, let's consider a similar problem.
I can prove the below equality by computing each of the two sides, directly. That means, there is an ...