Recently, Yan and the first named author investigated systematically the enumeration of inversion or ascent sequences avoiding vincular patterns of length $3$, where two of the three letters are required to be adjacent. They established many connections with familiar combinatorial families and proposed several interesting conjectures. The objective of this paper is to address two of their conjectures concerning the enumeration of $\underline{12}0$-avoiding inversion or ascent sequences.
Pattern-avoiding ascent sequences have recently been related to set-partition problems and stack-sorting problems. While the generating functions for several length-3 pattern-avoiding ascent sequences are known, those avoiding 000, 100, 110, 120 are not known. We have generated extensive series expansions for these four cases, and analysed them in order to conjecture the asymptotic behaviour. We provide polynomial time algorithms for the 000 and 110 cases, and exponential time algorithms for the 100 and 120 cases. We also describe how the 000 polynomial time algorithm was detected somewhat mec…
Ascent sequences were introduced by Bousquet-Melou et al. in connection with (2+2)-avoiding posets and their pattern avoidance properties were first considered by Duncan and Steingrimsson. In this paper, we consider ascent sequences of length $n$ avoiding two patterns of length 3, and we determine an exact enumeration for 16 different pairs of patterns. Methods include simple recurrences, bijections to other combinatorial objects (including Dyck paths and pattern-avoiding permutations), and generating trees. We also provide an analogue of the Erdos-Szekeres Theorem to prove that any sufficient…
The combined work of Bousquet-Mélou, Claesson, Dukes, Jelínek, Kitaev, Kubitzke and Parviainen has resulted in non-trivial bijections among ascent sequences, (2+2)-free posets, upper-triangular integer matrices, and pattern-avoiding permutations. To probe the finer behavior of these bijections, we study two types of restrictions on ascent sequences. These restrictions are motivated by our results that their images under the bijections are natural and combinatorially significant. In addition, for one restriction, we are able to determine the effect of poset duality on the corresponding ascent s…
Ascent sequences have received a lot of attention in recent years in connection with (2 + 2)-free posets and other combinatorial objects. Here, we first show bijectively that analogous repetition sequences are counted by the Bell numbers, and 021-avoiding repetition sequences by the Catalan numbers. Then we adapt a bijection of Chen et al and use it along with the "symbolic" method of Flajolet to find the 4-variable generating function for 021-avoiding ascent sequences by length, number of 0's, number of isolated 0's, and number of runs of 2 or more 0's. We deduce that 021-avoiding ascent sequ…
Ascent sequences were introduced by Bousquet-Mélou, Claesson, Dukes and Kitaev in their study of $(\bf{2+2})$-free posets. An ascent sequence of length $n$ is a nonnegative integer sequence $x=x_{1}x_{2}... x_{n}$ such that $x_{1}=0$ and $x_{i}\leq \asc(x_{1}x_{2}...x_{i-1})+1$ for all $1<i\leq n$, where $\asc(x_{1}x_{2}...x_{i-1})$ is the number of ascents in the sequence $x_{1}x_{2}... x_{i-1}$. We let $\cA_n$ stand for the set of such sequences and use $\cA_n(p)$ for the subset of sequences avoiding a pattern $p$. Similarly, we let $S_{n}(τ)$ be the set of $τ$-avoiding permutations in the s…
In this paper, we consider two sets of pattern-avoiding ascent sequences: those avoiding both 201 and 210 and those avoiding 0021. In each case we show that the number of such ascent sequences is given by the binomial convolution of the Catalan numbers. The result for $\{201, 210\}$-avoiders completes a family of results given by Baxter and the current author in a previous paper. The result for 0021-avoiders, together with previous work of Duncan, Steingrímsson, Mansour, and Shattuck, completes the Wilf classification of single patterns of length 4 for ascent sequences.
In this paper we introduce {\em weak ascent sequences}, a class of number sequences that properly contains ascent sequences. We show how these sequences uniquely encode each of the following objects: permutations avoiding a particular length-4 bivincular pattern; upper-triangular binary matrices that satisfy a column-adjacency rule; factorial posets that are weakly (3+1)-free. We also show how weak ascent sequences are related to a class of pattern avoiding inversion sequences that has been a topic of recent research by Auli and Elizalde. Finally, we consider the problem of enumerating these n…
In this paper, we construct a listing for the vertices of the augmentation graph of given size, and as a consequence, we obtain a Hamiltonian cycle for the square of the augmentation graph of given size. As applications, we have a Gray code for the $132$-$312$ avoiding permutations of given length such that two successive permutations differ by at most $2$ adjacent transpositions. Also we obtain Gray codes of strong distance $2$ for the $001$ avoiding ascent sequences and the $010$ avoiding ascent sequences of given length.
序列x= x_1x_2. x_n $ is said to be an ascent sequence of length $ n $ if it satisfies x_1=0 and $ 0 \leq x_i \leq asc（x_1x_2. x_{i-1}）+1 $ for all $ 2 \leq i \leq n $, where $ asc（x_1x_2. x_{i-1}）$ is the number of ascents in the sequence $ x_1x_2. x_{i-1} $. Recently, Duncan and Steingr\'ımsson proposed the conjecture that 210-avoiding ascent sequences of length $ n $ are equinumerous with 3-nonnesting set partitions of $ \{1，2，.，n\} $. In this paper, we confirm this conjecture by showing that 210-avoiding ascent sequences of length $ n $ are in bijection with 3-nonnesting set partitions of $ \…
We consider the distribution of ascents, descents, peaks, valleys, double ascents, and double descents over permutations avoiding a set of patterns. Many of these statistics have already been studied over sets of permutations avoiding a single pattern of length 3. However, the distribution of peaks over 321-avoiding permutations is new and we relate it statistics on Dyck paths. We also obtain new interpretations of a number of well-known combinatorial sequences by studying these statistics over permutations avoiding two patterns of length 3.
A Fishburn permutation is a permutation which avoids the bivincular pattern $(231, \{1\}, \{1\})$, while an ascent sequence is a sequence of nonnegative integers in which each entry is less than or equal to one more than the number of ascents to its left. Fishburn permutations and ascent sequences are linked by a bijection $g$ of Bousquet-Mélou, Claesson, Dukes, and Kitaev. We write $F_n(σ_1,\ldots,σ_k)$ to denote the set of Fishburn permutations of length $n$ which avoid each of $σ_1,\ldots,σ_k$ and we write $A_n(α_1,\ldots,α_k)$ to denote the set of ascent sequences which avoid each of $α_1,…
As shown by Bousquet-Mélou--Claesson--Dukes--Kitaev (2010), ascent sequences can be used to encode $({\bf2+2})$-free posets. It is known that ascent sequences are enumerated by the Fishburn numbers, which appear as the coefficients of the formal power series $$\sum_{m=1}^{\infty}\prod_{i=1}^m (1-(1-t)^i).$$ In this paper, we present a novel way to recursively decompose ascent sequences, which leads to: (i) a calculation of the Euler--Stirling distribution on ascent sequences, including the numbers of ascents ($\asc$), repeated entries $(\rep)$, zeros ($\zero$) and maximal entries ($\max$). In …
A sequence $(a_1, \ldots, a_n)$ of nonnegative integers is an {\em ascent sequence} if $a_0 =0$ and for all $i \geq 2$, $a_i$ is at most 1 plus the number of ascents in $(a_1, \ldots, a_{i-1})$. Ascent sequences were introduced by Bousquet-Mélou, Claesson, Dukes, and Kitaev, who showed that these sequences of length $n$ are in 1-to-1 correspondence with \tpt-free posets of size $n$, which, in turn, are in 1-to-1 correspondence with interval orders of size $n$. Ascent sequences are also in bijection with several other classes of combinatorial objects including the set of upper triangular matric…
Set partitions avoiding $k$-crossing and $k$-nesting have been extensively studied from the aspects of both combinatorics and mathematical biology. By using the generating tree technique, the obstinate kernel method and Zeilberger's algorithm, Lin confirmed a conjecture due independently to the author and Martinez-Savage that asserts inversion sequences with no weakly decreasing subsequence of length 3 and enhanced 3-nonnesting partitions have the same cardinality. In this paper, we provide a bijective proof of this conjecture. Our bijection also enables us to provide a new bijective proof of …
The systematic study of inversion sequences avoiding triples of relations was initiated by Martinez and Savage. For a triple $(ρ_1,ρ_2,ρ_3)\in\{<,>,\leq,\geq,=,\neq,-\}^3$, they introduced $\I_n(ρ_1,ρ_2,ρ_3)$ as the set of inversion sequences $e=e_1e_2\cdots e_n$ of length $n$ such that there are no indices $1\leq i<j<k\leq n$ with $e_i ρ_1 e_j$, $e_j ρ_2 e_k$ and $e_i ρ_3 e_k$. To solve a conjecture of Martinez and Savage, Lin constructed a bijection between $\I_n(\geq,\neq,>)$ and $\I_n(>,\neq,\geq)$ that preserves the distinct entries and further posed a symmetry conjecture of ascents on th…
We take the first steps in developing a theory of transport of patterns from Fishburn permutations to (modified) ascent sequences. Given a set of pattern avoiding Fishburn permutations, we provide an explicit construction for the basis of the corresponding set of modified ascent sequences. Our approach is in fact more general and can transport patterns between permutations and equivalence classes of so called Cayley permutations. This transport of patterns relies on a simple operation we call the Burge transpose. It operates on certain biwords called Burge words. Moreover, using mesh patterns …
Simultaneous avoidance of length-4 patterns in ascent sequences
College of Mathematical Sciences & Institute of Mathematics and Interdisciplinary Sciences, Tianjin Normal University（天津师范大学数学科学学院与数学及交叉科学研究所）；Department of Mathematics and Statistics, University of Strathclyde（斯特拉斯克莱德大学数学与统计系）
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 o…
