66 problems

Let $w$ range over permutations in $S_n$, and let $\mathfrak S_w(x_1,\dots,x_n)$ denote the Schubert polynomial indexed by $w$. A permutation is layered if it is a concatenation of…

Let Schubert polynomials be the polynomial family associated with permutations. Kohnert conjectured that each Schubert polynomial admits a generating-polynomial interpretation in t…

Let $w$ be a permutation, and let the Schubert polynomial corresponding to $w$ be considered for factorization into elementary symmetric polynomials. Pattern-avoidance conjecture.…

Let $S_n$ be the symmetric group, let $[ij]$ be the generators of the Fomin–Kirillov algebra, and let $\theta_i$ be the Dunkl elements … Let $\mathfrak{S}_w$ be the Schubert polyno…

Let $S_n$ be the symmetric group, and let $\psi_w$ denote the unnormalized steady state probability associated with a permutation $w\in S_n$ in the inhomogeneous TASEP. Assume that…

Let $\mathfrak{G}_w$ be a Grothendieck polynomial and let $\widehat{\mathfrak{G}}_w$ denote its top-degree component. Mészáros–Setiabrata–St Dizier's support conjecture. The suppor…

Let $b_{w,z}$ be the coefficients defined in Proposition 6.4 for permutations $w,z\in S_n$ occurring in the action of the Hecke algebra generators on the Schubert basis of the coin…

Let $r$ be an orbit of the equioriented type $A$ quiver representation space, with Thom polynomial $\operatorname{Tp}_r$, and let $x^i$ denote the Chern roots associated to the $i$…

For a permutation $w$, let $\mathfrak{S}_w(x)$ be the single Schubert polynomial and let $\mathfrak{G}_w(x)$ be the single Grothendieck polynomial. Lascoux's positivity conjecture.…

Let $u$ be a permutation. An affine key expansion conjecture asserts that there exist positive integers $c_1,\dotsc,c_t$ and affine weak compositions $N\gamma_j+\delta_j$, for…

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 $\mathfrak S_w(\mathbf x;\mathbf y)$ be…

Let $B$ be a $k$-row reduced bumpless pipe dream such that row $k$ has no blanks (“weighty tiles”). Let $\phi(B)$ be the RC graph corresponding to $B$ under the natural bijection b…

Bumpless pipe dreams are related to RC graphs by a bijection: for a bumpless pipe dream $B$ with $n$ rows, let $cphi(B)$ denote the corresponding RC graph, and let $\mathcal{Z}$ be…

Let $(\mathbf n,\mathbf m)$ be as above, and let $\boldsymbol\alpha_{\mathbf n}$ and $\boldsymbol\beta_{\mathbf m}$ be positive sequences satisfying the stated conditions and summa…

Layered exponential-growth conjecture. The two maxima have the same exponential growth rate:

Convergence-to-the-Schubert-permuton conjecture. The random permutations distributed according to this measure converge, as $n\to\infty$, to a deterministic permuton $\mu$ on…

The layered-permutation conjecture. The maximal value of $athfrak{S}_w(1^n)$ for $w\in S_n$ is achieved when $w$ is a layered permutation.

Guo–Lin's conjecture. There exist permutations that simultaneously maximize

Let $w$ be a permutation, let $\mathfrak{S}_w$ denote its Schubert polynomial, and consider its expansion in the SEM basis. A polynomial is a single SEM if it is one basis element…

Key-polynomial operator conjecture. For any composition $\alpha$,

Down-up Monk connectedness conjecture. For any $u,v\in S_{\mathbb Z}$, the set $\\{w\mid\overleftarrow{c_{u,v}^w}\ne0\\}$ is down-up Monk connected.

Commuting-map conjecture. There exists a map $\psi$ satisfying Proposition such that

Down-up Bruhat connectedness conjecture. For any $u,v\in S_{\mathbb Z}$, the set $\\{w\mid\overleftarrow{c_{u,v}^w}\ne0\\}$ is down-up Bruhat connected.

Li's conjecture. For all $u,v$, (a) $\operatorname{BS}(u,v)=\operatorname{St}(u,v)$; and (b) $\operatorname{BS}(u,v)\leq\max(\\{i\mid\theta_i(u)=1\text{ or }\theta_i(v)=1\\})$.

For $w\in S_n$, write … where $\theta_w$ is the number of supports of $\mathfrak{S}_w(x)$ and $p_u(w)$ counts occurrences of the permutation pattern $u$ in $w$. The coefficients ar…