Related papers: S-crucial and bicrucial permutations with respect …
A permutation is square-free if it does not contain two consecutive factors of length two or more that are order-isomorphic. A square-free permutation of length $n$ is $P$-crucial, where $P$ is a subset of $\{0,1,\ldots,n\}$, if any of its…
A square is a factor $S = (S_1; S_2)$ where $S_1$ and $S_2$ have the same pattern, and a permutation is said to be square-free if it contains no non-trivial squares. The permutation is further said to be bicrucial if every extension to the…
A pattern $\tau$ is a permutation, and an arithmetic occurrence of $\tau$ in (another) permutation $\pi=\pi_1\pi_2...\pi_n$ is a subsequence $\pi_{i_1}\pi_{i_2}...\pi_{i_m}$ of $\pi$ that is order isomorphic to $\tau$ where the numbers…
In this work, we prove that for every $k\geq 3$ there exist arbitrarily long bicrucial $k$-power-free permutations. We also show that for every $k\geq 3$ there exist right-crucial $k$-power-free permutations of any length at least…
A crucial permutation is a permutation that avoids a given set of prohibitions, but any of its extensions, in an allowable way, results in a prohibition being introduced. In this paper, we introduce five natural types of crucial…
A record in a permutation is a maximum or a minimum, from the left or from the right. The entries of a permutation can be partitioned into two types: the ones that are records are called external points, the others are called internal…
Superpermutations are words over a finite alphabet containing every permutation as a factor. Finding the minimal length of a superpermutation is still an open problem. In this article, we introduce superpermutations matrices. We establish a…
A permutation is said to be a square if it can be obtained by shuffling two order-isomorphic patterns. The definition is intended to be the natural counterpart to the ordinary shuffle of words and languages. In this paper, we tackle the…
Recently, Avgustinovich, Kitaev, and Taranenko defined five types of $(k, \ell)-$crucial permutations, which are maximal permutations that do not contain an increasing subsequence of length $k$ or a decreasing subsequence of length $\ell$.…
A permutation is said to be a square if it can be obtained by shuffling two order-isomorphic patterns. The definition is intended to be the natural counterpart to the ordinary shuffle of words and languages. In this paper, we tackle the…
We study decompositions of words into subwords that are in some sense similar, which means that one subword may be obtained from the other by a relatively simple transformation. Our main inspiration are shuffle squares, an intriguing class…
We describe the limit (for two topologies) of large uniform random square permutations, i.e., permutations where every point is a record. The starting point for all our results is a sampling procedure for asymptotically uniform square…
This paper discusses the permutations that are generated by rotating $k \times k$ blocks of squares in a union of overlapping $k \times (k+1)$ rectangles. It is found that the single-rotation parity constraints effectively determine the…
Permutations are usually enumerated by size, but new results can be found by enumerating them by inversions instead, in which case one must restrict one's attention to indecomposable permutations. In the style of the seminal paper by Simion…
Commonly the direct construction and the description of mutually orthogonal Latin squares (MOLS) makes use of difference or quasi-difference matrices. Now there exists a correspondence between MOLS and separable permutation codes. We like…
We complete the enumeration of Dumont permutations of the second kind avoiding a pattern of length 4 which is itself a Dumont permutation of the second kind. We also consider some combinatorial statistics on Dumont permutations avoiding…
A word is square-free if it does not contain a nonempty word of the form $XX$ as a factor. A famous 1906 result of Thue asserts that there exist arbitrarily long square-free words over a $3$-letter alphabet. We study square-free words with…
A permutation p is realized by the shift on N symbols if there is an infinite word on an N-letter alphabet whose successive left shifts by one position are lexicographically in the same relative order as p. The set of realized permutations…
Two permutations $s$ and $t$ are $k$-similar if they can be decomposed into subpermutations $s^1, \ldots, s^k$ and $t^1, \ldots, t^k$ such that $s^i$ is order-isomorphic to $t^i$ for all $i$. Recently, Dudek, Grytczuk and Ruci\'nski posed…
A word is square-free if it does not contain nonempty factors of the form $XX$. In 1906 Thue proved that there exist arbitrarily long square-free words over a $3$-letter alphabet. It was proved recently [7] that among these words there are…