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…

Let $A$, $B$, and $C$ be three log-concave nonnegative sequences without internal zeros. Write $A*C$ and $B*C$ for their convolution sequences, and let $A\sim_{\mathrm{w}} B$ mean…

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…

For $0\le r\le\min\{m,n\}$, let ${\mathcal{Z}}_{n,m,r}$ be the locus of $n\times m$ rook placements with exactly $r$ rooks, and let $R({\mathcal{Z}}_{n,m,r})$ denote its orbit harm…

Chen's conjecture. For any fixed $n$, the sequence $\ell_{n,1},\ell_{n,2},\ldots,\ell_{n,n}$ is log-concave. Equivalently, the distribution of $\ell_n(\sigma)$ for a uniformly chos…

Let $\mathrm{M}$ be a matroid on a finite set $E$, and write $r+1$ for its rank. Let $\chi_{\mathrm{M}}(\lambda)$ be its characteristic polynomial, and let $w_k(\mathrm{M})$ be the…

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…

Shifted-binomial log-concavity conjecture. The shifted polynomial

Chan–Pak conjecture. For every $A\subseteq P$, the sequence of probabilities of the minimum position is log-concave:

Let $\mathsf{PF}_n$ be the set of parking functions of length $n$, and define the sum statistic by $\mathsf{\sum}(\pi)=a_1+a_2+\ldots+a_n$ for…

Brenti's conjecture. The polynomial $W_{P,\omega}$ is log-concave with no internal zeroes.

Let $d_\ell(m)$ denote the coefficient of $x^\ell$ in the Boros-Moll polynomial $P_m(x)$, for $0\leq \ell\leq m$. A sequence is $k$-log-concave when its first $k$ iterates under th…

Heim–Neuhauser conjecture. For all $x\geq 2$,

Chern–Fu–Tang conjecture. Let $n>m\geq 1$ and $k\geq 2$. Except for $(k,n,m)=(2,6,4)$,

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