25 problems

Let $\Omega$ be a Gorenstein poset, and let its cd-index be the homogeneous noncommutative polynomial in $\mathbf{c}$ and $\mathbf{d}$ associated with its flag $f$-vector, where…

Let $P$ be a Gorenstein poset, and write its $cd$-index as a homogeneous polynomial in the noncommutative variables $c$ and $d$, of degrees $1$ and $2$, respectively. Bayer–Klapper…

Let $\mathbf{k}$ be a field, let $P$ be a Gorenstein$^*$ poset of rank $n+1$, and let $M$ be a degree-$n$ $cd$-monomial. Define the multidegree of $M$ by replacing each $c$ with…

Let $\Delta$ be a complete fan of dimension $n$, and let ${\mathcal A}(B\Delta)$ be its associated graded algebra with elements $L_i\in {\mathcal A}(B\Delta)$ of degree $e_i$, for…

Let $W$ be a Coxeter group, let $u,v\in W$ satisfy $u\leq v$ in the Bruhat order, and let $\Phi_{[u,v]}$ denote the cd-index of the interval $[u,v]$. Bruhat-interval non-negativity…

Let $P$ be a graded Eulerian poset, and let $\Phi_P$ denote its cd-index. A poset is Gorenstein when it has the homology type of a sphere; in the source this is described as $P$ tr…

Let $[u,v]$ be an interval in the Bruhat order on a Coxeter group, with $l(u)=k$ and $l(v)=d+k+1$. Let $\Phi_{[u,v]}$ denote its cd-index, and let a dual stacked polytope be a poly…

Let $\beta(L)$ denote the beta-value associated with a list $L$. A list is balanced if its entries are either $k$ or $k+1$ for some non-negative integer $k$. Let $L$, $M$, and…

Let $(m_1,n_1)$ and $(m_2,n_2)$ be pairs of non-negative integers, where $(m_1,n_1)$ is strictly better balanced than $(m_2,n_2)$, meaning that the first pair is strictly closer to…

Let $\beta(L)$ denote the beta-value associated with a list $L$, and let list concatenation be written by juxtaposition. Let $m,n$ be non-negative integers with $n>m$, and let $L$…

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…

Complete cd-index nonnegativity conjecture. Then

Billera–Brenti's strong conjecture. One has the coefficientwise inequality

Let $[u,v]$ be an interval in the Bruhat order of a Coxeter group. Its complete $cd$-index is the nonhomogeneous polynomial in $c$ and $d$ whose homogeneous part of top degree is t…

Let $(W,S)$ be a Coxeter system, and let $[u,v]$ be an interval in the Bruhat order of $W$. Write $k$ for the length of $u$ and suppose that the length of $v$ is $n+k+1$. The $cd$-…

A poset is Gorenstein when it is Cohen--Macaulay and Eulerian. Its $cd$-index is denoted by $\Phi_P(c,d)$; coefficientwise comparison means comparison of the coefficients of corres…

A regular CW sphere is a regular CW complex whose underlying space is a sphere, and its $cd$-index is the polynomial $\Phi_P(c,d)$ encoding its flag $f$-vector, with $c$ of degree…

Let $P$ be an Eulerian poset, and write its $cd$-index as a polynomial $\phi_P(c,d)$ in $c$ and $d$. A $cd$-index is nonnegative when all coefficients of this polynomial are nonneg…

Spanning conjecture. For all $n>0$, the complete $cd$-indices of all Bruhat intervals of rank $n+1$ span the whole space of $cd$-polynomials of degree bounded by $n$ whose nonzero…

Let $P$ be a Cohen–Macaulay CW-poset or a Gorenstein poset of rank $n$, let $\mathcal A_n$ be the indexing family for its flag $f$-vector coefficients, and let $\alpha_S(P)$ denote…

Let $G$ be a bounded labeled acyclic digraph with a balanced linear edge labeling. Non-negativity conjecture. The $\mathbf{c}\mathbf{d}$-index of $G$ is non-negative. This conjectu…

Let $P$ be an $(n-1)$-dimensional regular CW-sphere, or more generally a Gorenstein poset of rank $n+1$. Write its homogeneous cd-index as … where $w$ ranges over cd-monomials of d…

Sign conjecture. For every ${\bf cd}$-word $w$, one has

Caselli's conjecture. For each Bruhat interval $[u,v]$ and each $i=0,1,\dots,\lfloor n/2\rfloor$, one has

A poset $P$ is Gorenstein» if it is both Eulerian and Cohen–Macaulay. For every cd-word $w$, write $[w]_P$ for its coefficient in the cd-index of $P$. Stanley's conjecture. If $P$…