论文标题
具有约束等级和晶格路径的分区
Partitions with constrained ranks and lattice paths
论文作者
论文摘要
在本文中，我们研究了连续的等级属于给定集的分区。我们列举了此类分区，同时跟踪零件的数量，最大的部分，杜尔菲广场的侧面以及杜尔菲矩形的高度。我们还获得了Andrews和Bressoud的结果的新徒证明，即$ n $的分区数量至少$ 1- \ ell $等于$ n $的分区数量，没有等于$ \ ell+1 $的零件，对于$ \ ell \ ell \ ge0 $，这使我们可以通过上述统计数据进行完善。通过晶格路径来解释，将FOATA对单词的第二个基本转换与Greene和Kleitman的映射结合在一起，我们获得了连续等级满足某些约束的分区的枚举公式，例如由常数界定。
In this paper we study partitions whose successive ranks belong to a given set. We enumerate such partitions while keeping track of the number of parts, the largest part, the side of the Durfee square, and the height of the Durfee rectangle. We also obtain a new bijective proof of a result of Andrews and Bressoud that the number of partitions of $N$ with all ranks at least $1-\ell$ equals the number of partitions of $N$ with no parts equal to $\ell+1$, for $\ell\ge0$, which allows us to refine it by the above statistics. Combining Foata's second fundamental transformation for words with Greene and Kleitman's mapping for subsets, interpreted in terms of lattice paths, we obtain enumeration formulas for partitions whose successive ranks satisfy certain constraints, such as being bounded by a constant.
