Related papers: Shape and pattern containment of separable permuta…
We consider the problem of packing fixed-length patterns into a permutation, and develop a connection between the number of large patterns and the number of bonds in a permutation. Improving upon a result of Kaplansky and Wolfowitz, we…
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…
We consider a random permutation drawn from the set of 321-avoiding permutations of length $n$ and show that the number of occurrences of another pattern $\sigma$ has a limit distribution, after scaling by $n^{m+\ell}$ where $m$ is the…
A pattern class is a set of permutations closed under the formation of subpermutations. Such classes can be characterised as those permutations not involving a particular set of forbidden permutations. A simple collection of necessary and…
The theory of limits of permutations leads to limit objects called permutons, which are certain Borel measures on the unit square. We prove that permutons avoiding a given permutation of order $k$ have a particularly simple structure.…
We consider a random permutation drawn from the set of permutations of length $n$ that avoid some given set of patterns of length 3. We show that the number of occurrences of another pattern $\sigma$ has a limit distribution, after suitable…
A sequence of reversals that takes a signed permutation to the identity is perfect if at no step a common interval is broken. Determining a parsimonious perfect sequence of reversals that sorts a signed permutation is NP-hard. Here we show…
Comtet introduced the notion of indecomposable permutations in 1972. A permutation is indecomposable if and only if it has no proper prefix which is itself a permutation. Indecomposable permutations were studied in the literature in various…
For each integer k >= 2, let F(k) denote the largest n for which there exists a permutation \sigma \in S_n, all of whose patterns of length k are distinct. We prove that F(k) = k + \lfloor \sqrt{2k-3} \rfloor + e_k, where e_k \in {-1,0} for…
We conduct a computability-theoretic study of Ramsey-like theorems of the form "Every coloring of the edges of an infinite clique admits an infinite sub-clique avoiding some pattern", with a particular focus on transitive patterns. As it…
Much like sentences are composed of words, words themselves are composed of smaller units. For example, the English word questionably can be analyzed as question+able+ly. However, this structural decomposition of the word does not directly…
Given two permutations $\sigma$ and $\pi$, the \textsc{Permutation Pattern} problem asks if $\sigma$ is a subpattern of $\pi$. We show that the problem can be solved in time $2^{O(\ell^2\log \ell)}\cdot n$, where $\ell=|\sigma|$ and…
A permutation $\sigma\in S_n$ is said to be $k$-universal or a $k$-superpattern if for every $\pi\in S_k$, there is a subsequence of $\sigma$ that is order-isomorphic to $\pi$. A simple counting argument shows that $\sigma$ can be a…
Previous work has shown that the disarray (or displacement) of an (affine) (signed) permutation is bounded in terms of its Coxeter length. Here, we characterize the permutations for which the bound is sharp in two ways: in terms of a…
We define a class L_{n, k} of permutations that generalizes alternating (up-down) permutations and give bijective proofs of certain pattern-avoidance results for this class. As a special case of our results, we give two bijections between…
We prove that if $\sigma \in S_m$ is a pattern of $w \in S_n$, then we can express the Schubert polynomial $\mathfrak{S}_w$ as a monomial times $\mathfrak{S}_\sigma$ (in reindexed variables) plus a polynomial with nonnegative coefficients.…
We find generating functions for the number of words avoiding certain patterns or sets of patterns on at most 2 distinct letters and determine which of them are equally avoided. We also find the exact number of words avoiding certain…
Given a finite word $w$, Guibas and Odlyzko (J. Combin. Theory Ser. A, 30, 1981, 183-208) showed that the autocorrelation polynomial $\phi_w(t)$ of $w$, which records the set of self-overlaps of $w$, explicitly determines for each $n$, the…
The study of pattern containment and avoidance for linear permutations is a well-established area of enumerative combinatorics. A cyclic permutation is the set of all rotations of a linear permutation. Callan initiated the study of…
The Schur-positivity order on skew shapes is defined by B \leq A if the difference s_A - s_B is Schur-positive. It is an open problem to determine those connected skew shapes that are maximal with respect to this ordering. A strong…