Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBounded affine permutations II. Avoidance of decreasing patterns
77
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We continue our study of a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We focus on bounded affine permutations of size $N$ that avoid the monotone decreasing pattern of fixed size $m$. We prove that the number of such permutations is asymptotically equal to $(m-1)^{2N} N^{(m-2)/2}$ times an explicit constant as $Ntoinfty$. For instance, the number of bounded affine permutations of size $N$ that avoid $321$ is asymptotically equal to $4^N (N/4pi)^{1/2}$. We also prove a permuton-like result for the scaling limit of random permutations from this class, showing that the plot of a typical bounded affine permutation avoiding $mcdots1$ looks like $m-1$ random lines of slope $1$ whose $y$ intercepts sum to $0$.
rate research
Read More
We introduce a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We study pattern avoidance in bounded affine permutations. In particular, we show that if $tau$ is one of the finite increasing oscillations, then every $tau$-avoiding affine permutation satisfies the boundedness condition. We also explore the enumeration of pattern-avoiding affine permutations that can be decomposed into blocks, using analytic methods to relate their exact and asymptotic enumeration to that of the underlying ordinary permutations. Finally, we perform exact and asymptotic enumeration of the set of all bounded affine permutations of size $n$. A companion paper will focus on avoidance of monotone decreasing patterns in bounded affine permutations.
We consider permutations sortable by $k$ passes through a deterministic pop stack. We show that for any $kinmathbb N$ the set is characterised by finitely many patterns, answering a question of Claesson and Gu{dh}mundsson. Our characterisation demands a more precise definition than in previous literature of what it means for a permutation to avoid a set of barred and unbarred patterns. We propose a new notion called emph{$2$-avoidance}.
The maximum drop size of a permutation $pi$ of $[n]={1,2,ldots, n}$ is defined to be the maximum value of $i-pi(i)$. Chung, Claesson, Dukes and Graham obtained polynomials $P_k(x)$ that can be used to determine the number of permutations of $[n]$ with $d$ descents and maximum drop size not larger than $k$. Furthermore, Chung and Graham gave combinatorial interpretations of the coefficients of $Q_k(x)=x^k P_k(x)$ and $R_{n,k}(x)=Q_k(x)(1+x+cdots+x^k)^{n-k}$, and raised the question of finding a bijective proof of the symmetry property of $R_{n,k}(x)$. In this paper, we establish a bijection $varphi$ on $A_{n,k}$, where $A_{n,k}$ is the set of permutations of $[n]$ and maximum drop size not larger than $k$. The map $varphi$ remains to be a bijection between certain subsets of $A_{n,k}$. %related to the symmetry property. This provides an answer to the question of Chung and Graham. The second result of this paper is a proof of a conjecture of Hyatt concerning the unimodality of polynomials in connection with the number of signed permutations of $[n]$ with $d$ type $B$ descents and the type $B$ maximum drop size not greater than $k$.
A permutation whose any prefix has no more descents than ascents is called a ballot permutation. In this paper, we present a decomposition of ballot permutations that enables us to construct a bijection between ballot permutations and odd order permutations, which proves a set-valued extension of a conjecture due to Spiro using the statistic of peak values. This bijection also preserves the neighbors of the largest letter in permutations and thus resolves a refinement of Spiro s conjecture proposed by Wang and Zhang. Our decomposition can be extended to well-labelled positive paths, a class of generalized ballot permutations arising from polytope theory, that were enumerated by Bernardi, Duplantier and Nadeau. We will also investigate the enumerative aspect of ballot permutations avoiding a single pattern of length 3 and establish a connection between 213-avoiding ballot permutations and Gessel walks.
In this paper, we compute and demonstrate the equivalence of the joint distribution of the first letter and descent statistics on six avoidance classes of permutations corresponding to two patterns of length four. This distribution is in turn shown to be equivalent to the distribution on a restricted class of inversion sequences for the statistics that record the last letter and number of distinct positive letters, affirming a recent conjecture of Lin and Kim. Members of each avoidance class of permutations and also of the class of inversion sequences are enumerated by the $n$-th large Schroder number and thus one obtains a new bivariate refinement of these numbers as a consequence. We make use of auxiliary combinatorial statistics, special generating functions (specific to each class) and the kernel method to establish our results. In some cases, we utilize the conjecture itself in a creative way to aid in solving the system of functional equations satisfied by the associated generating functions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
