114 problems

Chen's conjecture. The third-order Turán inequality for $p(n)$ is valid for all $n\geqslant94$.

Partition function subalgebra uniqueness conjecture. The equality

Let $W_n^\beta$ denote the partition function of the time-inhomogeneous branching random walk at inverse temperature $\beta$, and let $\beta_c$ be its critical parameter. In the ti…

Formal convergence conjecture. In the $x_i$ variables, the partition functions converge as formal power series expansions.

Let $p_r(n)$ denote the $r$-colored partition function, and define $L_{p_r}(d)$ and $M_{p_r,k}(d)$ analogously to the corresponding quantities for the partition and overpartition f…

Threshold conjecture. For $1\leq m\leq 11$ and $1\leq k\leq 5$, $\Delta^k p(n)$, respectively $\Delta^k\bar{p}(n)$, satisfies the Laguerre inequality of order $m$ whenever…

Let $k\geq 3$, and let $\Delta_k(n)$ denote the broken $k$-diamond partition function. For a sequence $\{\alpha_n\}_{n\geq 0}$, define its Jensen polynomial of degree $d$ and shift…

Let $G$ be a $d$-regular graph, and let $M_G(\lambda)=\sum_k m_k(G)\lambda^k$ be its matching partition function, where $m_k(G)$ is the number of matchings of size $k$. Write…

The perturbative partition function of supersymmetric Yang–Mills theory on suitable toric Sasaki–Einstein manifolds admits a factorisation into contributions associated with the to…

Let $G$ be a $d$-regular graph on $n$ vertices, and let $H_{d,n}$ be the graph consisting of $n/(2d)$ copies of $K_{d,d}$. For each integer $k$ with $1\le k\le n/2$, write $m_k(G)$…

MRF partition-function conjecture. Polynomial-time algorithms for computing the partition function of an MRF can be constructed under a weaker assumption than $$ .

Let $\operatorname{Sing}_{n,q}^{\mathrm{per}}$ and $\operatorname{Sing}_{n,q}$ be the singular sets for the quantum permanent and quantum determinant, respectively. For…

Let $r\ge2$, and let $P_r(n)$ and $\widetilde P_r(n)$ denote the two partition-counting functions defined earlier in the paper. Qin's punctual-partition generating-function conject…

Ratio-limit conjecture. The existence of the limit

Let $e_{j}^{(\text{BIZ})}(t)$ denote the coefficients in the asymptotic expansion of the partition function $Z_{N}^{(\text{BIZ})}$, with $j$ indexing the expansion terms. Bessis–It…

Partition-function conjecture. The partition function of the ${\cal N}=4$ U(1) gauge theory in noncommutative $\mathbb{R}^4$ with topological term $\tau\int F\wedge F$ is

Let $G$ be a finite graph of maximum degree $\Delta$, let $Z(w)$ denote its hard-core lattice-gas partition function with equal fugacity $w$, and let $D_\Delta$ be a complex domain…

Chunk-energy sufficiency conjecture. Restricting weights to depend on $\mathbf{x}$ only through the chunk-energy vector is sufficient without loss of optimality for the step-$d$ sl…

Let $\Delta_k$ denote the optimal asymptotic MSE constant within the generalized grouping (GG) family for groups of size $k$. Monotonicity conjecture. … i.e., the optimal asymptoti…

Reconstruction conjecture. If

Let $\Delta_k(n)$ denote the broken $k$-diamond partition function. Dong, Ji and Jia's results for $k=1$ or $2$ show that $\Delta_k(n)$ satisfies the Turán inequalities of any orde…

Classification conjecture. The only primes $p>2$ for which $\mathfrak{p}(n,\pm(\tfrac{\cdot}{p}))$ vanishes on some arithmetic progressions modulo $2p$ are $p=5$ and $p=17$.

Let $pl(n)$ denote the plane partition function, counting plane partitions of $n$. A perfect power is an integer of the form $a^b$ with integers $a,b>1$. The plane-partition perfec…

For each positive integer $n$, let $p(n)$ denote the partition function, and let $\{r\}$ denote the fractional part of a real number $r$. The equidistribution conjecture. For each…

For each fixed integer $d\geq 0$, define … and let $M_k(d)$ be the largest $n$ for which $p(n)$ is within $d$ of a $k$th power. Merca–Ono–Tsai's stabilization conjecture. There is…