Enumeration of pyramids of one-dimensional pieces of arbitrary fixed integer length

TL;DR

This paper generalizes the enumeration of one-dimensional pyramids made of fixed-length pieces, establishing a formula for their count, relating them to a-ary trees, and analyzing their average width asymptotically.

Contribution

It provides a new enumeration formula for pyramids of fixed-length pieces and links them to a-ary trees, extending known results for the case a=2.

Findings

Number of pyramids of size m equals (am-1,m-1) for each a >= 2

Established a bijection between pyramids and a-ary trees

Average width of pyramids grows proportionally to the square root of size

Abstract

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.