On infinite permutations

Dmitri G Fon-Der-Flaass,Anna E Frid

doi:10.46298/dmtcs.3458

Abstract

We define an infinite permutation as a sequence of reals taken up to the order, or, equivalently, as a linear ordering of a finite or countable set. Then we introduce and characterize periodic permutations; surprisingly, for each period $t$ there is an infinite number of distinct $t$-periodic permutations. At last, we introduce a complexity notion for permutations analogous to subword complexity for words, and consider the problem of minimal complexity of non-periodic permutations. Its answer is different for the right infinite and the bi-infinite case.

Highlights

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We define an infinite permutation as a sequence of reals taken up to the order, or, equivalently, as a linear ordering of a finite or countable set
We introduce and characterize periodic permutations; surprisingly, for each period t there is an infinite number of distinct t-periodic permutations

Summary

Infinite permutations

We consider infinite permutations: a new object, naturally arising both from sequences of finite permutations and from sequences of reals. Let S be a finite or countable ordered set: we shall consider S equal either to N, or to Z, or to {1, 2, . Let AS be the set of all sequences of pairwise distinct reals defined on S. An S-permutation α can be interpreted as a sequence of reals taken up to their order and defined by any its representative sequence a; in this case, we write α = a. Any sequence of reals occurring in any problem can be considered as a representative of some infinite permutation. We can define an infinite permutation as a limit of a sequence of finite (usual) permutations. As we shall see below, some of these properties look to those of infinite words and some are not

Periodicity

Factors and Complexity