46 problems

Let $S=K[x_1,\ldots,x_n]$, let $J_{n,m}$ be the path ideal of the cycle graph on $n$ vertices with paths of length $m$, and set $d=\gcd(n,m)$. Let $t_0\leq n-1$ be maximal such tha…

Soleyman Jahan's conjecture. There always exists a Stanley decomposition $\mathcal D$ of $S/I$ such that

Let $S=\mathbb{K}[x_1,\dots,x_n]$ and let $\mathfrak{m}=(x_1,\ldots,x_n)$ be its maximal ideal. For a monomial ideal or module, write ${\rm sdepth}$ for Stanley depth. Cimpoeaş's c…

Let $S$ be a polynomial ring over a field and let $I\subset S$ be a monomial ideal. Write $\operatorname{depth}$ for homological depth and $\operatorname{sdepth}$ for Stanley depth…

Let $K$ be a field, let $S=K[x_1,\dots,x_n]$, and let $I_{n,1}=(x_1,\dots,x_n)$ be the squarefree Veronese ideal of degree $1$. Its Stanley depth is denoted by…

The conjectural Stanley-length formula. The Stanley length should satisfy

Stanley's conjecture. Every such module satisfies

Let $K$ be a field, let $S=K[x_1,\ldots,x_n]$, and let $\mathbf m=(x_1,\ldots,x_n)$ be its maximal graded ideal. For an integer $t\geq 1$, let $\operatorname{sdepth}(\mathbf m^t)$…

Stanley's depth conjecture. For any monomial ideals $0\subset I\subsetneq J\subset S$,

Let $S=K[x_1,\ldots,x_n]$, let $J_{n,m}$ be the path ideal of the cycle graph on $n$ vertices with paths of length $m$, let $d$ be the parameter defined by the paper for $n$ and…

Stanley's conjecture. Every $\mathbb{Z}^n$-graded module over an $n$-dimensional polynomial ring satisfies Stanley's inequality.

Let $K$ be a field, let $S=K[x_1,x_2,\ldots,x_n]$, let $I\subset S$ be a proper monomial ideal with linear quotients, and write $[n]=\\{1,\ldots,n\\}$. The stronger variable reduct…

Let $K$ be a field, let $S=K[x_1,x_2,\ldots,x_n]$, and let $I\subset S$ be a monomial ideal with linear quotients, meaning that its minimal generators admit an ordering…

Stanley's inequality. One should have

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$, and let $M$ be a finitely generated $\mathbb{Z}^n$-graded $S$-module. Stanley's conjecture. … This conjecture compa…

Let $G$ be a graph with $n$ vertices, let $J(G)$ be its cover ideal, and let $J(G)^{(k)}$ denote its $k$-th symbolic power. Let $\nu_o(G)$ denote the parameter used in the source f…

Let $S=\mathbb{K}[x_1,\dots,x_n]$, let $I$ be a polymatroidal ideal, and let $\ell(I)$ denote its analytic spread. Asymptotic Stanley-depth conjecture. For every sufficiently large…

Let $\mathbb{K}$ be a field, let $S=\mathbb{K}[x_1,\dots,x_n]$, and let $M$ be a finitely generated $\mathbb{Z}^n$-graded $S$-module. A Stanley decomposition of $M$ is a finite dir…

Stanley's conjecture. Every finitely generated $\mathbb{Z}^n$-graded $S$-module satisfies the Stanley inequality.

Let $K$ be a field, let $S=K[x_1,\dots,x_n]$, and let $I\subseteq S$ be a monomial ideal. Strict Stanley-depth inequality. One has … The paper studies this conjecture for squarefre…

Stanley's conjecture. One always has

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring, let $J_{n,m}$ denote the path ideal of the cycle graph on $n$ vertices with path length $m$, and let $\varphi(n,m)$ be the functio…

Let $S$ be a polynomial ring and let $I\subseteq S$ be a monomial ideal. The Betti poset of $I$ is the poset formed by the multidegrees supporting the multigraded Betti numbers of…

Stanley's conjecture. Every finitely generated $\mathbb{Z}^n$-graded $S$-module satisfies Stanley's inequality. This conjecture was recently disproved, so the asserted inequality d…

Let $R=\mathds{K}[X_1,\ldots,X_n]$ be a polynomial ring in an arbitrary positive number of variables, let $\mathfrak m=(X_1,\ldots,X_n)$ be its homogeneous maximal ideal, and let…