All Questions

The Eulerian numbers (OEIS A008292, not to be confused with the Euler numbers) pop up in numerous scenarios in combinatorics and advanced analysis, one as the components of the h-vectors of the ...

Let $\sigma \in S_n$ be a fixed permutation. Consider the polynomial $$ P_{\sigma}(t) = \sum_{\substack{\pi \in S_n \\ \pi \leq \sigma}} t^{\textrm{des}(\pi)} $$ where $\leq$ denotes Bruhat order, and ...

Define the associahedra partition polynomial $$ \begin{split} A(x) &= 1 + A_1(u_1) z + A_2(u_1,u_2) z^2 + A_3(u_1,u_2,u_3) z^3 + \cdots\\ & \qquad\qquad = 1 + \sum_{n \geq 1} A_n(u_1,...,u_n) ...

I'm looking for combinatorial proofs (using, e.g., trees) of the following particular matrix equation $(I)$ and also combinatoric operational analogs of its solution via matrix inversion and/or Cramer'...