任意固定整数长度的一维块的棱锥的Enumeration
Enumeration of pyramids of one-dimensional pieces of arbitrary fixed integer length
摘要：
我们考虑由固定整数长度a的一维片段组成的金字塔，它们可以有从1到a的整数长度的成对重叠。我们证明了大小为m的金字塔的数量，即由m个部分组成，对于每个a >= 2，等于（am-1，m-1）。这推广了a = 2的一个众所周知的结果。指出了所谓的右（或左）金字塔与a叉树之间的双射对应，并证明了金字塔的平均宽度与其大小的平方根成正比.
英文摘要：
We consider pyramids made of one-dimensional pieces of fixed integer length a and which may have pairwise overlaps of integer length from 1 to a. We prove that the number of pyramids of size m, i.e. consisting of m pieces, equals (am-1,m-1) for each a >= 2. This generalises a well known result for a = 2. A bijective correspondence between so-called right (or left) pyramids and a-ary trees is pointed out, and it is shown that asymptotically the average width of pyramids is proportional to the square root of the size.
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