55 problems

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…

Transposed Catalan triangle formula. The number of such permutations is

General Hankel-transform conjecture. The Hankel transform is given by

Let $C(x)=\sum_{n\geq 0}C_nx^n$ be the Catalan generating function, let $F(x,r)=C(x)^r$, and write $H_n(F(x,r))$ for the $n\times n$ Hankel determinant of the coefficients of…

Let $C_{n,k}:=\frac{n-2k}{n}{n\choose k}={n-1\choose k}-{n-1\choose k-1}$, with the Catalan triangle relation $B_{n,k}=C_{2n,n-k}$. Let $n$ and $a$ be positive integers with $n>a$,…

Catalan graph determinant conjecture. The determinant satisfies

Let $C_k$ denote the $k$th Catalan number, let $\operatorname{Nar}_{k}(t)$ denote the Narayana polynomial, and define … Let $W_{n,k}^{+}(t)$ and $W_{n,k}^{-}(t)$ be the symmetric-d…

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

Conjectures for $L_n$ and $M_n$. The following five propositions are conjectured:

Catalan valuation conjecture. Then $t=C_{1,n}$. The source presents this as an observed pattern in its empirical study of constant-term valuations; no resolution is given.

Catalan triangle Laurent polynomial and positivity conjecture. This expression is a Laurent polynomial in $q$, and it has non-negative integer coefficients when…

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

Chan–Robbins–Yuen volume conjecture. The relative volume of $P_n$ is

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…

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…

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 $G_n$ 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…

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

Alternating Eulerian–Catalan identity.

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 $a,b\geq 1$…

Catalan-convolution conjecture. For every $n\geq 1$,

Let $C_n$ denote the Catalan triangular array. A matrix is vertically recurrent when it satisfies the paper's vertical recurrence relation. The Catalan-array conjecture. The Catala…