30 problems

Let $a_0,\dots,a_n$ be non-negative integers, and let $\operatorname*{CT}_{x}$ denote the constant term in the variables $x_0,\dots,x_n$. Dyson's conjecture. … Dyson's conjecture i…

Let $R$ be a Weyl root system of rank $n$, with Weyl denominator $\Delta$, and let $d_1,\ldots,d_n$ be the degrees of the fundamental invariants of its Weyl group. For a Fourier po…

In the expansion … where $y_i=1-x_i$, write $c_\gamma$ for the coefficient of the monomial indexed by $\gamma$. Coefficient-vanishing conjecture. In the approach to Forrester's con…

For nonnegative integers $a_1, dots,a_n$, let $x^{(a)}$ be the alphabet consisting of $x_i,x_iq, dots,x_iq^{a_i-1}$ for each $i$, and write $|a|=a_1+ dots+a_n$. For a partition…

Kadell's orthogonality conjecture.

Andrews' q-Dyson conjecture.

Let $a=(a_0,\dots,a_n)$ be a sequence of non-negative integers, let $x^{(a)}$ be the alphabet defined by … and write $|a|=a_0+\cdots+a_n$. For a composition $v=(v_0,\dots,v_n)$, wr…

Let $n$ be a positive integer, let $a_1,\dots,a_n$ be nonnegative integers, and use the q-Dyson product $\mathcal{F}_n(\mathbf{x};\mathbf{a};q)$ and…

Let $A(1,x^{-1},\ldots,x^{-1})$ be the constant-term expression associated with the deformed staircase Macdonald polynomial, and let $N_{10}'(x^{-1},t_1,\ldots,t_{n-1})$ be the gen…

Negut's constant term conjecture.

Let $p$ be the parameter of the $D_2$ identity, let $t$ be an indeterminate, and let $\operatorname{Res}_{x_0,x_1,x_2,x_3,x_4}$ denote the iterated residue in the displayed variabl…

Let $m>2$, let $p$ be the parameter of the $D_m$ identity, let $t$ be an indeterminate, and let $\operatorname{Res}_{x_0,x_1,\ldots,x_{m+2}}$ denote the iterated residue in the dis…

Let $m$ and $p$ be the parameters of the $A_m$ constant-term identity, let $t$ be an indeterminate, and let $\operatorname{Res}_{x_0,x_1,\ldots,x_{m+1}}$ denote the iterated residu…

Let $h_{p,m}(t)$ and $\tilde{h}_{p,m}(t)$ be the polynomials defined by the factorizations of the residue polynomials $H_{p,m}(t)$ and $\tilde{H}_{p,m}(t)$ for the $D_m$ orbifold v…

Mészáros–Morales conjecture.

Rational Dyck-path constant-term conjecture. For any such $(u,v)$ and $\gamma$,

Let $n$ be a positive integer, let $a,b,k$ be arbitrary nonnegative integers, and let $m,n_0\leq n\leq m+n_0$. Theorem a-f asserts a specific constant-term product formula for…

Let $n$ be a positive integer, let $a,b,k$ be nonnegative integers, and let $\boldsymbol{x}=(x_1,\dots,x_n)$. Define … where $(t)_r=(1-t)(1-tq)\dots(1-tq^{r-1})$, and let…

Let $n$ be a positive integer, let $M\subset\{1,2,\dots,n\}$, and for each $s\in M$ choose an index $r_s$ such that $r_s\notin M$. Let $\boldsymbol{a}=(a_1,\dots,a_n)$ be nonnegati…

Let $n$ be a positive integer, let $\boldsymbol{a}=(a_1,\dots,a_n)$ be a sequence of nonnegative integers, and let $\boldsymbol{x}=(x_1,\dots,x_n)$ be indeterminates. For $k\geq 0$…

Andrews's q-Dyson conjecture. The constant term of $f_q(\boldsymbol{x})$ must be

Let $n\equiv\zeta\pmod 4$ with $\zeta\in\{0,1\}$, let $u\in\mathbb C$ satisfy … and let $\tau_{ij}$ and $\sigma_{ij}$ be signatures satisfying the paper's condition. Write…

Let $n$ be an odd positive integer, let $a,b$ be nonnegative integers, and let $u\in\mathbb C$ satisfy $\operatorname{Re}(1+nu/2)>0$. For $1\leq i<j\leq n$, let $\tau_{ij}$ be the…

Adamović–Milas logarithmic constant term conjecture. For odd positive integer $n$ and nonnegative integer $k$, set $m=(n-1)/2$ and $K=2k+1$. Then

Constant-term conjecture. This constant term equals