59 problems

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.…

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

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…

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

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 $\mathbb{T}$ denote the unit circle, let $X$ be sampled from the complex Gaussian ensemble $\mathcal{G}=\mathcal{N}_{\mathbb{C}}(0,1)^{n\times n}$, and let…

Let $p$ be an odd prime, let $\delta_{jk}$ be the Kronecker delta, and let $\operatorname{per}(M)$ denote the permanent of a matrix $M$. Absolute-difference permanent conjecture. ……

For a positive integer $n$, let $[j-k+\delta_{jk}]_{1\leqslant j,k\leqslant n}$ denote the $n\times n$ matrix whose $(j,k)$ entry is $j-k+\delta_{jk}$, where $\delta_{jk}$ is the K…

Extremal permanent conjecture. For all $M\in S(n,k)$ such that $T(M)=1$,

Let $\Pi$ be a partition of $n$, and let its Ferrers diagram be the associated diagram. For $A\in{\cal H}_n$, let the block of the Schur power matrix $\pi(A)$ associated with $\Pi$…

Let $A\in{\cal H}_n$, and let $A(i|j)$ denote the matrix obtained by deleting row $i$ and column $j$. Pate's inequality challenge. … The source presents this as a consequence of th…

Let $\mathcal{S}_n$ be the symmetric group and let $c:\mathcal{S}_n\to\mathbb{C}$ satisfy … for every complex-valued function $x$ on $\mathcal{S}_n$. For $A=[a_{i,j}]\in{\cal H}_n$…

Let ${\cal H}_n$ be the cone of positive-semidefinite Hermitian matrices, and let $A\circ B$ denote the Hadamard (entrywise) product. Chollet's conjecture. For $A,B\in{\cal H}_n$,…

Soules's Schur power matrix conjecture. The permanent $\mathop{\rm per} A$ is the maximum eigenvalue of $\pi(A)$. The conjecture is refuted: the source gives a positive-semidefinit…

Marcus's block permanent conjecture. Then

Let $A$ be partitioned into square blocks as … where $A$ is positive-semidefinite Hermitian. Marcus–Newman conjecture. For this partition, … The conjecture was solved by the coroll…

Let $n$ be a positive integer and define the $2n\times 2n$ matrix … The Luschny permanent conjecture. … Here $H_1,H_3,H_5,\ldots$ are the Genocchi numbers of the second kind, or me…