Self-modified difference ascent sequences

A new decomposition of ascent sequences and Euler–Stirling statistics

We solve two problems about ascent sequences: how to get the ascent sequence of the reflection of A with respect to its antidiagonal for a matrix A ∈ Intn and its ascent sequences, and how to determine the ascent sequence of A+B for k×k matrices A ∈ Intn and B ∈ Intm. We give the other definition of ascent sequence and get M-sequence. For the first question, we define M-sequence of A and rewrite the ascent sequences as another form. We build the bijection between M-sequences and ascent sequences and prove that our bijection is well-defined. For the second question, we define an operation on M-sequences. On the basis of the operation and the bijections, we get the ascent sequences of the sum of two matrices.

It is well known since the seminal work by Bousquet-Mélou, Claesson, Dukes and Kitaev (2010) that certain refinements of the ascent sequences with respect to several natural statistics are in bijection with corresponding refinements of \(({\bf2+2})\)-free posets and permutations that avoid a bi-vincular pattern. Different multiply-refined enumerations of ascent sequences and other bijectively equivalent structures have subsequently been extensively studied by various authors. In this paper, our main contributions area bijective proof of a bi-symmetric septuple equidistribution of Euler-Stirling statistics on ascent sequences, involving the number of ascents (\(\mathsf{asc}\)), the number of repeated entries (\(\mathsf{rep}\)), the number of zeros (\(\mathsf{zero}\)), the number of maximal entries (\(\mathsf{max}\)), the number of right-to-left minima (\(\mathsf{rmin}\)) and two auxiliary statistics; a new transformation formula for non-terminating basic hypergeometric \(_4\phi_3\) series expanded as an analytic function in base \(q\) around \(q=1\), which is utilized to prove two (bi)-symmetric quadruple equidistributions on ascent sequences. A by-product of our findings includes the affirmation of a conjecture about the bi-symmetric equidistribution between the quadruples of Euler-Stirling statistics \((\mathsf{asc},\mathsf{rep},\mathsf{zero},\mathsf{max})\) and \((\mathsf{rep},\mathsf{asc},\mathsf{max},\mathsf{zero})\) on ascent sequences, that was motivated by a double Eulerian equidistribution due to Foata (1977) and recently proposed by Fu, Lin, Yan, Zhou and the first author (2018). Mathematics Subject Classifications: 05A15, 05A19Keywords: Ascent sequences, equidistributions, Euler-Stirling statistics, Fishburn numbers, basic hypergeometric series

Refining the bijections among ascent sequences, (2+2)-free posets, integer matrices and pattern-avoiding permutations

Ascent sequences were introduced by Bousquet-Melou, Claesson, Dukes, and Kitaev in [1], who showed that ascent sequences of length n are in 1-to-1 correspondence with (2+2)-free posets of size n. In this paper, we introduce a generalization of ascent sequences, which we call p-ascent sequences, where p \geq 1. A sequence $(a_1, \ldots, a_n)$ of non-negative integers is a p-ascent sequence if $a_0 =0$ and for all $i \geq 2$, $a_i$ is at most p plus the number of ascents in $(a_1, \ldots, a_{i-1})$. Thus, in our terminology, ascent sequences are 1-ascent sequences. We generalize a result of the authors in [9] by enumerating p-ascent sequences with respect to the number of 0s. We also generalize a result of Dukes, Kitaev, Remmel, and Steingrimsson in [4] by finding the generating function for the number of p-ascent sequences which have no consecutive repeated elements. Finally, we initiate the study of pattern-avoiding p-ascent sequences.

Some enumerative results related to ascent sequences

This paper presents a bijection between ascent sequences and upper triangular matrices whose non-negative entries are such that all rows and columns contain at least one non-zero entry. We show the equivalence of several natural statistics on these structures under this bijection and prove that some of these statistics are equidistributed. Several special classes of matrices are shown to have simple formulations in terms of ascent sequences. Binary matrices are shown to correspond to ascent sequences with no two adjacent entries the same. Bidiagonal matrices are shown to be related to order-consecutive set partitions and a simple condition on the ascent sequences generate this class.

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 and a variety of other combinatorial structures. Let Fn denote the Fibonacci sequence given by the recurrence Fn = Fn-1 + Fn-2 if n ? 2, with F0 = 0 and F1 = 1. In this paper, we draw connections between ascent sequences and the Fibonacci numbers by showing that several pattern-avoidance classes of ascent sequences are enumerated by either Fn+1 or F2n-1. We make use of both algebraic and combinatorial methods to establish our results. In one of the apparently more difficult cases, we make use of the kernel method to solve a functional equation and thus determine the distribution of some statistics on the avoidance class in question. In two other cases, we adapt the scanning-elements algorithm, a technique which has been used in the enumeration of certain classes of pattern-avoiding permutations, to the comparable problem concerning pattern-avoiding ascent sequences.

Ascent sequences are sequences of nonnegative integers with restrictions on the size of each letter, depending on the number of ascents preceding it in the sequence. Ascent sequences have recently been related to $(2+2)$-free posets and various other combinatorial structures. We study pattern avoidance in ascent sequences, giving several results for patterns of lengths up to 4, for Wilf equivalence and for growth rates. We establish bijective connections between pattern avoiding ascent sequences and various other combinatorial objects, in particular with set partitions. We also make a number of conjectures related to all of these aspects.

In 2010, Bousquet-Mélou et al. defined sequences of nonnegative integers called ascent sequences and showed that the ascent sequences of length $n$ are in one-to-one correspondence with the interval orders, i.e., the posets not containing the poset $\mathbf{2}+\mathbf{2}$. Through the use of generating functions, this provided an answer to the longstanding open question of enumerating the (unlabeled) interval orders. A semiorder is an interval order having a representation in which all intervals have the same length. In terms of forbidden subposets, the semiorders exclude $\mathbf{2}+\mathbf{2}$ and $\mathbf{1}+\mathbf{3}$. The number of unlabeled semiorders on $n$ points has long been known to be the $n$th Catalan number. However, describing the ascent sequences that correspond to the semiorders under the bijection of Bousquet-Mélou et al. has proved difficult. In this paper, we discuss a major part of the difficulty in this area: the ascent sequence corresponding to a semiorder may have an initial subsequence that corresponds to an interval order that is not a semiorder.
 We define the hereditary semiorders to be those corresponding to an ascent sequence for which every initial subsequence also corresponds to a semiorder. We provide a structural result that characterizes the hereditary semiorders and use this characterization to determine the ordinary generating function for hereditary semiorders. We also use our characterization of hereditary semiorders and the characterization of semiorders of dimension $3$ given by Rabinovitch to provide a structural description of the semiorders of dimension at most $2$. From this description, we are able to determine the ordinary generating function for the semiorders of dimension at most $2$.

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 mechanically given an exponential time algorithm.
 For 120-avoiding ascent sequences we find that the generating function has stretched-exponential behaviour and prove that the growth constant is the same as that for 201-avoiding ascent sequences, which is known.
 The other three generating functions have zero radius of convergence, which we also prove. For 000-avoiding ascent sequences we give what we believe to be the exact growth constant. We give the conjectured asymptotic behaviour for all four cases.

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 Steingrímsson. 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 Erdős-Szekeres Theorem to prove that any sufficiently long ascent sequence contains either many copies of the same number or a long increasing subsequence, with a precise bound.

An ascent sequence is a sequence $a_1a_2\cdots a_n$ consisting of non-negative integers satisfying $a_1=0$ and for $1<i\leq n$, $a_i\leq \text{asc}(a_1a_2\cdots a_{i-1})+1$, where $\text{asc}(a_1a_2\cdots a_k)$ is the number of ascents in the sequence $a_1a_2\cdots a_k$. We say that two sets of patterns $B$ and $C$ are $A$-Wilf-equivalent if the number of ascent sequences of length $n$ that avoid $B$ equals the number of ascent sequences of length $n$ that avoid $C$, for all $n\geq0$. In this paper, we show that the number of $A$-Wilf-equivalences among triples of length-3 patterns is 62. The main tool is generating trees; bijective methods are also sometimes used. One case is of particular interest: ascent sequences avoiding the 3 patterns 100, 201 and 210 are easy to characterize, but it seems remarkably involved to show that, like 021-avoiding ascent sequences, they are counted by the Catalan numbers.

Vincular patterns in inversion sequences

We consider two order relations: that induced by the m-ary reflected Gray code and a suffix partitioned variation of it. We show that both of them when applied to some sets of restricted growth sequences still yield Gray codes. These sets of sequences are: subexcedant and ascent sequences, restricted growth functions and staircase words. In particular, we give the first suffix partitioned Gray codes for restricted growth f unctions and ascent sequences; these latter sequences code various combinatorial classes as interval orders, upper triangular matrices without zero rows and zero columns whose non-negative integer entries sum up to n, and certain pattern-avoiding permutations. For each Gray code we give efficient exhaustive generating algorithms and compare the obtained results.