211 problems

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…

Let $G$ be a very well-covered graph of order $n$, meaning that every maximal independent set has size $n/2$, and let … be its independence polynomial. Levit–Mandrescu's conjecture…

Let $F$ be a forest, and let $I_F(t)$ denote its independence polynomial, whose coefficients count independent sets of each cardinality. Alavi–Malde–Schwenk–Erdős conjecture. For e…

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 $P$ be a reflexive polytope, meaning a lattice polytope whose polar dual is also a lattice polytope. Let $h^*_P(x)$ denote its $h^*$-polynomial. Hibi's conjecture. Every reflex…

Let $P$ be a Gorenstein IDP polytope, meaning an IDP lattice polytope whose associated Ehrhart ring is Gorenstein. Let $h^*_P(x)$ denote its $h^*$-polynomial. Hibi–Ohsugi conjectur…

Stanton's unimodality conjecture. The coefficients of each polynomial $C_n(x)$ form a unimodal sequence.

Let $G$ be a well-covered graph, meaning that all maximal independent sets of $G$ have the same cardinality, and let $I_G(t)$ denote its independence polynomial. Brown–Dilcher–Nowa…

Let $T$ be a tree, and let $i_j(T)$ denote the number of independent sets of $T$ with cardinality $j$. Alavi–Erdős–Malde–Schwenk conjecture. The sequence of numbers of independent…

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 $M$ be a matroid of rank $r$. For $0\leq i\leq r$, let $W_i$ denote the number of rank $i$ flats of $M$. Rota's conjecture. The sequence $(W_i)_{0\leq i\leq r}$ is unimodal for…

Morier-Genoud–Ovsienko conjecture. The rank polynomial of $J(P)$ is unimodal.

Let … where $\lambda$ runs over Young tableaux of size $n$ and $H(\lambda)$ is the set of hook lengths associated with $\lambda$. Heim–Neuhauser conjecture. The polynomials…

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

Let $\mathrm{M}$ be a finite matroid, and let $f_i(\mathrm{M})$ denote the number of independent subsets of $\mathrm{M}$ of size $i$. Welsh's conjecture. The sequence…

For $n,r\in\mathbb{N}$, let $\mathfrak{S}_n^r$ be the set of $r$-stack-sortable permutations and let $A(\mathfrak{S}_n^r;x)$ denote its descent polynomial. Bona's conjecture. The p…

Let $V(m,n)$ and $V_d(m,n)$ denote the rank functions associated with the compositions considered in the paper, with $m\in\mathbb Z$ and $n\geq 0$. Zhou's unimodality conjecture. F…

For each positive integer $n$, let $\operatorname{av}_n^k(1324)$ be the number of 1324-avoiding permutations of length $n$ with $k$ inversions. Unimodality conjecture. The sequence…

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

Let $M$ be a matroid and let $\mathscr{P}(M)$ be its matroid polytope. If $\mathscr{P}$ is a lattice polytope of dimension $d$, its $h^*$-polynomial $h^*(\mathscr{P},x)$ is defined…

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…

For each integer $n\geq 1$, let $N_S(m,n)$ denote the number of $S$-partitions of size $n$ with spt-crank $m$. Andrews–Dyson–Rhoades' conjecture. The sequence … is unimodal for eve…

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 $P_n$ be the integral polytope of real $n\times n$ doubly stochastic matrices, and let $h_{P_n}^*(x)$ be its Ehrhart $h^*$-polynomial. Stanley's conjecture. The polynomial…