114 problems

Let $\mathfrak A_n$ denote the alternating group, let $f_n$ be the function defined by the paper's Algorithm 1, and let $\leq$ denote the Bruhat order on $\mathfrak A_n$. Bruhat-or…

Let $\Omega_n$ be the set of doubly-stochastic $n\times n$ matrices, and define … The authors exhibit doubly-stochastic matrices $A$ for which…

For $k,n\geq 1$, let $\Lambda(k,n)$ be the set of $n\times n$ matrices with nonnegative integer entries and all row and column sums equal to $k$, let…

Luschny's permanent conjecture. Luschny conjectured that

Let $A,B e 0$ be Hermitian positive semidefinite matrices of the same size, and let $Acirc B$ denote their Hadamard product. Chollet's conjecture. … The conjecture is a permanent a…

Let $A$ and $B$ be square matrices of the same size. Their permanents are multiplicative when … Marcus–Minc conjecture. Multiplicativity holds only when both $A$ and $B$ are produc…

Let $M$ be an $n\times n$ nonsingular matrix. The permanent rank of a matrix is the size of its largest square submatrix with nonzero permanent. Kahn's Permanent conjecture. The pe…

Pate's conjecture. The largest eigenvalue of $\mathscr{C}_k(A)$ is $\operatorname{per}(A)$. This conjecture is refuted; the paper states that its $k=1$ case has an $8\times8$ count…

Let $H$) be an $n\times n$ positive semi-definite Hermitian matrix, and let $\pi(H)$ denote its Schur power matrix, whose entries are indexed by permutations in $S_n$. Soules's per…

Let $A=(a_{ij})$ be a real $n\times n$ matrix with non-negative entries, and let $J$ denote the all-ones matrix. Assume that the entries of $A$ are weakly increasing down columns.…

Caputo–Carlen–Lieb–Loss conjecture. The maximum of the permanent of $A$ is attained either by $I$, with permanent $1$, or by the matrix $n^{-1/p}J$, with permanent $n!/n^{n/p}$. Eq…

The optimality conjecture. The particular polynomial $S_{n,n-m}$ is optimal in the sense that

The Asymptotic Lower $r$-Permanent Conjecture. Under these hypotheses,

Let $A$ be an $n\times n$ doubly stochastic matrix, and let $l$ be a positive integer. Its $l\times l$ permanental minors are the permanents of the submatrices obtained by choosing…

Let $M_{d,d}$ be the generic $d\times d$ matrix over a field $k$, and let $I$ be the ideal of $d\times d$ permanents of the $d\times2d$ matrix $(M_{d,d}\mid M_{d,d})$. De Loera's r…

Let $S_n$ be the symmetric group on \\{1,\ldots,n\}. For an $n\times n$ matrix $A=(a_{ij})$, define … Let $K_n$ be the set of all real nonnegative $n\times n$ matrices whose entr…

Let $\mathcal{C}:[0,1]\times[0,1]\to\mathbb{R}$ be twice continuously differentiable up to the boundary, symmetric in its two variables, zero on the diagonal, and satisfy…

Asymptotic uniformity conjecture. For every $x\inFbb{F}_{q}$,

Random permanent zero-probability conjecture.

Ryser's conjecture. The permanent of $J^*$ is maximal in $\mathscr{U}(v,k)$.

A $d$-dimensional polystochastic matrix of order $n$ is a nonnegative array whose sums over every line are equal to $1$. Its permanent is … where $D(A)$ is the set of diagonals of…

Navin-type conjecture. It holds that

For each positive integer $n$, let $A$ range uniformly over the matrices in $\mathbb{F}_3^{n\times n}$, and let $z(n)=|\{A\in\mathbb{F}_3^{n\times n}:\operatorname{perm}(A)=0\}|$.…

Permanental Sihong polynomial conjecture. The polynomial above is totally nonnegative for all $n$. This would provide a permanental analogue of the known totally nonnegative determ…

Let $\Omega_n$ denote the set of all doubly stochastic matrices of order $n$. For a matrix $A$, let $\sigma_k(A)$ denote its $k$th-order subpermanent. Holens–Doković conjecture. If…