Geodesic
Enumeration of factorizable multi-dimensional permutations
Journal of integer sequences, Tome 10 (2007) no. 5.

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Summary: A $d$-dimensional permutation is a sequence of $d+1$ permutations with the leading element being the identity permutation. It can be viewed as an alignment structure across $d+1$ sequences, or visualized as the result of permuting $n$ hypercubes of $(d+1)$ dimensions. We study the hierarchical decomposition of $d$-dimensional permutations. We show that when $d >= 2$, the ratio between non-decomposable or simple $d$-permutations and all $d$-permutations approaches 1. We also prove that the growth rate of the number of $d$-permutations that can be factorized into $k$-ary branching trees approaches $(k/e)^{d}$ as $k$ grows.
Classification : 05A16, 05A05, 05A15
Keywords: asymptotic enumeration, permutation 
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     author = {Zhang, Hao and Gildea, Daniel},
     title = {Enumeration of factorizable multi-dimensional permutations},
     journal = {Journal of integer sequences},
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     number = {5},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2007__10_5_a6/}
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Zhang, Hao; Gildea, Daniel. Enumeration of factorizable multi-dimensional permutations. Journal of integer sequences, Tome 10 (2007) no. 5. http://geodesic.mathdoc.fr/item/JIS_2007__10_5_a6/