Rigid honeycombs were introduced by Knutson et al. (J Am Math Soc 17:19–48, 2004), and they were shown in Bercovici et al. (J Funct Anal 258:1579–1627, 2010) to be sums of extreme rigid honeycombs, with uniquely determined summands up to permutations. Two extreme rigid honeycombs are essentially the same if they have proportional exit multiplicities and, up to this identification, there are countably many equivalence classes of such honeycombs. We describe two ways to approach the enumeration of these equivalence classes. The first method produces a (finite) list of all rigid tree honeycombs of fixed weight by looking at the locking patterns that can be obtained from a certain quadratic Diophantine equation. The second method constructs arbitrary rigid tree honeycombs from rigid overlays of two rigid tree honeycombs with strictly smaller weights. This allows, in principle, for an inductive construction of all rigid tree honeycombs starting with those of unit weight. We also show that some rigid overlays of two rigid tree honeycombs give rise to an infinite sequence of rigid tree honeycombs of increasing complexity but with a fixed number of nonzero exit multiplicities. This last result involves a new inflation/deflation construction that also produces other infinite sequences of rigid tree honeycombs.

Knutson 等人介绍了刚性蜂窝。（J Am Math Soc 17:19-48, 2004），它们在 Bercovici 等人中有所展示。(J Funct Anal 258:1579–1627, 2010) 是极端刚性蜂窝的总和，具有唯一确定的和直到排列。如果两个极端刚性的蜂窝具有成比例的出口多重性，那么它们本质上是相同的，并且直到这个识别，这种蜂窝的等价类可数很多。我们描述了两种方法来处理这些等价类的枚举。第一种方法通过查看锁定模式生成所有固定重量的刚性树蜂窝的（有限）列表可以从某个二次丢番图方程得到。第二种方法从两个具有严格较小权重的刚性树蜂窝的刚性覆盖层构造任意刚性树蜂窝。原则上，这允许从单位重量的那些开始的所有刚性树蜂窝的感应构造。我们还表明，两个刚性树蜂巢的一些刚性覆盖会产生无限的刚性树蜂巢序列，这些刚性树蜂巢的复杂性不断增加，但具有固定数量的非零出口多重性。最后一个结果涉及一个新的通货膨胀/通货紧缩结构，该结构还产生其他无限序列的刚性树蜂窝。