7 problems

Best-constant conjecture. The best constant $c_2$ in condition (c-2) for the Bell numbers $\{b_k(n)\}_{n=0}^{\infty}$ of any order $k\geq 2$ is $c_2=2$. The surrounding discussion…

Complementary Bell sign-run conjecture. The coefficient sequence of the power series $\exp(1-\exp(x))$ contains arbitrarily long subsequences of consecutive coefficients with the s…

Let $\wp_n^{(m)}$ be the family of sets obtained at iteration $m$ in the iterated partitioning process, and let $A_n^{(m)}$ denote the average cardinality of the sets in…

Let $B_m$ denote the $m$th Bell number. Bell-number congruence conjecture. There are infinitely many positive integers $n$ such that … The paper reports computational examples of s…

For $x>0$, define … so that $B(n)=B_n$ for nonnegative integers $n$, where $B_n$ is the $n$-th Bell number. Bell-function concavity conjecture. The function $B(x)^{1/x}$ is log-con…

A modified ascent sequence is obtained from an ascent sequence by the modification process described in the source. For patterns $101$, $0101$, $1021$, $1102$, $1120$, and $1210$,…

Let $B_n$ be the $n$th Bell number, and for a prime $p$ set … The Bell sequence is considered modulo $p$. Radoux's conjecture. The minimum period of $(B_n)_{n\geq 0}$ modulo $p$ is…