16 problems

Lazar and Wachs' cycle-distribution conjecture. The number of D-permutations on $[2n]$ with $k$ cycles equals the number of E-permutations on $[2n]$ with $k$ cycles for all $k$. Co…

Define the Gandhi polynomials by … with $\widetilde P_1(z)=1$, and let $G_{2r}$ denote the Genocchi numbers. Gandhi's conjecture. … The conjecture concerns a representation of the…

Let $G_{2n}$ denote the Genocchi number, and let $\mathcal{S}_{2n}$ be the symmetric group of permutations of $\{1,\ldots,2n\}$. In an occurrence of a parity-marked pattern, a supe…

The $q$-Genocchi triangle refinement conjecture. The terms of this triangle of $q$-integers refine the classical $q$-secant numbers.

Let $\mathcal Q_H(x)$, $\mathcal Q_{H^-}(x)$, and $\mathcal Q_{H^+}(x)$ count the prime divisors up to $x$ of the $H$-, $H^-$-, and $H^+$-sequences, respectively. Let…

Let $A$ be the Artin constant, let $\ell$ be a prime, and let $\mathcal P_G(x)$ count the $\ell$-Genocchi irregular primes up to $x$. Explicit ell-Genocchi asymptotic conjecture. A…

Let $\ell$ be a prime, let $\mathcal P_G(x)$ count the $\ell$-Genocchi irregular primes up to $x$, and let $\delta(\ell^2,2)$ be the explicit constant defined in the paper. ell-Gen…

Let $\ell$ be a prime and let $1\leq a<d$ with $(a,d)=1$. A prime is $\ell$-Genocchi regular if it is not $\ell$-Genocchi irregular. Positive-density conjecture. If $\ell$ is odd,…

Let $\ell$ be an odd prime, let $1\leq a<d$ with $(a,d)=1$, and let $\mathcal P_G(d,a)(x)$ count the primes $p\leq x$ satisfying $p\equiv a\pmod d$ that are $\ell$-Genocchi irregul…

Let $[2n]=\{1,2,\ldots,2n\}$, and consider cycles on $[2n]$. A drop is an occurrence of an entry followed by a smaller entry in the cyclic order; a drop is even-odd when its first…

A permutation has a descent from an even value to an even value when two consecutive entries form a descent and both values are even. The authors consider the sets of permutations…

Even-odd-drop and D-cycle correspondence conjecture. For every finite subset $A\subset\mathbb{Z}_{>0}$, the number of cycles on $A$ with only even-odd drops is equal to the number…

Genocchi cycle-counting conjecture. For all $n\geq 1$, $g_n$ is equal to the number of cycles on $[2n]$ with only even-odd drops.

Higher-order Genocchi branched continued-fraction conjecture. We have

For each $ngeq 1$, let $\mathcal{D}_n$ be the $n$-chord diagram in which every chord intersects every other chord, and let $D_n=varphi(\mathcal{D}_n)inmathbb{Z}[x]$. Write…

Hivert–Mallet's conjecture. For the relevant integers $k$, the Gandhi polynomial satisfies