67 problems

Let $n\geq 1$. For partitions $\pi,\mu,\nu\vdash n$, define the coefficient $f^\pi_{\mu,\nu}(\alpha)$ by … where $e_\pi$ denotes the elementary symmetric function. Monomial positiv…

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

Let $\lambda$ and $\mu$ be partitions, and let $m_i(\mu)$ denote the multiplicity of the part $i$ in $\mu$. Define … where $v_{\lambda\mu}(\alpha)$ are the coefficients defined in…

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 $J_\Lambda^{(1/\beta)}$ 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 $\lambda$ 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 $\Lambda$ 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…

Let shifted Jack Littlewood–Richardson coefficients be defined for partitions with $|\lambda|\leq|\mu|+|\nu|$. Shifted Jack congruence conjecture. The adjacent-triple congruence co…

Let $(\mu_1,\nu_1,\lambda_1)$ and $(\mu_2,\nu_2,\lambda_2)$ be adjacent triples of partitions corresponding to a pivot at $b$. Let $\mathcal W'$ and $\mathcal W$ be the associated…

Homogeneous positivity conjecture. The polynomial $\rho(\{y\},\beta)$ has non-negative coefficients as a homogeneous polynomial in $y_i$ and $\beta$, and it has degree $7$ in…

Let $\phi$ be the displayed map from the five auxiliary variables into the specified linear space, and write $\{m,n,r\}=(m_1,m_2,m_3,n_1,n_2,n_4,r_1,r_2)$. Stanley positivity conje…

Let $T$ be a triple of partitions and let $W$ be a window for $T$. A rule is a Stanley sum whose evaluation gives the corresponding Jack Littlewood–Richardson coefficient. Jack win…

For partitions $\mu,\nu,\lambda$, let $\mathsf{SD}_{\mu\nu}^{\lambda}$ be the set of Stanley diagrams and let a Stanley sum be an integer linear combination of these diagrams. Stan…

Let $c_{\mu\nu}^{\lambda}$ be the ordinary Littlewood–Richardson coefficient. For a triple of partitions $(\mu,\nu,\lambda)$, let $\mathsf{SD}_{\mu\nu}^{\lambda}$ denote the set of…

Let $(\mu_1,\nu_1,\lambda_1)$ and $(\mu_2,\nu_2,\lambda_2)$ be triples of partitions that differ by a single box move in one pair, with pivot boxes $b_1\in\sigma_1$ and…

Alexandersson–Haglund–Wang conjecture. The coefficients $a_k(\lambda,\mu)$ and $b_{d-k}(\lambda,\mu)$ are nonnegative integers. Moreover, the polynomials

Let $\lambda$ and $\mu$ be partitions, and let $\mathcal J_C^\sigma(\mathbb R_{\geqslant0})$ be the Jack cone generated by normalized Jack differences with parameter $\sigma$. Fix…

Fix $\tau_0\in[0,\infty]$, and let $\lambda$ and $\mu$ be partitions. Let $\mathcal M_S(\mathbb R_{\geqslant0})$ denote the Muirhead semiring over the nonnegative reals. Muirhead-s…

Let $\lambda$ and $\mu$ be partitions, and let $P_\lambda(x;\tau)$ denote the Jack polynomial. Write $\bm1=(1,\ldots,1)$. KT conjecture. The following are equivalent: 1. … 2. For s…

Let $\lambda$ and $\mu$ be partitions with $|\lambda|=|\mu|$. For Jack polynomials $P_\lambda(x;\tau)$, let $\mathbb F_{\geqslant0}^{\mathbb R}$ denote the real cone of rational fu…

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…

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…

Jack polynomial weak-dominance conjecture. The following statements are equivalent: $\lambda$ weakly dominates $\mu$; for $x\in[1,\infty)^n$,

Jack polynomial dominance conjecture. The following statements are equivalent: $\lambda$ dominates $\mu$; for every $x\in[0,\infty)^n$,

Let $k$ be a positive integer, let $\mathcal{Z}\mathcal{V}_k$ be the coset-type-constant subspace, let $\mathfrak{p}_\lambda$ be the $b$-deformed class vector for $\lambda\vdash k$…