35 problems

Planar-graph unimodality conjecture. For any planar graph $G$ of order $n$, the sequence $\{\st{G}{k}\}_{k=1}^n$ is unimodal. This property is known for paths, cycles, trees, lolli…

Central Stirling numbers conjecture. The binomial Ward numbers satisfy

Let $C^{(r)}$ be the Stirling cycle triangle of order $r$, let … be the triangle formed by reversing the rows of $C^{(r)}$, and let $c_{r,n}(x)$ be its row-generating polynomials.…

Let ${\cal P}_n^k$ denote the set of partitions of $[n]$ into $k$ blocks, and let $\mathop{\rm mak}$, $\mathop{\rm lmak}$, $\mathop{\rm mak}{}'$ and ${\mathop{\rm lmak}}'$ be the f…

Let $G$ be a graph, let $k$ be fixed, and let $\st{G}{k}_r$ denote the $r$-restricted graphical Stirling number, where the restricted-set size is indexed by $r$. Restricted-number…

For $n\geq 0$, let $s_{3,n}(x)$ denote the third-order Stirling subset polynomial. Third-order Stirling subset zero conjecture. The zeros of $s_{3,n}(x)$ lie in the left half-plane…

Stronger asymptotic error-bound conjecture. As $p\to\infty$,

Non-negative coefficient conjecture. For each integer $r\geq 1$, this power series has all non-negative coefficients.

Super Artin basis conjecture. The set ${\mathcal S}{\mathcal A}_{m,n}$ is a basis for $\operatorname{SRG}_{m,n}$.

Let $S(n,k)$ denote the number of partitions of an $n$-element set into exactly $k$ nonempty subsets, and let $v_2$ denote the $2$-adic valuation. For fixed $k\in\mathbb{N}$, consi…

For integers $n$ and $m$, consider the Diophantine equation … The displayed Stirling-number identity shows that this equation arises from equating $\genfrac\{\}{0pt}{}{n}{n-3}$ and…

Let $f(b,c)=\sigma_2(b)-\sigma_2(c)+\nu_2\left(\binom{2c-b-1}{c}\right)$, where $c3_2$ is the binary digit sum and $bd_2$ is the $2$-adic valuation. Let $a,b,c,h$ be positive integ…

Let $c$ be a positive integer, let $S(n,k)$ denote a Stirling number of the second kind, let $bd_2$ be the $2$-adic valuation, and let $c3_2(m)$ denote the sum of the binary digits…

Let $S^{(2)}_k$ denote the relevant sequence and let ${\rm Fail}(S^{(2)}_k)$ be its repair factor. For a prime $p$ and a positive integer $j$, the following assertions are conjectu…

Let $S^{(2)}_k$ denote the relevant sequence, let ${\rm Fail}(S^{(2)}_k)$ be its repair factor, and let ${\rm Rad}(n)$ denote the radical of $n$, the product of its distinct prime…

Let $S^{(2)}_k$ denote the relevant sequence and let ${\rm Fail}(S^{(2)}_k)$ be its repair factor. For a prime $p$, write $|\cdot|_p$ for the $p$-adic absolute value. The large-pri…

Let $S^{(2)}_k$ denote the relevant sequence and let ${\rm Fail}(S^{(2)}_k)$ be its repair factor. For a prime $p$, write $|\cdot|_p$ for the $p$-adic absolute value. The $p$-adic…

Let \genfrac\{\}{\}{0pt}{}{n}{k} denote the Stirling subset number, and define the reversed Stirling subset triangle by…

Let $p$ be an odd prime. Let $a,n,k$ be positive integers with $1\le a\le p-1$ and $2\le k\le a(p-1)p^{n-1}+1$. Define $\epsilon_k=0$ if $k$ is even and $\epsilon_k=1$ if $k$ is od…

Berrizbeitia–Medina–Moll–Moll–Noble conjecture. Fix a positive integer $k$ and a prime number $p$ such that $k<p$. Then there exist $m_0(k)\in\mathbb{N}_+$ and…

Amdeberhan–Manna–Moll conjecture. Fix a number $k\in\mathbb{N}_+$. Then there exist $m_0(k)\in\mathbb{N}_+$ and $\mu(k)\in\mathbb{N}$ such that for any integer $m\geq m_0(k)$ there…

For integers $n\geq k\geq1$, define … and let $\nu_p$ denote the $p$-adic valuation. The logarithmic lower-bound conjecture. For any prime number $p$ and any integer $k\geq1$, ther…

Higher-order Stirling-number conjecture. The following claims are valid: (1) for fixed $ell\in\mathbb{N}$ and $n\geq \ell+3$, the finite sequence…

Let $5\le\Delta\le8$ and $e\ge5$. For $j\ge0$, define … For $j>0$, $j\ne2^{e-1}$, and any integer $m$, define … Auxiliary valuation conjecture. The quantities satisfy all the follo…

Let $5\le\Delta\le8$, $0\le m<2^{e-1}$, and $e\ge4$. If … then the difference-valuation conjecture. For every $d\ge0$, … This is presented as a reduction step toward the one-zero c…