Counting Descent Pairs with Prescribed Tops and Bottoms

受限排列上的下降顶集与下降底集的分布

Distribution of sets of descent tops and descent bottoms on restricted permutations

本文研究模式避免排列上下降顶集与下降底集的分布，通过双射证明3字母模式下231、312避免排列的两类集合联合等分布，提出4字母模式相关猜想，且部分猜想已被他人证明。

The number of monotone triangles with prescribed bottom row

We show that the number of monotone triangles with prescribed bottom row (k_1,...,k_n) is given by a simple product formula which remarkably involves (shift) operators. Monotone triangles with bottom row (1,2,...,n) are in bijection with $n \times n$ alternating sign matrices.

An operator formula for the number of halved monotone triangles with prescribed bottom row

Monotone triangles are certain triangular arrays of integers, which correspond to $n \times n$ alternating sign matrices when prescribing $(1,2,...,n)$ as bottom row of the monotone triangle. In this article we define halved monotone triangles, a specialization of which correspond to vertically symmetric alternating sign matrices. We derive an operator formula for the number of halved monotone triangles with prescribed bottom row which is analogous to our operator formula for the number of ordinary monotone triangle.

Refined enumerations of alternating sign matrices: monotone (d,m)-trapezoids with prescribed top and bottom row

Monotone triangles are plane integer arrays of triangular shape with certain monotonicity conditions along rows and diagonals. Their significance is mainly due to the fact that they correspond to $n \times n$ alternating sign matrices when prescribing $(1,2,...,n)$ as bottom row of the array. We define monotone $(d,m)$--trapezoids as monotone triangles with $m$ rows where the $d-1$ top rows are removed. (These objects are also equivalent to certain partial alternating sign matrices.) It is known that the number of monotone triangles with bottom row $(k_1,...,k_n)$ is given by a polynomial $α(n…

具有指定底行的单调梯形的数量

The number of monotone trapezoids with prescribed bottom row

本文建立了具有指定底行的单调梯形数量的算子公式，推广交替符号矩阵，发现特殊情况下被消去的隐藏算子，涉及自由边界并揭示与共变代数的关系，提出相关猜想。

The operator formula for monotone triangles - simplified proof and three generalizations

We provide a simplified proof of our operator formula for the number of monotone triangles with prescribed bottom row, which enables us to deduce three generalizations of the formula. One of the generalizations concerns a certain weighted enumeration of monotone triangles which specializes to the weighted enumeration of alternating sign matrices with respect to the number of -1s in the matrix when prescribing $(1,2,...,n)$ as the bottom row of the monotone triangle.