7 problems

For $n\geq 1$, define $B(n,k)$ and $C(n,k)$ by … and … Here a simsun permutation is a permutation with the stated simsun property, and a big return is the stated statistic on an al…

Let $e_n=|E_{2n}|$ for $n\geqslant1$, where $E_n$ are the Euler numbers. A prime $q$ is Euler regular if $q$ does not divide any $e_n$ with $0<n<\frac{q-1}{2}$; otherwise it is Eul…

Let $n$ be a positive integer, and define the $2n\times 2n$ matrix … where $\operatorname{sgn}$ is the sign function. The Chen tangent-sign permanent conjecture. … Here $E_n$ are t…

Let $n$ be a positive odd integer, let $q$ be an indeterminate, and let $a$ be a parameter. Write $[m]=(1-q^m)/(1-q)$ for the $q$-integer, $(x;q)_k=\prod_{j=0}^{k-1}(1-xq^j)$ for t…

Alternating Turán-type inequality. For all $n\geq 0$ and $j,k\geq 1$, one has

Asymptotic density conjecture. For every $1\leq\alpha\leq 6$, the limit $p_{\infty}^{(\alpha)}=\lim_{n\rightarrow\infty}p_n^{(\alpha)}$ exists and satisfies

Let $p>3$ be a prime, and let $E_{p-3}$ denote the Euler number of index $p-3$. A prime $p$ is called Euler-irregular in this setting when $E_{p-3}\equiv 0\pmod p$. Infinitude conj…