Related papers: Counting Fixed-Length Permutation Patterns
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…
In this paper, we give a polynomial (O(n^8)) algorithm for finding a longest common pattern between two permutations of size n given that one is separable. We also give an algorithm for general permutations whose complexity depends on the…
We consider the distribution of cycles in two models of random permutations, that are related to one another. In the first model, cycles receive a weight that depends on their length. The second model deals with permutations of points in…
A sample of n generic points in the xy-plane defines a permutation that relates their ranks along the two axes. Every subset of k points similarly defines a pattern, which occurs in that permutation. The number of occurrences of small…
We consider uniform random permutations in classes having a finite combinatorial specification for the substitution decomposition. These classes include (but are not limited to) all permutation classes with a finite number of simple…
We consider two related problems arising from a question of R. Graham on quasirandom phenomena in permutation patterns. A ``pattern'' in a permutation $\sigma$ is the order type of the restriction of $\sigma : [n] \to [n]$ to a subset $S…
This paper studies permutation statistics that count occurrences of patterns. Their expected values on a product of $t$ permutations chosen randomly from $\Gamma \subseteq S_{n}$, where $\Gamma$ is a union of conjugacy classes, are…
We consider uniform random permutations in proper substitution-closed classes and study their limiting behavior in the sense of permutons. The limit depends on the generating series of the simple permutations in the class. Under a mild…
We study the complexity of the problem of searching for a set of patterns that separate two given sets of strings. This problem has applications in a wide variety of areas, most notably in data mining, computational biology, and in…
We investigate the typical cycle lengths, the total number of cycles, and the number of finite cycles in random permutations whose probability involves cycle weights. Typical cycle lengths and total number of cycles depend strongly on the…
Patterns are words with terminals and variables. The language of a pattern is the set of words obtained by uniformly substituting all variables with words that contain only terminals. In their original definition, patterns only allow for…
Length generalization, defined as the ability to extrapolate from shorter training sequences to longer test ones, is a significant challenge for language models. This issue persists even with large-scale Transformers handling relatively…
We initiate a general approach for the fast enumeration of permutations with a prescribed number of occurrences of `forbidden' patterns, that seems to indicate that the enumerating sequence is always P-recursive. We illustrate the method…
We develop the technique of reduced word manipulation to give a range of results concerning reduced words and permutations more generally. We prove a broad connection between pattern containment and reduced words, which specializes to our…
Packet reordering is an important property of network traffic that should be captured by analytical models of the Transmission Control Protocol (TCP). We study a combinatorial problem motivated by RESTORED, a TCP modeling methodology that…
In this article we propose a general method of obtaining infinite sums of products with functions that count patterns in numbers.
In a previous work, B\'ona and Pantone studied permutations that avoided all but one pattern of length $k$ that began with a length $k-1$ increasing subsequence. We draw the connection between that idea and distant patterns, first discussed…
We consider the problem of counting the number of possible sets of rankings (called ranking patterns) generated by unfolding models of codimension one. We express the ranking patterns as slices of the braid arrangement and show that all…
The use of patterns in predictive models is a topic that has received a lot of attention in recent years. Pattern mining can help to obtain models for structured domains, such as graphs and sequences, and has been proposed as a means to…
Two permutations are similar if they have the same length and the same relative order. A collection of $r\ge2$ disjoint, similar subsequences of a permutation $\pi$ form $r$-twins in $\pi$. We study the longest guaranteed length of…