Related papers: On restricted permutations on regular multisets
We analyze the structure and enumerate Dumont permutations of the first and second kinds avoiding certain patterns or sets of patterns of length 3 and 4. Some cardinalities are given by Catalan numbers, powers of 2, little Schroeder…
Any permutation statistic $f:\sym\to\CC$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= \Sigma_\tau\lambda_f(\tau)\tau$. To provide explicit expansions for certain…
Enumeration problems related to words avoiding patterns as well as permutations that contain the pattern $123$ exactly once have been studied in great detail. However, the problem of enumerating words that contain the pattern $123$ exactly…
We set up a new notion of local convergence for permutations and we prove a characterization in terms of proportions of \emph{consecutive} pattern occurrences. We also characterize random limiting objects for this new topology introducing a…
Jacobi permutations, introduced by Viennot in the context of Jacobi elliptic functions, are counted by the Euler numbers $E_{n}$ appearing in the series expansion $\sec x+\tan x=\sum_{n=0}^{\infty}E_{n}x^{n}/n!$. We conduct a systematic…
A permutation is called Grassmannian if it has at most one descent. In this paper, we investigate pattern avoidance and parity restrictions for such permutations. As our main result, we derive formulas for the enumeration of Grassmannian…
A simple permutation is one which maps no proper non-singleton interval onto an interval. We consider the enumeration of simple permutations from several aspects. Our results include a straightforward relationship between the ordinary…
We introduce an algorithm that conjectures the structure of a permutation class in the form of a disjoint cover of "rules"; similar to generalized grid classes. The cover is usually easily verified by a human and translated into an…
Motivated by a correlation between the distribution of descents over permutations that avoid a consecutive pattern and those avoiding the respective quasi-consecutive pattern, as established in this paper, we obtain a complete $\des$-Wilf…
Permutations avoiding all patterns of a given shape (in the sense of Robinson-Schensted-Knuth) are considered. We show that the shapes of all such permutations are contained in a suitable thick hook, and deduce an exponential growth rate…
In this paper we study the enumeration and the construction, according to the number of ones, of particular binary words avoiding a fixed pattern. The growth of such words can be described by particular jumping and marked succession rules.…
We study the expansions of permutation statistics in the basis of functions counting occurrences of a fixed pattern in a permutation. We show the finiteness of these pattern expansions for a class of permutation statistics including the…
The simple permutations in two permutation classes --- the 321-avoiding permutations and the skew-merged permutations --- are enumerated using a uniform method. In both cases, these enumerations were known implicitly, by working backwards…
Dokos et. al. studied the distribution of two statistics over permutations $\mathfrak{S}_n$ of $\{1,2,\dots, n\}$ that avoid one or more length three patterns. A permutation $\sigma\in\mathfrak{S}_n$ contains a pattern…
We compute the expected number of commutations appearing in a reduced word for the longest element in the symmetric group. The asymptotic behavior of this value is analyzed and shown to approach the length of the permutation, meaning that…
We review how the monotone pattern compares to other patterns in terms of enumerative results on pattern avoiding permutations. We consider three natural definitions of pattern avoidance, give an overview of classic and recent formulas, and…
Two $k$-ary Fibonacci recurrences are $a_k(n) = a_k(n-1) + k \cdot a_k(n-2)$ and $b_k(n) = k \cdot b_k(n-1) + b_k(n-2)$. We provide a simple proof that $a_k(n)$ is the number of $k$-regular words over $[n] = \{1,2,\ldots,n\}$ that avoid…
Pattern-avoiding permutations are a central object of study in both combinatorics and theoretical computer science. In this paper we design a data structure that can store any size-$n$ permutation $\tau$ that avoids an arbitrary (and…
In this paper we propose a notion of pattern avoidance in binary trees that generalizes the avoidance of contiguous tree patterns studied by Rowland and non-contiguous tree patterns studied by Dairyko, Pudwell, Tyner, and Wynn.…
We count the number of occurrences of restricted patterns of length 3 in permutations with respect to length and the number of cycles. The main tool is a bijection between permutations in standard cycle form and weighted Motzkin paths.