12 problems

Let $\CSPP{n}^{k}$ be the restricted class of cyclically symmetric plane partitions, let $\overline U_r(c)$ be its statistic for $1\leq r\leq n$, and define … where $\MT{n}^{k}$ is…

Let $\TSPP{n}^{k}$ be the subset of triangular shifted plane partitions whose entries in the first $n-1-k$ columns attain their maximal values $n$. Let $\MT{n}^{k}$ be the set of m…

Homotopy-type conjecture. The complex $\Delta(T',T)$ is contractible unless

Let $\alpha$ be the generalized monotone-triangle enumeration, let $A_{n-1}:=\alpha(n-1;1,2,\ldots,n-1)$, and let $n\geq k\geq4$. Polynomial-ratio conjecture. There exist polynomia…

Let $A_{n-1}:=\alpha(n-1;1,2,\ldots,n-1)$ and let $n\geq4$. Repeated-entry formula conjecture. The identity … should hold. The formula was obtained from computational experiments a…

Let $A_{n,j}$ denote the number of alternating-sign matrices of size $n$ whose unique $1$ in the first row is in column $j$, and let $n\geq1$ and $i=1,\ldots,n-1$. Hole-one-descent…

Let $A_n:=\alpha(n;1,2,\ldots,n)$, and let $n\geq1$, $i=1,\ldots,n-k+1$, and $k=1,\ldots,n$. Reverse-duplication conjecture. The identity … should hold. In words, the consecutive s…

Let $A_n:=\alpha(n;1,2,\ldots,n)$, and let $n\geq2$ and $i=1,\ldots,n-1$. One-descent conjecture. The identity … should hold. This is a computational conjecture related to a broade…

Block-repetition conjecture. The identities

Let $\alpha(n;k_1,\ldots,k_n)$ denote the generalized monotone-triangle enumeration, and let $n\geq1$. Even-argument reciprocity conjecture. The identity … should hold. The paper l…

Let $\alpha(n;k_1,\ldots,k_n)$ denote the generalized monotone-triangle enumeration, and let $n\geq1$. An alternating-sign matrix (ASM) is a square matrix with entries in…

Let $\alpha(n;k_1,\ldots,k_n)$ denote the number of monotone triangles with bottom row $(k_1,\ldots,k_n)$, extended to arbitrary integer bottom rows by the associated evaluation. F…