129 problems

Let $\lambda$ be a partition and let $\theta^{(\alpha)}_\mu(\lambda)$ be the Jack-character coefficient defined by the paper, with $b:=\alpha-1$. Let $m_1(\mu)$ denote the number o…

Let $c^rho{tau, u}(alpha)$ be the coefficients defined by … Here $b:=\alpha-1$ is the shifted Jack parameter. Goulden–Jackson's conjecture. The coefficients satisfy…

Macdonald–Stanley conjecture. The expansion coefficients $c_{\lambda,\mu}(\alpha)$ are polynomials in the parameter $\alpha$ with integer coefficients.

Let {\text{\boldmath p}}=(p_1,\ldots,p_m) and {\text{\boldmath q}}=(q_1,\ldots,q_m) be sequences of positive integers with $q_1\geq\cdots\geq q_m$, and let…

Let $\Lambda$ be a superpartition, let $D[\Lambda]$ be its diagram, and let $a_{\Lambda}(s)$ and $\ell_{\Lambda}(s)$ denote respectively the arm and leg statistics of a cell…

Fix $n\geq 1$. For partitions $\pi,\mu,\nu\vdash n$, define $d^\pi_{\mu,\nu}(\alpha)$ by … where $m_\nu$ denotes the monomial symmetric function in the alphabet $\boldsymbol{q}$. M…

Macdonald's conjecture. The integral identity

Let $n$ be a positive integer and let $\pi,\sigma,\lambda\vdash n$ be partitions. Let $\mathcal{G}^{\lambda}_{\pi,\sigma}$ denote the set of matchings $\delta$ satisfying…

Let $\lambda=\mathbf{r}^{\mathbf{p}}$ be a Young diagram in multirectangular coordinates. For a partition $\mu$, let $J^{\star,(\alpha)}_\mu$ be the shifted Jack function and let…

Let $S_\eta^{(L)}(y,\alpha)$ be a generalized Laguerre polynomial with prescribed symmetry, where $\eta=P^{-1}\kappa$ and $(\rho,\kappa)$ is the index data specified by. Let…

Let $S_\eta^{(H)}(y,\alpha)$ be a generalized Hermite polynomial with prescribed symmetry, where $\eta=P\kappa$ and $(\rho,\kappa)$ is the index data specified by. Let $H_\rho$ den…

Let $S_{(\rho,\mu)}(z,\alpha)$ be a Jack polynomial with prescribed symmetry, with $q=1$ and … where $\rho_1\geq\rho_2\geq\cdots\geq\rho_{N_0}\geq0$. Let $J_\rho^{(p+\alpha)}$ deno…

Let $N$ be the number of particles, let $\psi_{\vec n}(y)$ be the one-variable factor appearing in the separated eigenfunction, and let $\vec n=(n_1,\ldots,n_N)$. Set $t=e^{2iy}$.…

Lapointe–Vinet's Jack-operator conjecture. The creation operators share the following properties:

Let $\alpha$ and $\beta$ be independent parameters. For a rectangular partition $\lambda=p\times q$, define a two-parameter family by…

Let {\text{\boldmath p}}=(p_1,\ldots,p_m) and {\text{\boldmath q}}=(q_1,\ldots,q_m) be as above, let \lambda={\text{\boldmath p}}\times{\text{\boldmath q}}, let $\mu$…

Let $J_\Lambda^{(1/\beta)}$ be a Jack superpolynomial, and write its supermonomial expansion in the form … Here $\Omega\leq\Lambda$ denotes the ordering used for the triangular exp…

Let $\alpha\geq 1$ and let … For every real $x_0$, define the approximation error by comparing the distribution of $W_{\alpha}$ under Jack measure with the standard normal distribu…

Let $lambda$ be a partition and let $rgeq 0$. For the Pieri coefficients $c_i^{\alpha}(lambda)$, define … Let $d_\rho(lambda)$ and $z_\rho$ denote the quantities associated with a…

Let $J_\Lambda^{(1/\beta)}$ be a Jack superpolynomial, let $c_\Lambda^{\mathrm{min}}(\beta)$ be its minimal coefficient, and expand it in the monomial basis as … The integrality co…

Let $k$ be an integer and let $b1$ be a positive real. Write the monomial symmetric function indexed by the one-part partition $[k]$ as … Here $C_{\lambda}^{\alpha}$ are the Jack p…

The Jack polynomials form a one-parameter class depending on $b1$; at $b1=1$ they are the Schur functions. Jack's conjecture. For $b1=2$, the Jack polynomials are the zonal polynom…

Let $P_{\lambda}(\theta)$ and $P_{\mu}(\theta)$ be Jack polynomials, and write their product as … Here $c^{\nu}_{\lambda\mu}(\theta)$ denotes the corresponding structure constant.…

Let $\theta$ be a parameter, let $\pi_1$ and $\pi_2$ be rectangular Young diagrams, and let $\pi=\pi_1\cap\pi_2$. Write $I_{\Omega(\pi_1,\pi_2)}(\theta)$ and…

Let $F_{N,n}$ be the Hilbert space of states of the Spin Calogero–Sutherland model with coupling parameter $\beta>0$, and let $H_{\beta,N}$ be its Hamiltonian. For integer non-nega…