57 problems

Let $p$ be a prime, let $a$ be a positive integer with $(a,p)=1$, and let $k\geq3$ be odd. Write $s(N,K)$ for the Stirling number of the first kind and let $v_p$ denote the $p$-adi…

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…

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…

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…

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

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 $\operatorname{SRG}_{m,n}$.

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…

For integers $x,k,y,l$ with $\max\{k,l\}\leq 50$, the equation … has the known solutions $(x,k,y,l)=(5,2,6,5)$ and $(13,2,91,90)$. Frottier–Pintér–Pintér's conjecture. There is no…

Let $\xi_n$ be the random variable with … where $B_n=T_n(1)$ is the $n$-th Bell number, and let $\kappa^{(m)}(\xi_n)$ denote its $m$-th cumulant. Cumulant asymptotics conjecture. F…

Coefficientwise Hankel-total positivity conjecture. The row-generating polynomials of $[\mathbf{JS}_{n,k}]_{0\leq k\leq n}$ are coefficientwise Hankel-totally positive in $(x,y,q)$…

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 $S(n,k)$ denote a Stirling number of the second kind and let $bd_2$ be the $2$-adic valuation. Amdeberhan's conjecture. For every integer $h\ge 0$ and every integer $k$ with…

Let $S(n,k)$ denote a Stirling number of the second kind, let $u2$ be the $2$-adic valuation, let $c32(c)$ be the sum of the binary digits of $c$, and let $binom{n}{k}$ denote a b…

Let $S(n,k)$ denote a Stirling number of the second kind, let $u2$ be the $2$-adic valuation, and let $c32(a)$ denote the sum of the binary digits of $a$. **The Stirling valuation…

Hong–Qiu's conjecture.

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…