27 problems

Let $A$ and $D$ denote the Eulerian and Delannoy triangles, respectively, and let $A^2$ and $D^2$ be their matrix squares. For a lower triangular matrix $M$, write its $n$-th row g…

Dominance conjecture.

For $n\geq 3$, let $\Psi_{B_n}(t)$ denote the Ehrhart $h^*$-polynomial of the type $B_n$ hypersimplex, and let $E_{D_n}(t)$ denote the type $D_n$ Eulerian polynomial. Type B recipr…

Brenti's conjecture. The polynomial $W_{P,\omega}$ is log-concave with no internal zeroes.

For $n\geq 1$, write $Z_n(t)=\sum_k z(n,k)t^k$, and let … be the gamma-vector associated with $Z_n(t)$. A sequence is log-concave when $a_k^2\geq a_{k-1}a_{k+1}$ wherever the terms…

Sokal's reversed second-order Eulerian conjecture. The sequence $(E_n^{[2]*}(x))_{n\geq0}$ is coefficientwise Hankel-totally positive in $x$.

Let $n\geq 0$, and let $f$ be a polynomial of the form … Let $\operatorname*{\mathcal E}:\mathbb{R}[t]\to\mathbb{R}[t]$ be the Eulerian transformation defined by…

Let $\mathfrak{S}_n$ be the symmetric group, and define the two-sided alternating Eulerian polynomial by … Two-sided alternating gamma-expansion conjecture. For every $n\geq 1$, ……

Let $\mathfrak{S}_n$ be the symmetric group, and let $\operatorname{altdes}(\pi)$ and $\operatorname{altmaj}(\pi)$ denote the alternating descent number and alternating major index…

The signed log-concavity conjecture. The polynomial $\tilde{A}_n(t,-1)$ is log-concave for $n\geq1$.

For $n\equiv 0,1\pmod 4$, let $\mathcal{A}_n$ be the alternating subgroup of $\mathfrak{S}_n$, and define … For even $n$, define the type-$B$ polynomials $\mathrm{TSB}_n^+(s,t)$ an…

Let $m\geq 1$, let ${\mathbf{x}}=(x_0,\ldots,x_{m-1})$ and ${\mathbf{c}}=(c_L)_{L\geq 0}$ be indeterminates, and let ${\mathcal{P}}^{(m)-}_n({\mathbf{x}},{\mathbf{c}})$ be the exte…

Let $\mathcal{F}_n^\alpha(x)$ be the flag descent polynomial associated with the quotient $\mathbb{Z}_{\alpha}\wr \mathfrak{S}_n/H$. Unimodality and palindromicity conjecture. The…

Let $W$ be a finite Coxeter group of rank $n$, and let … be its two-sided Eulerian polynomial. Generalized Gessel's conjecture. There exist nonnegative integers $\gamma_{a,b}^W$ su…

Euler-zigzag stability threshold conjecture. For any $n\geq 3$, the polynomial

Hyatt's conjectures. For $n\geq 1$, $B_n^+(x)$ interlaces $x^nB_n^+(1/x)$, and hence $B_n(x)=B_n^+(x)+x^nB_n^+(1/x)$ has only real zeros. For $n\geq 2$, $D_n^+(x)$ interlaces…

Let $\{x_n\}_{n\geq 0}$ be a sequence of nonnegative numbers, and define $z_n=\sum_{k=0}^n A_{n,k}x_k$ using the Eulerian triangle $\{A_{n,k}\}_{0\leq k\leq n}$. A sequence…

Dilks–Petersen–Stembridge conjecture. For any irreducible finite Weyl group $W$, the affine Eulerian polynomial $\widetilde{W}(x)$ has only real zeros. The conjecture is an affine…

Let $A_n(s,t)$ denote the two-sided Eulerian polynomial. For $n\geq 1$, the polynomial is considered in the basis … with indices satisfying $0\leq i-1$, $j$, and $j+2i\leq n+1$. Ge…

Let $P^{\pm}(\{1,1,2,2,\ldots,n,n\})$ be the signed permutations of the multiset $\{1,1,2,2,\ldots,n,n\}$, and let $\mathfrak I_{2n}^{(1,4,3,8,\ldots,2n-1,4n)}$ be the correspondin…

For $n\geq 2$, let $\widetilde{B}_n(\mathbold{x},\mathbold{y})$ be the multivariate refinement defined by … Here $\mathbb{N}[\mathbold{x},\mathbold{y}]$ denotes the polynomials in…

For $n\geq 2$, let $\widetilde{B}_n(\mathbold{x},\mathbold{y})$ be the multivariate refinement defined by … Write $\mathfrak{S}_{\mathbb{R}}[\mathbold{x},\mathbold{y}]$ for the cla…

For $n\geq 2$, let $D_n(\mathbold{x},\mathbold{y})$ be the multivariate refinement defined by … Here $A_{n-2}$ and $B_n$ are the stable multivariate refinements of the type $A$ and…

Eulerian descent-set distribution conjecture. For any $n$ and any subset $D\subseteq\{1,2,\dots,n-1\}$,

Power-sum character formula conjecture. The Eulerian quasisymmetric functions satisfy