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A permutation on a set is equivalently
a bijection from to itself,
a pair of linear orderings on ;
an element in the symmetric group of .
In the case of a set that is already equipped with a natural ordering (such as a finite ordinal ), these two ways of defining permutations are interchangeable, but they correspond to different abstract structures that are more salient in different contexts, and to different natural ways of comparing permutations. Typically, the definition of permutations-as-automorphisms is important in group theory, while the definition of permutations-as-linear-orders is more common in combinatorics. Both views of permutations are relevant to the theory of symmetric operads.
As automorphisms in Set, the permutations of naturally form a group under composition, called the symmetric group (or permutation group) on . This group may be denoted , , or . When is the finite set , then its symmetric group is a finite group of cardinality = “ factorial”, and one typically writes or .
Two permutations and are said to be conjugate (written ) just in case there is a bijection such that , or equivalently, when there is a commuting square of bijections:
Observe that conjugacy is an equivalence relation, and that conjugacy classes of permutations of are in one-to-one correspondence with partitions of (see below).
If is a set equipped with a linear ordering , then a permutation of is the same thing as a second (unrelated) linear ordering on . Indeed, suppose we label the elements of according to the order as , and according to the order as . Then we can define a bijection as the function sending , , and so on. Conversely, any bijection induces a linear ordering on by defining iff .
This way of viewing permutations naturally gives rise to “array notation” (or “one-line notation”), where a permutation is represented as the list of elements , and it may be contrasted with “cycle notation” (see below). Whereas cycle notation makes it easy to compare permutations for conjugacy, array notation leads to a different natural way of comparing permutations known as pattern containment.
Given two linearly ordered sets and , a permutation (or “pattern”) is said to be contained in a permutation (written ) just in case there exists a monotone injective function such that
for all , or in other words, if contains a subsequence of elements whose relative ordering (in ) is the same as the relative ordering (in ) of the sequence . Note that this definition of is equivalent to asking for the existence of a commuting square:
where and are monotone injections (each uniquely determined by the other).
Whereas conjugacy defines an equivalence relation on the permutations of a particular set , pattern containment defines a partial order on the permutations of arbitrary linearly ordered sets (which restricts to the discrete order on the permutations of ).
In combinatorics, one sometimes also wants a slight generalisation of the notion of permutation-as-linear-order: for any natural number , an -permutation of is an injective function . An -permutation corresponds to a list of distinct elements of , so that for a finite set of cardinality , an -permutation is the same thing as an ordinary permutation of (it is surjective and therefore bijective, since Set is a balanced category). More generally, the number of -permutations of a set of cardinality is counted by the falling factorial .
As an element of the symmetric group , every permutation generates a cyclic subgroup , and hence inherits a group action on . The orbits of this action partition the set into a disjoint union of cycles, called the cyclic decomposition of the permutation .
For example, let be the permutation on defined by
The domain of the permutation is partitioned into three -orbits
corresponding to the three cycles
We can express this more compactly by writing in “cycle notation”, as the composition , or leaving implicit the action of the identity (1).
Narthana Epa, Nora Ganter, Platonic and alternating 2-groups, Higher Structures 1(1):122-146, 2017 (arXiv:1605.09192)
Michael Hopkins, Mark Mahowald, Hal Sadofsky, Constructions of elements in Picard groups, Contemporary mathematics, Volume 158, 1994 in Eric Friedlander, Mark Mahowald (eds.), Topology and Representation theory (doi:10.1090/conm/158/01454)
For the operadic structure of permutations, see Volume 1, Chapter 1 of:
For more on permutation patterns, see:
Wikipedia, Permutation pattern.
Étienne Ghys, A Singular Mathematical Promenade, August 2017. arXiv:1612.06373 author pdf
Last revised on May 3, 2022 at 22:25:44. See the history of this page for a list of all contributions to it.