53 problems

Let $I_\infty$ be the involutions of the infinite symmetric group, let $\mathrm{Des}_V(z)$ be the visible descent set, and let $\mathfrak G^{\mathsf{O}}_z$ be the orthogonal involu…

Let $I^{\mathsf{fpf}}_\infty$ denote the fixed-point-free involutions, let $\mathfrak G^{\mathsf{Sp}}_z$ be the corresponding symplectic involution Grothendieck polynomial, and let…

K-jdt–Buch insertion conjecture. The K-jdt rectification of $T \ast S$ is the same tableau as the Buch insertion of $T$ into $S$.

Let $S_n$ be the symmetric group, let ${\cal G}_w(x_1,\ldots,x_n)$ be the Grothendieck polynomial associated with $w\in S_n$, and let $G_i^{A_{n-1}}$ be the elements defined by…

Let $u,v\in S_n$ be vexillary permutations, let $\phi(w)$ be the Wachs flag of $w$, and write $a\preceq b$ when $a$ is a prefix of $b$. Let $\mathfrak G_w^{(\beta)}(\mathbf x)$ be…

Let $\mathcal P$ denote the set of partitions, let $\widetilde{G}_{\lambda}$ be the modified stable Grothendieck symmetric function, and let $\leqslant_{\operatorname{g}}$ denote G…

Robichaux's ghost Kohnert conjecture. For every permutation $w\in S_n$ and $\alpha\in\mathbb{Z}_{\geq 0}^n$,

Ross–Yong's Grothendieck-polynomial conjecture. For every permutation $w\in S_n$,

Grothendieck inclusion–exclusion positivity conjecture. For every permutation $w\in S_n$ and every subword $u$ of $w$,

Dennin's conjecture. For every permutation $w$,

Gao's conjecture. For any permutation $w$, $c_w\geq 0$. Equivalently,

Almost vexillary support conjecture. For every almost vexillary permutation $w$, there exists a subset $A\subset D(w)$ such that

Let $\beta$ be a formal variable, let $u,v,w\in S_\infty$, and let $\mathfrak{G}_w(\mathbf{x};\mathbf{y})$ denote the double Grothendieck polynomial. Define the coefficients…

Let $\delta_n=(n,n-1,\ldots,1)$ be the staircase partition, and let $\rho\subseteq\delta_n$. Write $\rho'$ for the conjugate partition. Stembridge-type coincidence conjecture. One…

Let $\Lambda=\bigoplus_{\lambda}\mathbb{Q}(\alpha,\beta)s_{\lambda}(\mathbf{x})$ be the ring of symmetric functions over the field of rational functions in $\alpha$ and $\beta$, an…

Let $I^{\mathsf{vex}}_\infty$ be the set of vexillary involutions, and let $\mathcal{B}_{\mathsf{inv}}(z)$, $\mathcal{B}_{\mathsf{inv}}^+(z)$, and…

Fix $w\in S_n$ and write … Let $\operatorname{supp}(\mathfrak{G}_w^{\mathrm{top}}(x))$ denote the exponents of monomials of highest degree in $\mathfrak{G}_w(x)$, and order exponen…

Fix $w\in S_n$ and write … For a monomial exponent $\alpha$, write $|\alpha|=\alpha_1+\cdots+\alpha_n$, and order exponents componentwise: $\alpha<\beta$ when $\alpha_i\leq\beta_i$…

Let $w\in S_n$ and $\gamma\in\mathbb{Z}_{\geq 0}^n$. Let $\overline{{\sf KKoh}}(w,\gamma)$ denote the modified family of diagrams associated with $w$ and weight $\gamma$, and let…

Let $w\in S_n$ and $\gamma\in\mathbb{Z}_{\geq 0}^n$. Write $g_{w,\gamma}=\#\{P\in {\sf Pipes}(w)\mid {\tt wt}(P)=\gamma\}$, and let ${{\sf KKoh}}(w,\gamma)$ be the set of diagrams…

Let $w\in S_n$ be a vexillary permutation. Write $\mathfrak G_w^\mathrm{top}$ for the top-degree homogeneous component of its Grothendieck polynomial, and let $D^\mathrm{top}(w)$ b…

Fix $w\in S_n$ and any term order with $x_1>x_2>\cdots>x_n$. Let $\mathfrak{G}_w^{(k)}$ denote the degree-$k$ homogeneous component of $\mathfrak{G}_w$, and let $m_k(\bm{x})$ be it…

Fix $w\in S_n$. A heavy climbing chain of $w$ is a climbing chain together with its marked-link data $M(C)$; write its overlined and ordinary weights as…

Fix $w\in S_n$ and any term order with $x_1<x_2<\cdots<x_n$. Let $\mathfrak{G}_w^{(k)}$ denote the degree-$k$ homogeneous component of $\mathfrak{G}_w$, and let $m_k(\bm{x})$ be it…

Gao's beta-Grothendieck conjecture. For all permutations $w$, $c^{(\beta)}_w\in\mathbb Z_{\geq0}[\beta]$.