13 problems

Let $F(x,r)=M(x)^r$, where $M(x)$ is the generating function of the Motzkin numbers, and let $H_n(F(x,r))$ be its Hankel determinants. The conjecture asserts that, for…

Let $M(x)=\sum_{n\geq 0}M_nx^n$ be the generating function of the Motzkin numbers, let $F(x,r)=M(x)^r$, and write $H_n(F(x,r))$ for the Hankel determinant of the coefficients of…

Let $M(x)=\sum_{n\geq 0}M_nx^n$ be the generating function of the Motzkin numbers, let $F(x,r)=M(x)^r$, and write $H_n(F(x,r))$ for the Hankel determinant of the coefficients of…

Let $a,b\in\mathbb{Z}$, let $p$ be a prime, and define … Assume that $p$ does not divide any member of $S_p$. Multiplicative generation conjecture. The set $S_p$ generates…

Rowland–Yassawi's conjecture. For each $p\in\{31,37,61\}$ and every $n\in\mathbb{Z}_{\geq 0}$,

Let $\operatorname{\mathbf{Tam}}_n$ be the Tamari lattice, and consider its intervals. For an interval, measure degree separately in the variable pairs $(x,y)$ and…

Deutsch–Sagan's divisibility conjecture. The number $M_n$ is divisible by $4$ if and only if

Let an even thread be a thread of even type and an odd thread be a thread of odd type. For an even thread $\alpha$ and an odd thread $\beta$, consider the threads entangled with ea…

Nonvanishing conjecture. For all $n\geq 0$,

Let $n$ and $k$ be positive integers. Define the Motzkin-triangle entries by … For a $2n\times 2n$ skew-symmetric matrix, let $\operatorname{Pf}$ denote its Pfaffian. Generalized M…

Let $M_n$ denote the $n$th Motzkin number. For each partition $\lambda$ with at most three parts, consider standard Young tableaux of skew shape $\lambda/(2,1)$ having $n$ entries.…

Regev's conjecture. The cardinality satisfies

Schork's conjecture. The Motzkin numbers of rank $r$ satisfy a recursion relation of order $2r+1$.