55 problems

Additional enumeration conjectures. The following identities are conjectured:

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

Let $\mathcal{J}_n$ be the Jones monoid, let $E_n$ be its set of idempotents, let $\operatorname{rank}(\alpha)$ denote the rank of $\alpha\in E_n$, let $e_n=|E_n|$, and let $d_n^*$…

Decreasing normalized count conjecture. The sequence $\{a_k,\ k=10,11,12,\dots\}$ is decreasing. This conjecture concerns the growth of the number of inequivalent binary self-dual…

Determinant-evaluation conjecture. The following identities should hold:

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…

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…

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…

Enumeration conjecture.

Let $G=G(n)$ be a sequence of $d$-regular graphs of order $n$, with even degree $d=d(n)$. Let $\rho(G)$ denote the residual entropy, … where $\operatorname{EO}(G)$ is the number of…

Frozen-corner determinant conjecture. The number $B_{n,s}$ may be evaluated as

Behrend–Fischer–Koutschan symmetry conjecture. Given $n$, for any $\rho$ and $\tau$, we have

Let an off-diagonally symmetric alternating sign matrix (OSASM) be an alternating sign matrix invariant under reflection in the off-diagonal axis, and let $n$ be a nonnegative inte…

Let $\mathcal{Q}_n$ denote the set of Stirling permutations of order $n$, and let $\operatorname{flat}_3(\mathcal{Q}_n)$ be the set of those with exactly three runs. Three-run enum…

For each $d$, let $t_{n,d}$ denote the number of tiles contained in $[n]^d$, and let $v_n$ denote the volume parameter used in the paper. Polynomial-factor conjecture. There is a c…

Let $\pi$ be a plane partition inside an $(a,a,c)$-box. It is quasi transpose complementary of second kind (QTC2) if it is transpose complementary except along the diagonal, namely…

Let $\operatorname{qspp}(a,c)$ denote the number of quasi symmetric plane partitions inside an $(a,a,c)$-box. Let $p_a(c)$ be a polynomial in $\mathbb{Q}[c]$. Quasi symmetric plane…

Enumeration conjecture. For every $n\geq 0$,

Enumeration conjecture. For every $n\geq 0$,

Fishburn–classical enumeration conjecture. For every $n\geq 1$,