143 problems

For each integer $m\geq 4$, let $f_m$ and $h_m$ be the polynomials defined in the paper, and let the ULC inequality refer to the ultra-log-concavity inequality for their coefficien…

For $n=2$, write the Chern class in the Schur basis as … and expand each coefficient in the binomial basis: … A polynomial is binomially positive when all its binomial-basis coeffi…

Let $C$ be the $\mathbf{\ell}_1$ ball, let $p(w\mid\xi)$ be the posterior density under the uniform prior on $C$, and write $\operatorname{Cov}_{\mathrm{Uni}(C)}(w)$ for the covari…

Log-Concavity Genus Distribution conjecture. For every graph $G$, the genus polynomial is log-concave.

Let $A\subset\mathbb{Z}_{>0}$, let $p^{(k)}(n)$ be the generalized partition functions associated with $A$, and define … Assume that $A$ has properties $P_{k+1},P_{k+2},\dots$. Eve…

Let $\lambda$ be a partition and let $R=((m_1^{l_1}), \dots, (m_k^{l_k}))$ be a sequence of rectangular partitions. For an integer $N \geq 1$, define … Let $K_{\lambda,R}$ denote t…

Let $c_{\mu, \nu}^{\lambda}$ denote the Littlewood–Richardson coefficient associated to partitions $\lambda$, $\mu$, and $\nu$. For a partition $\alpha$ and a positive integer $N$,…

Let $\mu$ be a measure on a Boolean lattice. It is ULC+ when every measure obtained from it by imposing an external field and projecting is ULC. ULC implication conjecture. If…

In the random-cluster (RC) model and the competing urns model, let $X_e$ denote the indicator associated with an edge or site, and let $S$ be any subset of the relevant index set.…

For a measure on a Boolean lattice, its rank sequence is the sequence of probabilities of the possible values of the total number of occupied coordinates. It is ULC (ultra-log-conc…

Stanley's conjecture. The sequence $(h_0,h_1,\dots,h_s)$ is log-concave, meaning that $h_{i-1}h_{i+1}\le h_i^2$ for $2\le i\le s-1$.

Let $g_\ell(n)$ denote the number of subgroups of index $n$ in $\mathbb{Z}^\ell$, and write … Call a sequence $(a_k)_{k\in\mathbb{N}}$ log-concave at $k\geq 1$ when…

Let $P_n^\sigma(X)=\sum_{k=0}^n p_n^\sigma(k)X^k$ be the D'Arcais polynomial defined by … and call a sequence $(a_k)_{k\in\mathbb{N}}$ log-concave at $k\geq 1$ when…

Let $G$ be a graph, and let $g_k(G)$ denote the number of equivalence classes of 2-cell embeddings of $G$ on the orientable surface of genus $k$. The sequence…

Shifted-binomial log-concavity conjecture. The shifted polynomial

Let $\lambda=(2^{e_2},\ldots,m^{e_m})$ be a partition without parts equal to $1$, set $N:=|\lambda|$ and $M:=|\widetilde\lambda|$, and write … For each admissible pair…

Shifted-binomial log-concavity conjecture. For every partition $\lambda$ without parts equal to $1$, and every admissible index $i$, the sequence…

Let $\operatorname{den}(n,x)$ be the ordinary denominator of the reciprocal of the partition polynomial for $n$. Ballantine–Beck–Feigon–Maurischat's log-concavity conjecture. The s…

Generalized Glasby–Paseman sequence conjecture. This sequence satisfies the following properties:

Let $C$ be an integral projective locally planar curve over $\mathbf{k} = \mathbb{C}$, let $g=g(C)$ and $\widetilde{g}=\widetilde{g}(C)$ be its arithmetic and geometric genera, and…

Let $(C,0)$ be a reduced planar curve singularity over $\mathbf{k} = \mathbb{C}$, let $\delta$ be its $\delta$-invariant, and let $n_h(C,0)$, $0\leq h\leq\delta$, be the BPS invari…

For integers $n$ and $m$, let $Pi{n,m}$ be the locus of unordered set partitions of $[n]$ with block sizes bounded by $m$, and let $R( Pi{n,m})$ be its orbit-harmonics graded…

For each positive integer $n$, let $I(n)$ and $J(n)$ be the ideals in the polynomial ring ${\mathbb C}[{\mathbf x}_{\binom{[n]}{2}}]$ on variables indexed by the two-element subset…

Let $\lambda$, $\mu$, $\lambda'$, and $\mu'$ be partitions in $\mathbb{Z}^n$. For vectors $x,y\in\mathbb{Z}^n$, define … A symmetric polynomial is Schur nonnegative if it is a nonn…

Let $M=(E,\mathcal{I})$ be a matroid with rank $r_M$, and let $f_k(M)$ be its degree-$k$ independent-set polynomial … For $0<k<r_M$, the strengthened polynomial log-concavity conje…