254 problems

Let the $n$th row of Pascal's triangle be the sequence … A sequence $(a_k)$ is log-concave if $a_k^2\geq a_{k-1}a_{k+1}$ for every relevant index $k$. A sequence is infinitely log-…

For each $n$, let $a_{n,k}$ be the number of permutations in ${\mathfrak{S}}_n$ whose longest increasing subsequence has length $k$. A sequence $(a_1,a_2,\dots,a_N)$ of positive re…

Let ${\mathcal{UZ}}_{n,m,r}$ be the upper rook placement locus and let $R({\mathcal{UZ}}_{n,m,r})$ be its orbit harmonics ring, with the natural…

Fix $p\in(0,1)$ and $b>1$. For $a>0$, let $q(a)=I^{-1}_p(a,b)$ be the unique value satisfying … and define $\phi(a)=-a\log q(a)$. Concavity conjecture. The function $\phi$ is conca…

Let $M$ be a matroid of rank $r(M)$, with characteristic polynomial … The integers $w_i(M)$ are the Whitney numbers of the first kind. Heron–Rota–Welsh conjecture. The sequence … o…

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 $G$ 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…

Okounkov's conjecture. The function

Let $\mathsf{M}$ be a matroid, and write $Q_{\mathsf{M}}(t)=\sum_i a_i t^i$. A sequence is log-concave with no internal zeros if $a_i^2\geq a_{i-1}a_{i+1}$ whenever defined, and it…

Let $G$ be a group and $(V_1,\dots,V_N)$ a sequence of $G$-modules. The sequence is $G$-log-concave if there are $G$-module embeddings … for every $1<i<N$. For a graded $G$-module…

Let $p_k(n)$ denote the number of $k$-coloured partitions of $n$. Chern–Fu–Tang's conjecture. For $1\leq\ell<n$ and $k\geq 2$, except for $(k,n,\ell)=(2,6,4)$, one has … This conje…

Let $G$ be a finite graph and let $\chi_G(x)$ be its chromatic polynomial. Read's conjecture. The absolute values of the coefficients of $\chi_G(x)$ are unimodal. This conjecture w…

Let $\operatorname{{\mathsf M}}$ be a matroid of rank $r+1$, with characteristic-polynomial coefficients…

Gedeon–Proudfoot–Young's conjecture. For every equivariant matroid $W\curvearrowright M$, both the equivariant characteristic polynomial and the equivariant Kazhdan–Lusztig polynom…

Let $X$ and $Y$ be independent identically distributed log-concave random variables in dimension $1$, and let $h$ denote differential entropy. Ball–Nayar–Tkocz conjecture. The map…

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…

For each positive integer $n$, let $b_{n,k}$ denote the number of involutions in ${\mathfrak{S}}_n$ whose longest decreasing subsequence has length $k$, for $1\leq k\leq n$. A fini…

Let $M$ be a matroid, and let $a_0,a_1,\ldots,a_n$ be the absolute values of the coefficients of its characteristic polynomial. A finite nonnegative sequence is log-concave if…

Almost-log-concavity conjecture. The Hilbert series of $\mathbb{C}[\mathbf{x}_{n\times n}]/I_{\mathfrak{S}_{n,r}}$ is almost log-concave: log-concavity may fail for some small indi…

Let $\mathrm{M}$ be a matroid, and define $h_i(\mathrm{M})$ by … where $f_i(\mathrm{M})$ counts independent subsets of size $i$. Dawson's conjecture. For any $i$ and any matroid…

For $k\in\mathbb{N}$, consider the Pascal-triangle column sequence $\{\binom{n+k}{k}\}_{n=0}^{\infty}$. McNamara–Sagan conjecture. This sequence is infinitely log-concave, meaning…

Let $P_m(a)=\sum_{i=0}^m d_i(m)a^i$ be the Boros-Moll polynomial. For $1\leq i\leq m$, define $u_i(m)=i(i+1)\bigl(d_i(m)^2-d_{i-1}(m)d_{i+1}(m)\bigr)$, with the coefficient sequenc…

Shifted-binomial log-concavity conjecture. The shifted polynomial

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 $cmu$ be an even log-concave measure on $cmathbb{R}^d$, let $clambdacin(0,1)$, and let $K,Lcsubsetcmathbb{R}^d$ be convex and centrally symmetric. Gardner–Zvavitch's dimensiona…