9 problems

Let $\mathsf{M}$ be a loopless matroid of rank $k$ on a ground set of size $n$. Let $\mathsf{U}_{k,n}$ be the uniform matroid of rank $k$ on $n$ elements, and write…

Let $m_n(k,l)$ be the number of all $(k,l)$-uniform matroids on the set $[n]$. Asymptotic uniformity conjecture. If $(k,l)\neq (1,1)$, then either … or, dually, … This conjecture p…

Let $M$ be a matroid of rank $d$ on $m+d$ elements, and write its Kazhdan–Lusztig polynomial as … Let $U_{m,d}$ denote the uniform matroid of rank $d$ on $m+d$ elements, and let…

For integers $1<k\le n$, let $U_{n,k}$ be the uniform matroid and let its independence complex be the simplicial complex whose facets are the bases of $U_{n,k}$. A partial shelling…

Let $M$ be a matroid of rank $k$ on $n$ elements, and let $d4c9_p(M)$ denote its associated monoid. Write $\operatorname{Irr}(\mathcal{Q}_p(M))$ for the set of irreducibles of this…

For integers $m$ and $d$, and $0\leq i\leq d$, define … A finite sequence of real numbers is an $n$-sequence when the associated polynomial has real zeros of one common sign. The…

For integers $m$ and $d$, and $0\leq i\leq d$, define … A finite sequence of real numbers is an $n$-sequence when the associated polynomial has real zeros of one common sign. The…

Let $M$ be a 2-uniform matroid, given by a representation over the 2-uniform partial field. Basis-counting conjecture. There is a polynomial-time algorithm to count the number of b…

A 2-uniform matroid is a matroid in the class of matroids represented over the 2-uniform partial field, and a QU matroid is a quaternionic unimodular matroid. 2-uniform matroid con…