142 problems

Let $\pow\geq2$, $k\geq1$, and let $a_1,\ldots,a_k,b$ be positive integers. Write $[r]_q:=1+q+\dots+q^{r-1}$ and, for an integer $\pow$, write…

Planar-graph unimodality conjecture. For any planar graph $G$ of order $n$, the sequence $\{\st{G}{k}\}_{k=1}^n$ is unimodal. This property is known for paths, cycles, trees, lolli…

Let $calpha>2$, and for each $n$ consider the coefficient sequence $(c_{n,k})_{k\in\{1,2,\ldots,n\}}$. Unimodality conjecture. For every choice of $calpha>2$, the sequence … is uni…

Almost-all-graphs unimodality conjecture. For almost all graphs $G$, the sequence $c^{\mathcal{P}}_i(G)$, $0\leq i\leq n(G)$, is unimodal.

Let $l$ be a positive integer, and let $i$ and $j$ be non-negative integers with $j>i$. For $1\leq k\leq l$, let $L_{i,j}^k$ be the list of length $l$ whose entries are $i$ except…

Alavi–Malde–Schwenk–Erdős conjecture. The independence polynomial of every tree is unimodal.

Let $P$ be a lattice polytope with the integer decomposition property (IDP), meaning that every lattice point in $mP$ is a sum of $m$ lattice points in $P$. Let $h^*_P(x)$ be its…

Let $\rho$ be a permutation, and let $D(\rho)$ be the principal downset of all permutations contained in $\rho$, ordered by permutation-pattern containment and ranked by permutatio…

Almkvist's conjecture. For each $m$, the Hilbert function $H(m,n)$ is unimodal for all $n\geq 11$. Moreover, if $m$ is even, then $H(m,n)$ is unimodal for all $n$.

Let $P$ be a lattice polytope. It is Gorenstein if its associated Ehrhart ring is Gorenstein, and it has the integer decomposition property (IDP) if every lattice point in $mP$ is…

Zhang and Zhang's conjecture. The polynomial

Parity-unimodality conjecture. The fake-degree polynomials $f^{clambda}(q)$ are parity-unimodal for all partitions $clambda$.

Let $\mathcal{P}$ be a lattice polytope. It is Gorenstein if some positive integer dilate of $\mathcal{P}$ is reflexive, and it has the integer decomposition property if every latt…

A tree is a connected acyclic graph, and a forest is an acyclic graph. For a graph $G$, its independence sequence $(i_0(G),i_1(G),\ldots,i_{\alpha(G)}(G))$ records the numbers of i…

Unimodality conjecture. The $h^*$-vector of $P_M(W_n)$ is unimodal for every $n\geq 4$.

Let $\bar{a}$ be a tuple of positive integers, and let $\overline{F}(\bar{a})$ be the corresponding circular fence poset with rank polynomial $\overline{R}(\bar{a};q)$. A polynomia…

Near-unimodality conjecture. The polynomials $\SYT(\lambda)^{\maj}(q)$ are nearly unimodal but not unimodal for partitions $\lambda$ or $\lambda'$ in the following cases: any parti…

Kirillov's unimodality conjecture. For any $n\geq 1$, the Ehrhart $\delta$-polynomial $\delta(\mathcal{P}_n;t)$ is unimodal.

Let $q$ be a positive integer, let $\pi$ be a permutation of $\{\lceil q/2\rceil,\lceil q/2\rceil+1,\ldots,q\}$, and let $i_k(G)$ denote the number of independent sets of cardinali…

Let $(W,S,*)$ be a triple consisting of a Coxeter system $(W,S)$ and an $S$-preserving involution $*\in\operatorname{Aut}(W)$, with corresponding twisted involutions $\mathbf I_*$.…

Unimodality conjecture. If $bc$ and $bd$ are symmetric unimodal distributions on $a4$, then $bc\boxtimesbd$ is also unimodal.

Unimodality conjecture. For each fixed $(m2^a1^b)$-type $(S,s)$, the sequence

Unimodality conjecture. For fixed $n,k$, the numbers $\mathsf{Dist}_n(k,\ell)$ as $\ell$ increases are unimodal.

Gaussian-product unimodality conjecture. For every $0\leq k<n$, this polynomial is symmetric and unimodal. The source states that, using known results together with its preceding t…

Nonnegative decomposition conjecture. For $n\geq 9$ and $k\geq 1$, the coefficients $b_{2n,k}$ are nonnegative integers.