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Combinatorial aspects of pyramids of one-dimensional pieces of fixed integer length

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 give a combinatorial proof that the number of pyramids of size $m$, i.e., consisting of $m$ pieces, equals $\binom{am-1}{m-1}$ for each $a \geq 2$. 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 equals $\sqrt{\frac{\pi}{2} a(a-1)m}$.
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Bergfinnur Durhuus, Søren Eilers. Combinatorial aspects of pyramids of one-dimensional pieces of fixed integer length. 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), 2010, Vienna, Austria. pp.143-158. ⟨hal-01185593⟩

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