40 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…

Let $P=\{P_n(x)\}_{n\geq 0}$ be a polynomial sequence with $\deg(P_n(x))=n$, and write … The associated Eulerian transformation is the linear transformation obtained by taking…

Let $P$ be a poset, and let $W_P(t)$ denote its $W$-polynomial, also called its $P$-Eulerian polynomial. Neggers–Stanley conjecture. The polynomial $W_P(t)$ is real rooted. This co…

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

Dominance conjecture.

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…

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

Let $\mathcal{M}_n$ be the set of matchings on $[2n]$, let $\operatorname{lne}(M)$ denote the number of left-nestings of a matching $M$, and let $Q_n(x)$ be the associated polynomi…

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…

For a positive-integer sequence $s=(s_1,s_2,\ldots)$, let $E_{2n}^s$ be the corresponding $s$-Eulerian polynomial, and let $A_{2\mathbf{1}}^{2\mathbf{1}}$ be the Type-B multiset Eu…

Let $A_{2\mathbf{1}}^{2\mathbf{1}}$ denote the Type-B analog of the multiset Eulerian polynomial, and call a polynomial $s$-Eulerian if it equals an $s$-Eulerian polynomial for som…

Gamma-positivity conjecture. Let $p>1$ and $q\geqslant 1$ be two given real numbers. Then the two polynomials in the symmetric decomposition of…

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…

Let $Z_n(t)$ be the zig-zag Eulerian polynomial, and let $\operatorname{\mathcal{C}}(Z_n)$ be the chain polytope of the zig-zag poset $Z_n$. Its $h^*$-polynomial is…

Let $A^{(2)}_n(z)$ denote the $1/2$-Eulerian polynomial, and let $\Delta(\mathcal{Q}^+(v))$ be the positive signed sum system associated to a generic weight $v$. Tolosa Villareal's…

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 $n\geq1$.

Loday's conjecture. The signed Eulerian descent enumerator satisfies

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…

Sun's conjecture. For every integer $n\geq 1$, all coefficients $c_{n,j}$ are positive integers.

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…

Positive gamma-expansion conjecture. For any $n\geq 1$, all $c_j$ are positive integers in the expansion