Self-complementary totally symmetric plane partitions
Self-complementary totally symmetric plane partitions
复制标题
自补全对称平面分区
DOI：
10.1016/0097-3165(86)90098-1
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发表时间：
1986
期刊：
影响因子：
--
通讯作者：
H. Rumsey
中科院分区：
文献类型：
--
作者：
W. Mills;D. P. Robbins;H. Rumsey
全对称平面剖分是指其三维Ferrers图包含在方框Xn=[1，n]×[1，n]×[1，n]中的平面剖分，它在坐标轴的所有排列下都映射到自身。当从顶点(n+1，n+1，n+1)的有利位置观察时，这种平面划分(即boxXn中不属于Ferrers图的格点的集合)的Ferrers图的补也是完全对称的。一个完全对称的平面划分是自补的，如果它(在几何意义上)同余于它的补。除非n=2m是偶数，否则这不可能发生。本文给出了关于自补全对称平面划分的几个猜想和几个定理。特别地，我们描述了与M个交错符号矩阵有密切关系的证据。在前面的一篇文章中，我们描述了交错符号矩阵与不超过m的降维平面划分之间的密切联系。因此，我们只剩下三类明显相互关联的对象。仍然有许多问题没有解决，其中最简单的是证明任何两个对象具有相同的基数。
Atotally symmetric plane partitionof sizenis a plane partition whose three-dimensional Ferrers graph is contained in the boxXn= [1,n] × [1,n] × [1,n] and which is mapped to itself under all permutations of the coordinate axes. The complement of the Ferrers graph of such a plane partition (that is, the set of lattice points in the boxXnthat do not belong to the Ferrers graph) is again totally symmetric when viewed from the vantage point of the vertex (n+ 1,n+ 1,n+ 1). A totally symmetric plane partition isself-complementaryif it is congruent (in the geometrical sense) to its complement. This cannot occur unlessn= 2mis even. In this paper we give several conjectures and a few theorems concerning self-complementary totally symmetric plane partitions. In particular we describe evidence which indicates a close relationship withmbymalternating sign matrices. In an earlier paper we described the close connection betweenmbymalternating sign matrices and descending plane partitions with no parts exceedingm. We are thus left with three classes of objects which are all apparently interrelated. There remain many unsolved problems, the simplest of which is to prove that any two of the objects have the same cardinality.