Block-Decomposition of Permutations
Ruth Hoffmann
Supervisor: Prof. Steve Linton
University of St Andrews, School of Computer Science
Motivation
Decomposing permutations into smaller more computable
permutations is beneficial when working on a big scale. It can provide
more insight into the properties of large sets of permutations closed
downwards under the containment order.
The set of permutations closed under the block-decomposition, also
called the wreath closed class, connects the simple permutations in a
class with global properties of the class. [1]
Background
A permutation is a bijective function from a set {1, 2, ..., n} to
itself, where n ∈ N.
A permutation can be represented as the list of its values.
The plot of a permutation π is the generic point set {(i , π(i ))} in
the plane. [2]
An interval in a permutation π is a set of consecutive indices such
that the set of their values is contiguous.
Every permutation of length n ∈ N has intervals of length 1 and n,
at least.
Plot of π = 327986145 with valid and invalid intervals
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2
1
A permutation π of length n is said to be simple if it contains
intervals of length 0, 1 and n, and no other. [3]
For example π = 481375926 is a simple permutation.
Given a permutation π of length m and nonempty permutations
α1, . . . , αm the inflation of π by α1, . . . , αm , written as
π[α1, . . . , αm ], is the permutation obtained by replacing each entry
π(i ) by an interval that is order isomorphic to αi .
Conversely a block-decomposition or deflation [1] of σ is any
expression of σ as an inflation σ = π[α1, . . . , αm ]. [4]
The inflation of 324615 by 12, 1, 1, 21, 12, 1 is
324615[12, 1, 1, 21, 12, 1] = 453698127 and so conversely a
block-decomposition of 453698127 is 324615[12, 1, 1, 21, 12, 1].
Plot of 324615[12, 1, 1, 21, 12, 1] = 453698127
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1 2 3 4
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1 2 3 4 5 6 7 8 9
Conclusion
Using the idea of intervals being maximal under the set inclusion
order, we have found an algorithm to compute the
block-decomposition of a permutation with complexity O (n3).
Computing block-decomposition of permutations is the first step
towards working with grid classes of permutations [5, 6]. A grid class
is defined by a matrix, where each cell is a permutation class which
restricts the form of the permutations in the class.
Block-decompositions may also assist a range of other computations.
References
1 2 3 4 5 6 7 8 9
Theorem [1]
Let σ ∈ C. There is a unique simple permutation π ∈ C and a sequence
α1, . . . , αn ∈ C such that
σ = π[α1, . . . , αn ].
π(3)π(4)π(5) = 798 is an interval whereas π(7)π(8)π(9) = 145 is
not.
Plot of unique decomposition 453698127 = 2413[2314, 21, 12, 1]
Inflation and block-decomposition
1 2 3 4 5 6
1 2 3 4 5 6 7 8 9
Example
If π 6= 12, 21 then α1, . . . , αn are also uniquely determined by σ.
[1] M. Albert and M. Atkinson, “Simple permutations and pattern restricted
permutations,” Discrete Mathematics, vol. 300, no. 1–3, 2005.
[2] R. Brignall, “Grid classes and partial well order,” Journal of Combinatorial Theory,
Series A, vol. 119, no. 1, 2012.
[3] R. Brignall, N. Ruškuc, and V. Vatter, “Simple permutations: decidability and
unavoidable substructures,” Theoretical Computer Science, vol. 391, no. 1, pp.
150–163, 2008.
[4] R. Brignall, “A survey of simple permutations,” Permutation Pattern, vol. 376, pp.
41–65, 2010.
[5] V. Vatter, “Small permutation classes,” Proceedings of the London Mathematical
Society, vol. 103, no. 5, pp. 879–921, 2011.
[6] M. H. Albert, M. D. Atkinson, M. Bouvel, N. Ruškuc, and V. Vatter, “Geometric
grid classes of permutations,” ArXiv e-prints, Aug. 2011.