Related papers: Five subsets of permutations enumerated as weak so…

In the set of all patterns in $S_n$, it is clear that each k-pattern occurs equally often. If we instead restrict to the class of permutations avoiding a specific pattern, the situation quickly becomes more interesting. Mikl\'os B\'ona…

We consider permutations avoiding a pattern of length three under the family of Mallows distributions. In particular, for any pattern $\tau\in S_3-\{321\}$, we obtain rather precise results on the asymptotic probability as $n\to\infty$ that…

The large Schroder numbers are known to count several classes of permutations avoiding two 4-letter patterns. Here we show they count another family of permutations, those whose left to right minima decomposition, when reversed, is…

Given a permutation w, we show that the number of repeated letters in a reduced decomposition of w is always less than or equal to the number of 321- and 3412-patterns appearing in w. Moreover, we prove bijectively that the two quantities…

A permutation $\pi \in S_n$ is said to {\it avoid} a permutation $\sigma \in S_k$ whenever $\pi$ contains no subsequence with all of the same pairwise comparisons as $\sigma$. For any set $R$ of permutations, we write $S_n(R)$ to denote the…

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…

We completely classify the asymptotic behavior of the number of alternating sign matrices classically avoiding a single permutation pattern, in the sense of [Johansson and Linusson 2007]. In particular, we give a uniform proof of an…

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…

A permutation $\pi$ contains a pattern $\sigma$ if and only if there is a subsequence in $\pi$ with its letters are in the same relative order as those in $\sigma$. Partially ordered patterns (POPs) provide a convenient way to denote…

A {\it superpattern} is a string of characters of length $n$ that contains as a subsequence, and in a sense that depends on the context, all the smaller strings of length $k$ in a certain class. We prove structural and probabilistic results…

The Euler number $E_n$ (resp. Entringer number $E_{n,k}$) enumerates the alternating (down-up) permutations of $\{1,\dots,n\}$ (resp. starting with $k$). The Springer number $S_n$ (resp. Arnold number $S_{n,k}$) enumerates the type $B$…

The subject of pattern avoiding permutations has its roots in computer science, namely in the problem of sorting a permutation through a stack. A formula for the number of permutations of length n that can be sorted by passing it twice…

We identify the structure of the lexicographically least word avoiding 5/4-powers on the alphabet of nonnegative integers. Specifically, we show that this word has the form $p \tau(\varphi(z) \varphi^2(z) \cdots)$ where $p, z$ are finite…

An ascent sequence is one consisting of non-negative integers in which the size of each letter is restricted by the number of ascents preceding it in the sequence. Ascent sequences have recently been shown to be related to (2+2)-free posets…

In this paper, we characterize and enumerate pattern-avoiding permutations composed of only 3-cycles. In particular, we answer the question for the six patterns of length 3. We find that the number of permutations composed of $n$ 3-cycles…

In this note, we prove some and conjecture other results regarding the distribution of descent top and descent bottom sets on some pattern-avoiding permutations. In particular, for 3-letter patterns, we show bijectively that the set of…

Inspired by the definition of modified ascent sequences, we introduce a new class of integer sequences called revised ascent sequences. These sequences are defined as Cayley permutations where each entry is a leftmost occurrence if and only…

Frequent pattern mining is a flagship problem in data mining. In its most basic form, it asks for the set of substrings of a given string $S$ of length $n$ that occur at least $\tau$ times in $S$, for some integer $\tau\in[1,n]$. We…

In 1916, MacMahon showed that permutations in $S_n$ with a fixed descent set $I$ are enumerated by a polynomial $d_I(n)$. Diaz-Lopez, Harris, Insko, Omar, and Sagan recently revived interest in this descent polynomial, and suggested the…

We study the descent distribution over the set of centrosymmetric permutations that avoid the pattern of length 3. Our main tool in the most puzzling case, namely, $\tau=123$ and $n$ even, is a bijection that associates a Dyck prefix of…