10 problems

Divisibility conjecture. One has

Let $s$ and $t$ be positive integers with $\gcd(s,t)=1$, and let $f_t(s)$ be the polynomial appearing in Fayers' formula for the number of $(s,s+t,s+2t,\dots)$-core partitions. Fay…

Schützenberger's divisibility conjecture. For every positive integer $n$, the polynomial $E_{2n+1}(q)$ is divisible by

The principal-polynomial support-reduction conjecture. If $E^\circ$ is a principal polynomial, then either

The divisibility and support descent conjecture. The following implications hold: for $w=(0,1)$ and $i\in[m(n-m)(a-1)/a+1,m(n-m)]$, the $i$-th $w$-divisibility conditions imply the…

Terminal-pattern divisibility conjecture. The following implications hold:

Divisibility conjecture. If $H\sqsubseteq\mathcal{H}$ are $k$-trees for $k\geq 3$, then

Generalized Prouhet–Thue–Morse divisibility conjecture. If $k=2$, then

Let $n=2pq$, where $p$ and $q$ are distinct primes, and let $H_n$ denote the set of $0,1$-polynomials divisible by $\Phi_n(x)$ as defined in the paper. Lowest-degree three-term pol…

Let $n>1$, let $\Phi_n(x)$ be the $n$-th cyclotomic polynomial, and let $p$ be the smallest prime dividing $n$. Consider monic polynomials with nonnegative coefficients that are di…