23 problems

Let $C$ denote the set of small conditionals, let $J-1$ be the dimension of the underlying lattice, and consider the coefficient matrix of the Diophantine equation governing the ma…

Let $R=(r_1,\ldots,r_m)$ and $C=(c_1,\ldots,c_n)$ be positive integer vectors satisfying … A contingency table with margins $(R,C)$ is an $m\times n$ non-negative integer matrix wi…

Békessy–Békessy–Komlós conjecture. The relative error of this approximation is $O\left(N^{-1}\right)$ as $N\longrightarrow +\infty$.

Contingency-table norm conjecture. For arbitrary table dimensions $m$ and $n$, table sum $\Sigma$, and row and column sums $r_k$ and $c_p$, this multi-decomposition is non-degenera…

Let $\boldsymbol \pi \in \Delta_{pq-1}$ be the probability table of a $p\times q$ contingency table in the independence model, so that its entries are $\pi_{ij}$ and the first row…

Pak–Panova's conjecture. With high probability,

Logarithmic maximum-entry conjecture. With high probability,

Barvinok's marginal-distribution conjecture. The limiting marginal distribution of an entry is Bernoulli when $b=1$ and geometric when $b=\infty$, with mean given by the correspond…

Barvinok's phase-transition conjecture. There is a sharp phase transition in the behavior of the typical table for some $\lambda$ between $2$ and $3$, and the uniformly random cont…

Erdős–Gallai–tameness conjecture. The quadratic-gap Erdős–Gallai condition is equivalent to $\delta$-tameness in the sense of Barvinok and Hartigan.

Let $\mathcal{M}(\mathbf{r},\boldsymbol c)$ be the set of $m\times n$ contingency tables with nonnegative integer entries and prescribed row margins $\mathbf{r}=(a_1,\ldots,a_m)$ a…

Let $\mathbf d=(d_1,\ldots,d_n)\in\mathbb N^n$ satisfy $0\le d_i\le n-1$, and let $c(\mathbf d)$ be the number of simple graphs on vertices $v_1,\ldots,v_n$ with $\deg(v_i)=d_i$ fo…

Let $\mathbf a\in\mathbb N^r$ and $\mathbf b\in\mathbb N^s$ have common size $n=|\mathbf a|=|\mathbf b|$, and let $\operatorname{CT}(\mathbf a,\mathbf b)$ be the number of continge…

Let $\mathbf a=(a_1,\ldots,a_r)\in\mathbb N^r$ and $\mathbf b=(b_1,\ldots,b_s)\in\mathbb N^s$ satisfy $|\mathbf a|=|\mathbf b|$, and let $\operatorname{CT}(\mathbf a,\mathbf b)$ de…

Non-emptiness conjecture. The polytope $\mathcal{P}_{I,J,1}$ is non-empty if and only if, for every such $A$ and $B$, all four inequalities hold:

Compatibility conjecture. The displayed condition is equivalent to compatibility of the margins for all $r\times s\times2$ examples.

Let $p_{ijk\ell}$ be entries of a $2\times2\times2\times2$ table sampled uniformly from the probability simplex, with $i,j,k,\ell\in\{0,1\}$ and $p_{ijk\ell}\geq0$. Consider the tw…

Let $M$ be the matrix in the equation defining the space of tables, and let $A$, $B$, and $C$ denote the index sets used to specify the table and its conditionals. A total Markov b…

Consider two-way contingency tables under the independence model, with some cell counts subject to upper bounds and other cell counts unbounded. In the unbounded case, the Markov b…

Let $m$ covariates index combinations of levels, and let ${\cal B}_{\Lambda(A_1 \otimes \dots \otimes A_m)}$ be the set of moves … Consider the subset for which every element of…

Let $m$ covariates index combinations of levels, and let $\Bj=(j_1,\dots,j_m)$ denote such a combination. Let $\bm{e}_{\Bj}$ be the array with $1$ at cell $(1,\Bj)$ and $-1$ at cel…

Let $m$ covariates index combinations of levels, and let ${\cal B}_{\Lambda(A_1 \otimes \dots \otimes A_m)}$ be the set of moves defined by arrays of the form … where $\Bj_i$ are c…

Let $M(m,s;n,t)$ denote the number of $m\times n$ nonnegative integer matrices with every row sum equal to $s$ and every column sum equal to $t$, where $ms=nt$. Under the asymptoti…