17 problems

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…

Generalized Möbius polynomial valuation conjecture. The generalized Möbius polynomial is a valuation on matroids.

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 $k\le n$, 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,…

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 $(\pi)$ be a family of permutations, and let $\mu(1,\pi)$ denote the Möbius function of the permutation pattern poset from the minimum permutation $1$ to $\pi$. Brignall–Marcha…

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 conjecture. Even without the assumption that $L_G$ is a lattice, if the maximal tubings containing $\mathcal{X}$ form an interval $[\mathcal{Y},\mathcal{Z}]$, then

Let $[H,G]$ be an interval of finite groups. Its dual Euler totient is … An interval $[H,G]$ is Boolean when its subgroup-lattice interval is a Boolean lattice. Nonvanishing dual E…

The extremal Möbius-value conjecture. For every separable permutation $\pi$ of length $n\ge3$, the maximum value of $\mu(\sigma,\pi)$ is attained by a permutation of this form, wit…

Let $P$ be a poset and let $P^*$ be the Kleene closure of the alphabet consisting of the elements of $P$, ordered by generalized subword order: $u\leq w$ when a subword…

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

The 132-avoidance conjecture. If every permutation in $[\sigma,\tau]$ avoids the pattern $132$ (equivalently, if $\tau$ avoids $132$), then

Let $P:= \{ a,\ b,\ c,\ |\ a<c,\ b<c \}$, and let $P^{*}$ be the poset of finite words over $P$ with generalized subword order. Let $\mu$ be a Mbius function of $P^{*}$, and let…