18 problems

A permutation matrix is a square $0$-$1$ matrix with exactly one $1$ in each row and column. The saturation function $\operatorname{sat}(n,P)$ is the minimum number of $1$ entries…

Let $I_k$ be the $k\times k$ identity matrix. For an $n\times n$ matrix, let $\mathrm{M}(n,I_k)$ denote the maximum possible number of $1$-entries in a strongly $I_k$-forcing matri…

Cigler's conjecture. If a permutation-like matrix group contains a maximal cycle, then it is a permutation matrix group.

Let $A\in \mathcal{CP}(n)$ and let $L$ be its corresponding polygonal line. If $n$ is even, consider the endpoints of $L$ ordered by increasing $x$-coordinate. Endpoint monotonicit…

Let $PL_d$ denote the set of permutation-like matrices of degree $d$, and let $A\in PL_d$ be a permutation-like matrix with $d>1$. A permutation-like matrix eigenvalue conjecture.…

Let $DS_n$ be the set of complex numbers that occur as eigenvalues of $n$-by-$n$ doubly stochastic matrices. For permutations $P,Q$, let $I$ denote the identity permutation matrix,…

Let $DS_n$ be the set of complex numbers that occur as eigenvalues of $n$-by-$n$ doubly stochastic matrices, and let a convex combination of permutation matrices mean a matrix of t…

Let $D$ be a doubly stochastic matrix of order $n$, meaning that it has nonnegative real entries and every row and column sums to $1$. Its binary diagonal X-ray is the vector…

Let $S_{2n}$ be the symmetric group, and let $l_t(\pi)$ denote the Toeplitz characteristic of a permutation. A fixed-point-free involution is a permutation $\pi \in S_{2n}$ satisfy…

A binary diagonal X-ray of an $n \times n$ permutation matrix is its diagonal X-ray with entries in $\{0,1\}$. A tournament is a loopless directed graph in which exactly one of…

A $d$-dimensional $0$-$1$ matrix $A$ contains another $0$-$1$ matrix $P$ if $A$ has a submatrix that can be transformed into $P$ by changing any number of ones to zeros; otherwise,…

Let $\mathcal{C}_{N+1}$ be the set of signed permutation matrices of size $2N\times2N$ invariant under diagonal and anti-diagonal reflections and avoiding $(-2,-1)$. Let $B(c)$ be…

Let $\mathcal{C}_n$ be the set of signed permutation matrices of size $2(n-1)\times2(n-1)$ invariant under diagonal and anti-diagonal reflections and avoiding the pattern $(-2,-1)$…

Archimedean distribution conjecture. The nonzero entries of this partial permutation matrix are distributed according to the Archimedean distribution.

Let $P$ be a finite permutation matrix of bandwidth $w>0$. A greedy bubble matrix of $P$ is a product of reducing and overtaking swaps of $P$ such that $P$ and the resulting matrix…

Let $P$ be a finite permutation matrix of bandwidth $w>0$, where bandwidth $1$ means tridiagonal. A bandwidth--1 permutation matrix has nonzero entries only on the main diagonal an…

Let a permutation matrix have bandwidth $w$, meaning that its nonzero entries lie within $w$ diagonals of the main diagonal. A bandwidth-$1$ permutation is a product of mutually no…

Let ${\mathfrak S}_n$ be the symmetric group on $n$ elements. For $\sigma\in{\mathfrak S}_n$, let ${\operatorname{Des}}(\sigma)$ be its set of descents, and let…