21 problems

Let $P_n$ be the algebra containing the element $f$, let $\mathrm{Pfaf}'_D$ denote the diagram pfaffinants, and let $A_{\lambda/\mu}$ be a generalized $Q$-Jacobi-Trudi matrix assoc…

Pfaffian product conjecture. At $\alpha=q^{1/2}t^{1/2}$,

Let $n$ be the number of bodies and consider a non-singular collinear configuration with the Newtonian potential, whose associated inverse-problem matrix produces the resulting Pfa…

Let $n$ be a positive integer and let $p$ be a non-negative integer. For a partition $\lambda=(\lambda_1,\ldots,\lambda_n)$, let $m_r(\lambda)$ denote the number of parts equal to…

Let $n$ be a positive integer and let $p$ be a non-negative integer. For a partition $[200~\lambda=(\lambda_1,\ldots,\lambda_n)$, let $m_r(\lambda)$ denote the number of parts equa…

Pfaffian generator conjecture. A Pfaffian generator exists if and only if either ${\mathscr{Y}}_{N,\ell}^{+}(\sigma)$ has $N$ even and $\ell$ odd, or…

Let $X\in\mathfrak{so}(2N)$ be a nilpotent element corresponding to a special D-partition $P=(p_1\geq p_2\geq\dots\geq p_{2m})$ of $2N$, and let $g\in\mathfrak{so}(2N)$. Define … F…

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…

Generalized product formula conjecture. There exists a sequence of polynomials $o_n(k,t)\in\mathbb{Z}[k,t]$ such that

Let $\mathrm{Cat}(2,3)$ be the catalecticant variety, let $\mathbb{P}\\,\mathrm{Cat}(2,3)^{-1}$ denote its reciprocal variety, and let a cubic Pfaffian mean the Pfaffian of a…

Let $C_1$ denote the rank-one locus of the relevant catalecticant variety, let $A$ be a point on $C_1$, and let $\phi(A)$ be its image under the map $\phi$. Seven-Pfaffian conjectu…

Let $S$ be the symplectic manifold with symplectic form $\Omega$, let $\chi$ denote the relevant matrix of logarithmic vector fields, and let…

Asymptotic Pfaffian-sum conjecture. A general form of degree $k$ can be expressed as a sum of $s$ pfaffians of such matrices, for

Let $n$ be a positive integer, let $x_1,\dots,x_{2n}$ be variables, and let $P_{\lambda}(x_1,\dots,x_{2n};t)$ be the Hall–Littlewood polynomial indexed by a partition $\lambda$. Wr…

Let $n$ and $k$ be positive integers. Define the Motzkin-triangle entries by … For a $2n\times 2n$ skew-symmetric matrix, let $\operatorname{Pf}$ denote its Pfaffian. Generalized M…

Let the representations in Eqs. (2.11), (5.3), and (5.4) be the representations shared by random matrix ensembles with Dyson index $\beta=2$, and let the corresponding orthogonal p…

Let $n$ be a positive integer, and consider the Laplacian-parametrized $n\times n$ matrix and symmetric-matrix Cayley identities, with Bernstein–Sato polynomials $b(s)$. Minimality…

Let $a_n=\frac{1}{2n+1}\binom{3n}{n}=\frac{1}{3n+1}\binom{3n+1}{n}$. Vertically symmetric alternating-sign-matrix Pfaffian conjecture. For every integer $n\geq1$, … The sequence…

Let $M_n=\sum_{k=0}^n\binom{n}{2k}C_k$ be the Motzkin numbers, $D_n=\sum_{k=0}^n\binom{n}{k}\binom{n+k}{k}$ the central Delannoy numbers, $S_n=\sum_{k=0}^n\binom{n+k}{2k}C_k$ the S…

Let $U_n^{(a)}(x;q)$ be the Al-Salam–Carlitz polynomials, and let $G_n(a;q)=\sum_{k=0}^n\left[{{n}\atop{k}}\right]_q a^k$ be their moment sequence, where…

Power-of-two Pfaffian conjecture. The absolute value of the Pfaffian satisfies