Bergfinnur Durhuus ; Søren Eilers

Combinatorial aspects of pyramids of onedimensional pieces of fixed integer length
dmtcs:2794 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2010,
DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10)

https://doi.org/10.46298/dmtcs.2794
Combinatorial aspects of pyramids of onedimensional pieces of fixed integer lengthArticle
1 Department of Mathematical Sciences [Copenhagen]
We consider pyramids made of onedimensional 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 $\binom{am1}{m1}$ for each $a \geq 2$. This generalises a well known result for $a=2$. A bijective correspondence between socalled right (or left) pyramids and $a$ary trees is pointed out, and it is shown that asymptotically the average width of pyramids equals $\sqrt{\frac{\pi}{2} a(a1)m}$.
Volume: DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10)
Bergfinnur Durhuus;Søren Eilers, 2013, On the entropy of LEGO ®, Journal of Applied Mathematics and Computing, 45, 12, pp. 433448, 10.1007/s1219001307309.