142 problems

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 $\pow\geq2$, $k\geq1$, 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…

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…

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.

The conjecture. For all $n\geq 4$:

Let $P$ be a convex $d$-polytope with $f$-vector $(f_0,f_1,\ldots,f_{d-1})$, where $f_k$ denotes the number of $k$-dimensional faces. Bjrner's quarter-monotonicity conjecture. The…

Let $P$ be a convex $d$-polytope with $f$-vector … where $f_k$ is the number of $k$-dimensional faces of $P$. The unimodality conjecture. For each $d$-polytope $P$ there is an inte…

Zero-divisor graph unimodality conjecture. The independence polynomial $I\bigl(\varGamma(\mathbb{Z}_n),x\bigr)$ is unimodal. The paper presents this as a conjecture following compu…

For binary partitions, let $\operatorname{num}_{\mathcal B}(n,x)$ denote the binary numerator. Ballantine–Beck–Feigon–Maurischat's binary unimodality conjecture. For every $n>5$, t…

Write the ordinary partition numerator as $\operatorname{num}(n,x)=\operatorname{num}_0(n,x)+\operatorname{num}_1(n,x)$, where $\operatorname{num}_0(n,x)$ contains the even powers…

For $m,n\in\mathbb{N}$, define the $q$-Fibonomial coefficient by … where $[F_n]^!_q:=\prod_{k=1}^n[F_k]_q$ and $[n]_q:=1+q+\dots+q^{n-1}$. Bergeron–Ceballos–Küstner's conjecture. T…

Let $G$ be a finite undirected graph on $n$ vertices. A dominating set is a set of vertices such that every vertex is either in the set or adjacent to a vertex in the set. For…

Generalized Glasby–Paseman sequence conjecture. This sequence satisfies the following properties:

Let $w$ be a composition, ordered by the subword order, and let its principal downset consist of all compositions contained in $w$. A composition is associated with a layered permu…

For permutations $\rho$ and $\tau$ with $\rho oeq \tau$ in the permutation pattern poset, let $[\rho,\tau]$ denote the interval of permutations $u$ satisfying $\rho oeq u oeq \tau$, ran…

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…

Let $T$ be a tree, let $\alpha=\alpha(T)$ be its independence number, and let $a_i$ denote the number of independent sets of cardinality $i$, so that $a_0=1$. The Tree Unimodality…

Let $k\geq 2$ be an integer and let $n_k,\ldots,n_1$ be non-negative integers. Let $b(n_k,\ldots,n_1)$ denote the polynomial constructed in Theorem 3. Unimodality conjecture. For e…

Let $(\boldsymbol{a},\boldsymbol{b})$ be tuples of positive integers satisfying Landau's criterion, namely … Let $D(\boldsymbol{a},\boldsymbol{b};q)$ be the corresponding $q$-facto…