9 problems

Let $a_n$ denote the number of Motzkin paths of length $n$ satisfying the rule specified in the source. For $n\geq 1$, let A005802 and A216947 denote the cited OEIS sequences, with…

Periodic Motzkin ground-state conjecture. The periodic Motzkin chain with $N$ sites has $2N+1$ degenerate ground states with zero eigenvalue, with independent states…

Equinumeracy conjecture. The set $\mathcal{B}^{\text{tiny Fib}}(3,k)$ is equinumerous with the set of topside peakless Motzkin paths of length $k$.

Terminal-pattern divisibility conjecture. The following implications hold:

Possible-roots conjecture. For $M$, the rational roots of $w_l(M)$ lie in

Marczinzik's conjecture. The number of Dyck paths such that $\mathcal N\cap\mathcal D$ is contained in $\mathcal F$ equals the number of Motzkin paths of length $n$. Moreover, the…

Let $d$ be a positive integer, and let the admissible unit steps be … where $\{\epsilon_1,\dots,\epsilon_{d+1}\}$ is the standard basis of $\mathbb{R}^{d+1}$. Consider paths from t…

Let $e_n$ denote the number of set partitions of $[n]$. Let $b=(b_0,b_1,b_2,\ldots)$ and $\lambda=(\lambda_1,\lambda_2,\ldots)$ be sequences defining the weight of a Motzkin path a…

Let $p_n(x)$ be the polynomial sequence used in the paper, and let $M'(k)$ denote the number of peakless Motzkin paths to $(k,0)$, where a peakless Motzkin path contains neither th…