12 problems

Möbius support conjecture. For any base polymatroid $\mathscr{P}$, the Möbius support $\mu\text{-supp}(\mathscr{P})$ is a generalized polymatroid. The theorem preceding this conjec…

Mann's conjecture. The quantity $|\mu(H,G)|$ is bounded by a polynomial function of $|G:H|$, and the number of subgroups $H$ of $G$ of index $m$ satisfying $\mu(H,G)\neq 0$ grows a…

For $k\le n$, let $\pi_{n,k}$ be the subpermutation of $\kappa_n$ induced by the values $n+1,n+3,\dots,n+2k-1,1,3n+1,2,3n+2,\dots,n,4n,n+2,n+4,\dots,n+2k$. In particular,…

Möbius–Chebyshev conjecture. For all integers $i\leq j$, $\mu(a^i,c^j)$ is the coefficient of $x^{j-i}$ in $T_{i+j}(x)$.

Let $P$ be a locally finite poset, and let $\mu_P(x,y)$ denote its Möbius function. Say that $P$ has the Möbius uncertainty property when this property holds as defined in the pape…

Let $\lambda$ be a partition with repeating elements, and let ${\mathbb D}_{\mathsf{S}}(\lambda)$ be the associated poset of staircase matrices. Denote its Möbius function by…

Let $A\in\T\left(\mathbf{1}^n\right)$ and let $\mu$ be the Möbius function for the corresponding Tesler poset. O'Neill's conjecture. … This conjecture concerns the growth of Möbiu…

Even-odd-drop and D-cycle correspondence conjecture. For every finite subset $A\subset\mathbb{Z}_{>0}$, the number of cycles on $A$ with only even-odd drops is equal to the number…

For each positive integer $n$, let $\kappa_n\in {\mathcal S}_{4n}$ be the permutation … It is $321$-free and can be split into four quadrants, each an increasing subsequence of len…

Let $n>0$ and let $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_r)$ be a composition of $n$ into $r$ parts. Let $H_{\alpha}(x,y)$ be the polynomial defined on order ideals of the releva…

Möbius function formula conjecture. If $\pi$ begins with $12$, ends with $(n-1)n$, or contains a triple adjacency, then

Principal-interval Möbius conjecture. If $\tau$ avoids the pattern $132$ (equivalently, $312$, $213$, or $231$), then