79 problems

Chan–Robbins–Yuen conjecture. The volume of $\mathcal{F}_{K_n}$ is given by the product above.

Let $C_k$ denote the $k$th Catalan number, and let $p>3$ be a prime. Sun's Catalan congruences. … and … The paper states that these two conjectures are confirmed by its theorem for…

Catalan-number odd-moment conjecture. The average value of the main term of the $(2k+1)$st moment is

Enumeration conjecture for ${\cal A}_n^j$. The number of elements of ${\cal A}_n^j$ is

Narayana refinement conjecture. The number $D_{nk}$ is divisible by

Let $n\ge 2$, and let $F_n$ be the $n\times n$ matrix whose $(i,j)$ entry is $1$ when $j\le i+1$ and $0$ otherwise. The matrix $F_n$ is a union of permutation matrices correspondin…

For $l,m\in\mathbb{N}$, define the indicator $[A]$ of an assertion $A$ by … Here $\lceil\cdot\rceil$ denotes the ceiling function, and binomial coefficients are defined for arbitra…

Enumeration conjecture.

The $(q,t)$ Catalan character conjecture. In particular, $\chi_{\mathbf e\cdot L(\mathsf{triv})}(q,t)=C_n^{(k)}(q,t)$, the $(q,t)$ Catalan number. This conjecture refines the corre…

Roques's conjecture. For walks of length $2n+1$, the number of walks is

Roques's conjecture. The number of such walks of length $2n+1$ is

For each positive integer $k$, let $D'_{k,u,n}$ denote the number of $k$-dimensional balanced ballot paths of length parameter $n$ having semisymmetric height $u$, and consider the…

Transposed Catalan triangle formula. The number of such permutations is

Let $(C_n)_{n\in\mathbb{N}}$ be the Catalan numbers, and let $d\in\mathbb{N}\!\setminus\!\{0\}$. A sequence is D-algebraic of order $2$ and degree $2d$ if it satisfies a polynomial…

Let $\vec{b}=(1^2,3^2,5^2,\dots)$, and let $C_n^{\vec{b}}$ denote the corresponding weighted Catalan number, which counts combinatorial types of Morse links of order $n$. For a pos…

Let $G_n$ be the action graph associated with $S(0,n)$, and let $K_{\ell,v,n}$ be the number of paths of length $\ell$ in $G_n$ that start at a vertex labeled $v$ and end at a vert…

Let $G_n$ be the graph associated with $S(0,n)$, and let $K_{\ell,v,n}$ denote the number of paths of length $\ell$ in $G_n$ that start at a vertex labeled $v$ and end at a vertex…

Aliniaeifard and Li's identity. For any odd positive integer $n$,

Alternating Eulerian–Catalan identity.

Conjecture 1. If $b\mid\sigma(bk-1)$, then $b\in B$; furthermore, there are no other values of $b$ satisfying this property.

Dyck-path strengthening. There exists a set $\Pi=\{\pi_1,\pi_2,\ldots,\pi_{C_k}\}$ of $C_k$ permutations of $\mathbb{Z}_{2k+1}$, each fixing $0$, and a bijection…

Let $C_n=\frac{1}{n+1}\binom{2n}{n}$ denote the $n$th Catalan number, and let $\nu_p(m)$ be the exponent of the prime $p$ in the prime factorization of $m$. Assume that $a,b\geq 1$…

For $n\geq 1$, let $N={2n+2\choose n+1}$, and let $\alpha$ be the flip sequence of a star transposition ordering of all $(n+1,n+1)$-combinations, with entries in $[2n+1]$. Write ……

The super FiboCatalan numbers are defined by replacing the binomial and Catalan-number expressions in the corresponding super Catalan identities with their FiboCatalan analogues. A…

Let $A^{c,p}_{n}$ be the set of fully commutative elements of size $p$ in $W(A_n)$, and let $\mathcal C_n^p=|A^{c,p}_{n}|$ for $0\le p\le n$, with $\mathcal C_n^p=0$ when $n<0$,…