Related papers: On Pattern Avoiding Alternating Permutations

Rudolph conjectures that for permutations $p$ and $q$ of the same length, $A_n(p) \le A_n(q)$ for all $n$ if and only if the spine structure of $T(p)$ is less than or equal to the spine structure of $T(q)$ in refinement order. We prove one…

An infinte word w avoids a pattern p with the involution t if there is no substitution for the variables in p and no involution t such that the resulting word is a factor of w. We investigate the avoidance of patterns with respect to the…

An occurrence of a classical pattern p in a permutation \pi is a subsequence of \pi whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be…

Using techniques from Poisson approximation, we prove explicit error bounds on the number of permutations that avoid any pattern. Most generally, we bound the total variation distance between the joint distribution of pattern occurrences…

Let $S_n$ denote the set of permutations of $[n]:=\{1,\cdots, n\}$, and denote a permutation $\sigma\in S_n$ by $\sigma=\sigma_1\sigma_2\cdots \sigma_n$. For $l\ge2$ an integer, let $A^{(n)}_{l;k}\subset S_n$ denote the event that the set…

It is shown that the maximum number of patterns that can occur in a permutation of length $n$ is asymptotically $2^n$. This significantly improves a previous result of Coleman.

For a permutation $\pi$ the major index of $\pi$ is the sum of all indices $i$ such that $\pi_i > \pi_{i+1}$. It is well known that the major index is equidistributed with the number of inversions over all permutations of length $n$. In…

In this paper, we study pattern avoidances of generalized permutations and show that the number of all generalized permutations avoiding $\pi$ is independent of the choice of $\pi\in S_3$, which extends the classic results on permutations…

A permutation is (1-23-4)-avoiding if it contains no four entries, increasing left to right, with the middle two adjacent in the permutation. Here we give a 2-variable recurrence for the number of such permutations, improving on the…

We study the longest increasing subsequence problem for random permutations avoiding the pattern $312$ and another pattern $\tau$ under the uniform probability distribution. We determine the exact and asymptotic formulas for the average…

We study questions of even-Wilf-equivalence, the analogue of Wilf-equivalence when attention is restricted to pattern avoidance by permutations in the alternating group. Although some Wilf-equivalence results break when considering…

We investigate the avoidability of unary patterns of size of four with morphic permutations. More precisely, we show that, for the positive integers $i,j,k$, the sizes of the alphabets over which a pattern $x \pi ^ {i} (x) \pi^{j}(x)…

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…

Recently, Babson and Steingrimsson have introduced generalised permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We consider pattern avoidance for such patterns, and…

Let $\pi$ be a permutation of $[n]=\{1,\dots,n\}$ and denote by $\ell(\pi)$ the length of a longest increasing subsequence of $\pi$. Let $\ell_{n,k}$ be the number of permutations $\pi$ of $[n]$ with $\ell(\pi)=k$. Chen conjectured that the…

The concept of pattern avoidance respectively containment in permutations can be extended to permutations on multisets in a straightforward way. In this note we present a direct proof of the already known fact that the well-known…

We use the theory of symmetric functions to enumerate various classes of alternating permutations w of {1,2,...,n}. These classes include the following: (1) both w and w^{-1} are alternating, (2) w has certain special shapes, such as…

The length is(w) of the longest increasing subsequence of a permutation w in the symmetric group S_n has been the object of much investigation. We develop comparable results for the length as(w) of the longest alternating subsequence of w,…

Andr\'e proved that the number of alternating permutations on $\{1, 2, \dots, n\}$ is equal to the Euler number $E_n$. A refinement of Andr\'e's result was given by Entringer, who proved that counting alternating permutations according to…

Let $\pi$ be a permutation of the set $[n]=\{1,2,\dots, n\}$. Two disjoint order-isomorphic subsequences of $\pi$ are called twins. How long twins are contained in every permutation? The well known Erd\H{o}s-Szekeres theorem implies that…