112 problems

Let $\mathfrak{S}_n$ be the symmetric group on $[n]=\{1,\dots,n\}$. For a fixed permutation $\pi$, let $\operatorname{Av}_n(\pi)$ be the set of permutations in $\mathfrak{S}_n$ tha…

Let $\mathfrak{S}_n$ be the set of permutations of $[n]$, let $\mathcal{S}$ be the stack-sorting operation, and let $\mathfrak{S}_n^2$ denote the permutations sorted by applying…

For a fixed $n\in\mathbb{N}$, let $f_n(p)$ denote the number of isomorphism classes of finite $p$-groups of order $p^n$. Higman's PORC conjecture. There exist a fixed integer $N$ a…

Chan–Robbins–Yuen conjecture. The volume of $\mathcal{F}_{K_n}$ is given by the product above.

Fine-sequence conjecture. For every $n\geq 1$,

Grassmannian enumeration conjecture. For every $n\geq 1$,

Let $M_{n,r}$ denote the number of matroids on a fixed $n$-element set having rank $r$. Welsh's conjecture. The sequence $(M_{n,r})_r$ is unimodal in rank. The paper notes that the…

Let $[m]$ denote the list $1,\dotsc,m$, and let $F_{[m],[n]}(x)$ be the polynomial defined through the factorisation and recurrence relations for the tiling generating functions of…

Determinant-evaluation conjecture. The following identities should hold:

Profile quasi-polynomial conjecture. If $R$ has bounded signature or finite kernel and $\varphi_R$ is bounded by some polynomial, then $\varphi_R$ is a quasi-polynomial.

Let $P_n(x)$ be the generating function for alternating sign matrices invariant under flips in the vertical and horizontal axes, with the weight defined in the source. Such matrice…

Consider self-complementary cyclically symmetric plane partitions whose Ferrers graph lies in ${\cal X}_{2n}=[1,2n]\times[1,2n]\times[1,2n]$, with a part $a_{ij}$ called special wh…

Let $Q_n(x,y)$ be the generating function for quarter-turn-symmetric $n$ by $n$ alternating sign matrices, and let $H_n$ and $A_n$ denote the half-turn-symmetric and unrestricted a…

Let $F_n(x)$ be the weight-generating function for flip-symmetric alternating sign matrices, with the exponent recording the number of $-1$ entries in the first half of the columns…

Let $A_n(x,y)$ be the weight-generating function for $n$ by $n$ alternating sign matrices, where a matrix with $r$ entries equal to $-1$ and its top-row $1$ in position $s$ has wei…

Let $\Psi^{(L)}(\tau)$ be the homogeneous open boundary qKZ solution, let $L$ be its size, and let $N_8(L+1;\tau)$ and $A_V(L+1;\tau)$ denote the generating polynomials for the cor…

Let $x$ be the fugacity associated with the walk length, let $x_c$ be the critical fugacity, and let $\mu$ be the growth constant for self-avoiding walks on the square lattice. For…

Odd half-turn enumeration conjecture. The number $A_{\mathrm{HT}}(L)$ is

Boundary-correction structure conjecture. For every $d\ge4$ and every $m,n$ with exactly one of them below $d-1$, the count $E_d(m,n)$ differs from $p_d(m,n)$ by a polynomial corre…

Let $E_d(m,n)$ be the count and let $p_d(m,n)$ denote its closed-form polynomial in the threshold regime. For $s=\min(m,n)<d-1$, define the boundary correction, with the other vari…

Let $d\ge5$, let $(a,b)$ satisfy $a+b=d$ and $a\le b$, and let $R_{(a,b)}(m,n)$ be the interior remainder. Denote by $[m^{d-3}]R_{(a,b)}(m,n)$ the coefficient of $m^{d-3}$; by symm…

Let $d\ge5$ and let $(a,b)$ be a split with $a\le b$. Write $R_{(a,b)}$ for the remainder after removing the single-side contributions from the count associated with the split. Deg…

Greedy $m$-Tamari intervals are intervals in the greedy $m$-Tamari poset, and planar $(m+1)$-constellations are planar maps with the relevant face-degree profile. Bousquet-Mélou an…

Enumeration conjecture.

Let $\operatorname{Av}(2413,3142)$ be the class of permutations avoiding $2413$ and $3142$, and let $S_n$ denote the large Schröder number of semilength $n$. Shapiro–Getu conjectur…