33 problems

Let $\lambda$ be a partition with $n$ outside corners. Choose a corner $k$ and corresponding weight $\omega_\ell$, and retain the notions of Theorem … 1leq i1<cdots<irleq kleq jr<c…

Given an ordered pair $(\lambda,\mu)$ of partitions with the same number of parts, define … and … For any partition $\nu$, the starred-transformation conjecture. … This is a combin…

Witness-elimination conjecture. With

Linear-independence conjecture. Fix a rectangle $R$. The products

Let $\lambda=(2^{e_2},\ldots,m^{e_m})$ be a partition without parts equal to $1$, set $N:=|\lambda|$ and $M:=|\widetilde\lambda|$, and write … For each admissible pair…

Shifted-binomial log-concavity conjecture. For every partition $\lambda$ without parts equal to $1$, and every admissible index $i$, the sequence…

The CII maximal-involution conjecture.

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

Let $G_w$ be the Stanley symmetric function of a permutation $w$. Call $G_w$ multiplicity-free when, in its Schur expansion, every coefficient is either $0$ or $1$. Billey–Pawlowsk…

Purity and exterior-power basis conjecture. For every $w\in Z_{r+1}$, the cohomology of $(\Omega^{\bullet}(Y),\nabla_{\tilde{\omega}})$ is pure,

Regular factorization conjecture. For certain factorizations $\kappa$, there exist factorization solutions for which

Let $\tau$ be a triangular partition of size $n$, and let $\mathcal{A}_{\tau}(q,t)$ be the associated polynomial. For each partition $\lambda$ of length at most $2$, write…

Let $G(k,m)\in\Lambda$ be the Petrie symmetric function, let $p_2$ be the second power-sum symmetric function, let $\operatorname{Par}$ denote the set of partitions, and let…

Let $m,k\in\mathbb{Z}_{>0}$, let $\mu=(m^k)$, and let $\nu=(m+k-1,m+k-3,\dots,|m-k|+1)$. For partitions $\lambda$ and $\mu$ with $\mu\subseteq\lambda$, let…

Fourfold product stabilization conjecture.

Let $\rho_k=(k,k-1,\ldots,2,1)$ be the staircase partition, and let $gp_{\rho_k}({\bf y})$ and $g_{\rho_k}({\bf y})$ denote the tableau-defined and stable dual Grothendieck functio…

The conjectural combinatorial formula. One has

Near-equality classification conjecture. Up to reversal of $\alpha$ and $\beta$, the compositions $\alpha$, $\beta$ and partition $\nu$ are one of the five cases of Theorem 1.1, on…

Let $a,b$ be nonnegative integers with $a<b$, and let $[s_{2^a1^b}]_{3,n}$ denote the coefficient of $s_{2^a1^b}$ in the Schur expansion for the $(3,n)$ case. Write $[r]_{q,t}$ for…

For nonnegative integers $a,b,n$, let $[s_{2^a1^b}]_{3,n}$ denote the coefficient of the Schur function $s_{2^a1^b}$ in the relevant Schur expansion for the $(3,n)$ case. The…

The exact two-Schur correlator conjecture. The average of the two Schur-polynomial insertions satisfies

Let $\lambda$ and $\mu$ be partitions, and let $\bm{x}=(x_1,\ldots,x_n)$ have nonnegative entries. For a partition $\lambda$, let…

Let $Y_n$ denote the set of Young diagrams with $n$ boxes, and let $s_\lambda$ be the Schur symmetric function indexed by $\lambda$. Let $\lambda,\hat\lambda\in Y_n$ and…

Let $b4$ be a diagram with $n$ cells, let $f_b4$ be an injective function as in the injective Yamanouchi-word conjecture, and let $c0$ range over standard Yamanouchi words of parti…

Let $1\leq m\leq n$. Let $\lambda\vdash n(n-1)$ satisfy $n-1\leq\ell(\lambda)\leq n$ and … Let $\mu\vdash n(n+1)$ satisfy … and, if $m<n$, $\mu_j=\lambda_{j-1}$ for…