Related papers: Pattern avoidance in permutations and their square…
Circular permutations on {1,2,...,n} that avoid a given pattern correspond to ordinary (linear) permutations that end with n and avoid all cyclic rotations of the pattern. Three letter patterns are all but unavoidable in circular…
Using generating functions and some trivial bijections, we show in this paper that the binomial coefficients count the set of (123,132) and (123,213)-avoiding permutations according to the number of crossings. We also define a q-tableau of…
We study the distribution of the statistics 'number of fixed points' and 'number of excedances' in permutations avoiding subsets of patterns of length 3. We solve all the cases of simultaneous avoidance of more than one pattern, giving…
Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize 3-crossings and 3-nestings,…
We consider the distribution of the length of the longest subsequence avoiding a given pattern in a random permutation of length n. The well-studied case of a longest increasing subsequence corresponds to avoiding the pattern 21. We show…
A permutation is said to be \emph{alternating} if it starts with rise and then descents and rises come in turn. In this paper we study the generating function for the number of alternating permutations on $n$ letters that avoid or contain…
In this paper, we investigate pattern avoidance of parity restricted (even or odd) Grassmannian permutations for patterns of sizes 3 and 4. We use a combination of direct counting and bijective techniques to provide recurrence relations,…
In this work we obtain recurrent formulae for the number of permutations with either increasing or monotonic (i.e., both increasing and decreasing) runs of bounded length. Our formulae allow one to efficiently compute the number of such…
Refining an existing counting argument, we provide an improved upper bound for the number of 1324-avoiding permutations of a given length.
We count permutations avoiding a nonconsecutive instance of a two- or three-letter pattern, that is, the pattern may occur but only as consecutive entries in the permutation. Two-letter patterns give rise to the Fibonacci numbers. The…
We study groups generated by sets of pattern avoiding permutations. In the first part of the paper we prove some general results concerning the structure of such groups. In the second part we carry out a case-by-case analysis of groups…
Fix a strong rectangulation pattern $P$ of size $L$. We show that the growth constant of the class of strong rectangulations avoiding $P$ is strictly smaller than $\Lambda =27/2$, the growth constant for all strong rectangulations. More…
A 321-k-gon-avoiding permutation pi avoids 321 and the following four patterns: k(k+2)(k+3)...(2k-1)1(2k)23...(k+1), k(k+2)(k+3)...(2k-1)(2k)123...(k+1), (k+1)(k+2)(k+3)...(2k-1)1(2k)23...k, (k+1)(k+2)(k+3)...(2k-1)(2k)123...k. The…
Permutations that avoid given patterns are among the most classical objects in combinatorics and have strong connections to many fields of mathematics, computer science and biology. In this paper we study fixed points of both 123- and…
We use various combinatorial and probabilistic techniques to study growth rates for the probability that a random permutation from the Mallows distribution avoids consecutive patterns. The Mallows distribution behaves like a $q$-analogue of…
Previous work has studied the pattern count on singly restricted permutations. In this work, we focus on patterns of length 3 in multiply restricted permutations, especially for double and triple pattern-avoiding permutations. We derive…
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…
We construct an injection from the set of permutations of length $n$ that contain exactly one copy of the decreasing pattern of length $k$ to the set of permutations of length $n+2$ that avoid that pattern. We then prove that the generating…
For permutations avoiding consecutive patterns from a given set, we present a combinatorial formula for the multiplicative inverse of the corresponding exponential generating function. The formula comes from homological algebra…
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…