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It appears that the zeta map which I used to answer Vince Vatter's initial question and which I describe in my answer there, see also page 50 of Jim Haglund's book, indeed solves also this problem: A …

Below my modified answer containing a complete bijection between the above sequences and Dyck paths: Let $a = (a_1,\ldots,a_n)$ be a sequence of $n$ integers. $a$ satisfies Property $A$ if it satisfi …

Let $E_n(z)$ be the Eulerian polynomial $$E_n(z) = \sum_{\tau \in \mathfrak{S}_n} z^{\operatorname{des}(\tau)}$$ where $\mathfrak{S}_n$ denotes the set of all permutations of $\{1,\ldots,n\}$ and $\op …

This is a known open problem (for isometric regions), which, as far as I know, is still not settled. The dimension 3 case was proved affirmatively in https://arxiv.org/abs/1501.05991, where also some …

As Vasu commented already: there is not "the" q-analogue of the Catalan numbers. And indeed, you're mixing two different here. Your first q-Catalan numbers defined by the $q$-binomials is MacMahon's …

I recommend reading Macdonald's volume University Lecture Series Vol 12 Symmetric functions and orthogonal polynomials. It is a rather short introduction to Macdonald polynomials for the symmetric gr …

This is more of an expansion of Sam's comments, but too long for a comment itself: As pointed out by Sam in that Theorem 4.2.2.2, tilting modules of the linear type $A_n$ quiver are in natural corres …

As a follow-up of Allen's question Coxeter exchanges in non-reduced words, I wonder whether it is known that the Demazure product is well-defined in Artin groups. This is: Let $(W,S)$ be a Coxeter sy …

Let $P$ be a finite poset which is a lattice with $0,1 \in P$. An atom in $P$ is an upper cover of $0$ and a coatom is a lower cover of $1$. $P$ is atomic if every element is a join of atoms and coat …

Increasing parking functions are in (more or less canonical) bijection with Dyck paths (see, e.g., here), so your question can be rephrased as Is there a direct bijection between (n,n+1)-cores and Dy …

Thanks for asking this question! It really is a perfect occation for me advertising (once again) the combinatorial statistic finder http://www.FindStat.org! To search the database for partitions, see …

I still hope there is a complete (and easy) answer to the question, but as mentioned in my comment above, and since no one else answered so far, I give a description of the inverse map that is not com …

Here is a bijection to Dyck paths (or to well-formed bracketings): Take a SYT of shape $2\times n$ (so it contains the numbers $\{1,\ldots,2n\}$, and we aim to form a word of $n$ up-steps (opening br …

As you find in "Humphreys, Reflection and Coxeter groups" (link behind paywall) in Section 5.7, the set of reflections of a Coxeter system $(W,S)$ is given by $R = \{ wsw^{-1} : w \in W, s \in S\}$, t …

The Catalan numbers have a famous generalization associated to finite irreducible reflection groups. Afaik, the formula $$\operatorname{Catalan}(W)=\prod_{i=1}^n \frac{d_i+h}{d_i}$$ appeared first in …