## Enumeration of Factorizable Multi-Dimensional Permutations

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@MISC{Zhang_enumerationof,

author = {Hao Zhang and Daniel Gildea},

title = {Enumeration of Factorizable Multi-Dimensional Permutations},

year = {}

}

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### Abstract

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 d e as k grows. 1

### Citations

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(Show Context)
Citation Context ...eered the combinatorial study of permutation factorization, studying the one-dimensional case. Albert et al. first reported the sequence of the number of simple permutations of n (sequence A111111 of =-=[11]-=-) and showed the fundamental role of simple permutations for generating all permutations. In this paper, we take a similar analytical approach to study the generalized problems of the factorization of... |

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Citation Context ... factor and left-most bracketing structure wherever there are multiple bracketings. The left-most preference for resolving binarization ambiguity is also used by both Shapiro and Stephens [10] and Wu =-=[14]-=- for the case of k = 2 and d = 1. 3 The number of k-ary parsable d-permutations In this section, we develop the recurrence and the generating function for k-ary parsable d-permutations by counting the... |

206 |
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- 1972
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42 | Synchronous binarization for machine translation - Zhang, Huang, et al. - 2006 |

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Citation Context ...logy, the original problem was the enumeration of all common intervals between genetic sequences aligned according to a specified permutation [12], which was then generalized to multiple permutations =-=[5]-=-. In linguistics, permutations play an essential role in the definition of Syntax Directed Translation Schemata [1], or equivalently Synchronous Context Free Grammars (SCFG) [9], which model language ... |

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Citation Context ...ls of gene sequences [6], and also as a method to model language translation using synchronous grammars with limited-length rules, which in turn admit efficient parsing algorithms [15]. Albert et al. =-=[2]-=- pioneered the combinatorial study of permutation factorization, studying the one-dimensional case. Albert et al. first reported the sequence of the number of simple permutations of n (sequence A11111... |

24 |
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Citation Context ...,k[n − 1]. 5 Growth Rates for Factorizable d-permutations Since Sd,k represents a monotonically increasing combinatorial sequence with an algebraic generating function (by Eq. 6), a standard argument =-=[4]-=- shows that the ratio between successive numbers in Sd,k approaches a constant, which we call the growth rate Gd,k. A more careful analysis of Eq. 6 shows that it is a single-equation algebraically ap... |

23 | Some computational complexity results for synchronous context-free grammars
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Citation Context ... multiple permutations [5]. In linguistics, permutations play an essential role in the definition of Syntax Directed Translation Schemata [1], or equivalently Synchronous Context Free Grammars (SCFG) =-=[9]-=-, which model language translation by generating strings in two languages in parallel. Melamed [7] generalized SCFG to “multitexts” of more than two languages, which also require multiple permutations... |

19 | Revisiting T. Uno and M. Yagiura’s algorithm - Bui-Xuan, Habib, et al. - 2005 |

18 |
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- 1991
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Citation Context ...ed branching factor and left-most bracketing structure wherever there are multiple bracketings. The left-most preference for resolving binarization ambiguity is also used by both Shapiro and Stephens =-=[10]-=- and Wu [14] for the case of k = 2 and d = 1. 3 The number of k-ary parsable d-permutations In this section, we develop the recurrence and the generating function for k-ary parsable d-permutations by ... |

16 | Gene proximity analysis across whole genomes via PQ trees
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Citation Context ... 1s2 3 1 4 2 1 3 4 1 2 3 4 3 4 2 1 2 1 3 4 1 2 3 4 Figure 1: 2-dimensional permutations ((2, 1, 3, 4), (2, 3, 1, 4)) (left) and ((2, 1, 3, 4), (3, 4, 2, 1)) (right) common intervals of gene sequences =-=[6]-=-, and also as a method to model language translation using synchronous grammars with limited-length rules, which in turn admit efficient parsing algorithms [15]. Albert et al. [2] pioneered the combin... |

9 | Factorization of synchronous context-free grammars in linear time
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Citation Context ...ight) common intervals of gene sequences [6], and also as a method to model language translation using synchronous grammars with limited-length rules, which in turn admit efficient parsing algorithms =-=[15]-=-. Albert et al. [2] pioneered the combinatorial study of permutation factorization, studying the one-dimensional case. Albert et al. first reported the sequence of the number of simple permutations of... |

8 |
Note on runs of consecutive elements
- Wolfowitz
- 1944
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Citation Context ...p1,n,k k=3 n! = O(n−1 ) [2]. However, the series p1,n,2 grows rapidly as n increases. The counting of p1,n,2 is a problem studied in an earlier context of runs of consecutive elements in permutations =-=[13]-=-. It turns out the numbers of consecutive runs follow a Poisson distribution of mean value 2. p1,n,2 n! is the probability mass contributed by the permutations containing a non-zero number of consecut... |

6 |
Yagiura M: Fast algorithms to enumerate all common intervals of two permutations
- Uno
(Show Context)
Citation Context ...omputational biology and computational linguistics. In biology, the original problem was the enumeration of all common intervals between genetic sequences aligned according to a specified permutation =-=[12]-=-, which was then generalized to multiple permutations [5]. In linguistics, permutations play an essential role in the definition of Syntax Directed Translation Schemata [1], or equivalently Synchronou... |