13 problems

the observed rank-divisibility conjecture. The numbers of partitions of $An+B$ into two part sizes for which $rk_2(\lambda)$ is odd and for which $rk_2(\lambda)$ is even are both d…

Let $(\mX,Y)$ be the random pair underlying the rank-based Azadkia–Chatterjee statistic $\xi_n$, and let $\sigma_2^2$ denote the variance term appearing in the paper's limiting dis…

Let $K$ be fixed. Let $U$ be the population eigenvector matrix and let $\widehat{U}$ be the corresponding sample eigenvector matrix. For a matrix $M$, write…

Let $u(m,n)$ denote the number of strongly unimodal sequences of weight $n$ and rank $m$, and let $N(0,n)$ denote the auxiliary counting function used in the paper. Rank-difference…

Let $N(k,11,n)$ and $M(k,11,n)$ denote the rank and crank counting functions modulo $11$. For integers $a_k,b_k$ with $0\leq k\leq5$, suppose … For a residue $m$ modulo $11$, defin…

Let $N(r,11,n)$ and $M(r,11,n)$ denote the numbers of partitions of $n$ with rank and crank, respectively, congruent to $r$ modulo $11$. For each residue $m$ modulo $11$, write…

Let $N(m,n)$ denote the number of partitions of $n$ with rank $m$. Eventual rank unimodality conjecture. For $n\geq39$ and $0\leq m\leq n-3$, one has … By the symmetry…

Rank-ordering conjecture. For every integer $n\geq 11$, these inequalities and equalities hold.

Let $n$ be a nonnegative integer, and let $\mathcal{D}_{3}^{0}(n)$ denote the number of 3-marked odd Durfee symbols of $n$. Andrews's conjecture. $\mathcal{D}_{3}^{0}(n)$ is even i…

For bivariate random variables $X$ and $Y$, let $\tau^*(X,Y)$ denote the Bergsma–Dassios sign-covariance defined through concordant and discordant quadruples. Bergsma–Dassios' cons…

Let $\boldsymbol{H}$ be the hat matrix of a regression model, let $\boldsymbol{\boldsymbol{\epsilon}}=(\epsilon_1,\ldots,\epsilon_n)$ be a vector of independent identically distrib…

Let $X$ and $Y$ be real random variables, and define … where $(X_i,Y_i)$ are iid copies of $(X,Y)$ and … The paper proves the claim when $X$ and $Y$ are real continuous random vari…

For an integer $k\geq 1$, let $M_{2k}(n)$ and $N_{2k}(n)$ be the $2k$-th crank and rank moments, respectively, let $p(n)$ be the partition function, and define … The constants sati…