Ascent Sequences in Combinatorial Structures

• Ascent sequences are finite nonnegative integer sequences starting with 0, where each entry is bounded by the number of previous ascents plus one.
• They form a combinatorial framework bijectively linked to (2+2)-free posets, Fishburn permutations, and upper-triangular matrix models.
• Recent work leverages refined statistics, generating functions, and pattern avoidance to deepen enumeration and structural analyses.

An ascent sequence is, in the standard convention, a finite sequence $x=x_1x_2\cdots x_n$ of nonnegative integers such that $x_1=0$ and, for each $i\ge 2$, $x_i \le \operatorname{asc}(x_1x_2\cdots x_{i-1})+1$, where $\operatorname{asc}$ counts adjacent rises. Introduced in the study of $(2+2)$-free posets, ascent sequences are counted by the Fishburn numbers and occupy a central position in the combinatorics of interval orders, Fishburn permutations, upper-triangular matrix models, set partitions, and several pattern-avoidance theories (Duncan et al., 2011, Pudwell, 2014, Egge, 2022).

1. Definitions, conventions, and intrinsic structure

For a finite integer sequence $x_1x_2\cdots x_k$, an ascent is an index $j$ with $x_j<x_{j+1}$. The defining inequality for ascent sequences is recursive: the set of admissible values at position $i$ depends on the ascent set of the prefix. In the standard nonnegative convention, the first term is forced to be $x_1=0$0; some later work uses a shifted positive-integer convention with $x_1=0$1, which is suited to generalized difference-ascent frameworks (Duncan et al., 2011, Cerbai et al., 2024).

Several statistics recur throughout the literature. Besides $x_1=0$2, one frequently tracks the number of repeated entries $x_1=0$3, the number of zeros $x_1=0$4, the number of maximal entries $x_1=0$5, and the number of right-to-left minima $x_1=0$6. A basic structural fact is that in any ascent sequence all maximal entries occur in the initial strictly increasing prefix $x_1=0$7, where $x_1=0$8 in the sense of the statistic $x_1=0$9 (Fu et al., 2019). Another standard refinement is the class of primitive ascent sequences, meaning ascent sequences with no two consecutive equal entries (Yan, 2012).

Pattern containment is defined by reduction: a word $i\ge 2$0 contains a pattern $i\ge 2$1 if some subsequence of $i\ge 2$2 is order-isomorphic to $i\ge 2$3, with equal letters in $i\ge 2$4 realized by equal letters in the subsequence. This extends classical permutation patterns to words with repetition. A useful criterion is that the avoidance class $i\ge 2$5 consists entirely of restricted growth functions exactly when $i\ge 2$6 is a subpattern of $i\ge 2$7 (Duncan et al., 2011).

2. Bijections and the Fishburn framework

The foundational result of Bousquet-Mélou, Claesson, Dukes, and Kitaev is that ascent sequences are in bijection with unlabeled $i\ge 2$8-free posets, equivalently interval orders; this identifies the number of ascent sequences of length $i\ge 2$9 with the $x_i \le \operatorname{asc}(x_1x_2\cdots x_{i-1})+1$0-th Fishburn number (Yan, 2012, Keller et al., 2018). Through the same program, ascent sequences were also related to upper triangular matrices with nonnegative entries and no zero row or column, certain pattern-avoiding permutations, and Stoimenow-type structures (Yan, 2012, Pudwell, 2014).

A complementary permutation model is given by Fishburn permutations. The BMCDK bijection $x_i \le \operatorname{asc}(x_1x_2\cdots x_{i-1})+1$1 sends a Fishburn permutation to an ascent sequence by recording the labels of active sites at the successive insertions of $x_i \le \operatorname{asc}(x_1x_2\cdots x_{i-1})+1$2. This makes ascent sequences a canonical coordinate system for Fishburn structures: the recursive growth of the combinatorial object is encoded directly as a word satisfying the ascent bound (Egge, 2022).

The interval-order viewpoint is especially important for semiorders. Although unlabeled semiorders are counted by the Catalan numbers, their image under the ascent-sequence bijection is subtle: a semiorder can correspond to an ascent sequence whose initial prefixes do not all correspond to semiorders. This obstruction motivates hereditary subclasses and refined structural theories rather than a naive characterization by a local forbidden-pattern condition on the sequence alone (Keller et al., 2018).

3. Pattern avoidance and major enumerative classes

Systematic study of pattern avoidance in ascent sequences began with patterns of lengths up to $x_i \le \operatorname{asc}(x_1x_2\cdots x_{i-1})+1$3, Wilf equivalence, and growth questions (Duncan et al., 2011). The subject rapidly developed into a catalog of Catalan, Narayana, Fibonacci, Bell, and binomial-convolution phenomena.

Pattern class	Enumeration	Representative consequence
$x_i \le \operatorname{asc}(x_1x_2\cdots x_{i-1})+1$4	$x_i \le \operatorname{asc}(x_1x_2\cdots x_{i-1})+1$5	First basic Wilf class (Duncan et al., 2011)
$x_i \le \operatorname{asc}(x_1x_2\cdots x_{i-1})+1$6	$x_i \le \operatorname{asc}(x_1x_2\cdots x_{i-1})+1$7	Noncrossing partitions; ascents have Narayana distribution (Duncan et al., 2011)
$x_i \le \operatorname{asc}(x_1x_2\cdots x_{i-1})+1$8	$x_i \le \operatorname{asc}(x_1x_2\cdots x_{i-1})+1$9	Nonzero entries are weakly increasing; linked to 132-avoiding permutations (Chen et al., 2012)
$\operatorname{asc}$0	$\operatorname{asc}$1	$\operatorname{asc}$2 matches $\operatorname{asc}$3 on 132-avoiding permutations (Mansour et al., 2012)
$\operatorname{asc}$4	$\operatorname{asc}$5	Ternary encoding with an even number of $\operatorname{asc}$6's (Duncan et al., 2011)
$\operatorname{asc}$7	3-nonnesting set partitions	Explicit bijection via growth diagrams (Yan, 2012)
$\operatorname{asc}$8	$\operatorname{asc}$9	Binomial convolution of Catalan numbers (Mansour et al., 2012, Pudwell, 2014)

Among these, the class $(2+2)$0 is especially influential. A key characterization is that 021-avoidance is equivalent to the condition that all nonzero entries are weakly increasing from left to right (Chen et al., 2012, Callan, 2019). This simple description underlies several Dyck-path bijections, generating-function constructions, and refined zero-statistics.

For the pattern $(2+2)$1, avoidance is equivalent to avoiding $(2+2)$2, and the resulting ascent sequences are exactly restricted growth functions encoding noncrossing set partitions (Duncan et al., 2011). This identifies a Catalan class internal to ascent-sequence theory that is simultaneously linked to noncrossing partitions and to 312-avoiding permutations.

Longer patterns and simultaneous avoidance lead to Wilf classification questions. The class of 210-avoiding ascent sequences is in bijection with 3-nonnesting set partitions via growth diagrams for 01-fillings of triangular Ferrers shapes, confirming a conjecture of Duncan and Steingrímsson (Yan, 2012). For length-$(2+2)$3 single patterns, the pair $(2+2)$4 and $(2+2)$5 is Wilf-equivalent and counted by the binomial convolution of the Catalan numbers, completing the single-pattern length-$(2+2)$6 classification when combined with earlier work (Mansour et al., 2012, Pudwell, 2014). For pairs of patterns of length $(2+2)$7, eight specific pairs are Catalan-enumerated, and several of them refine to Narayana numbers by number of ascents (Callan et al., 2014).

4. Refined statistics, generating functions, and Dyck-path correspondences

A major line of work studies multivariate generating functions for statistics on unrestricted or restricted ascent sequences. A new recursive decomposition of ascent sequences yields a generating function

$(2+2)$8

together with an explicit closed expression that refines the Fishburn series. This framework proves, among other things, that $(2+2)$9 are equidistributed on ascent sequences, confirming and extending a conjecture of Dukes and Parviainen, and constructs a bijection $x_1x_2\cdots x_k$0 giving the quadruple equidistribution

$x_1x_2\cdots x_k$1

It also yields an extension of Levande’s conjecture relating ascent sequences to $x_1x_2\cdots x_k$2-avoiding inversion sequences (Fu et al., 2019).

For 021-avoiding ascent sequences, a Dyck-path bijection $x_1x_2\cdots x_k$3 supports a more specialized but very detailed analysis. Using a decomposition due to Chen et al. and Flajolet’s symbolic method, one obtains a 4-variable generating function

$x_1x_2\cdots x_k$4

tracking length, number of zeros, number of isolated zeros, and number of runs of zeros of length at least $x_1x_2\cdots x_k$5. The corresponding Dyck-path statistics are good peaks and good runs of peaks. Specializations of $x_1x_2\cdots x_k$6 imply that 021-avoiding ascent sequences with no consecutive zeros and those with no isolated zeros satisfy closely related Catalan-like recurrences (Callan, 2019).

Statistics-preserving bijections with permutation classes are another recurrent theme. A recursive bijection

$x_1x_2\cdots x_k$7

preserves both $x_1x_2\cdots x_k$8 and the number of right-to-left minima, proving the equidistribution of $x_1x_2\cdots x_k$9 on 021-avoiding ascent sequences and 132-avoiding permutations (Chen et al., 2012). For 0012-avoidance, the pair $j$0 on ascent sequences has the same distribution as $j$1 on 132-avoiding permutations, so the ascent statistic is Narayana-distributed on $j$2 (Mansour et al., 2012).

5. Variants, analogues, and structural subclasses

One productive direction replaces the ascent statistic in the defining bound by other local statistics. Repetition sequences use the number of equal adjacencies, and descent sequences use the number of descents. Repetition sequences are counted by the Bell numbers, and their 021-avoiding subclass is counted by the Catalan numbers via a bijection to Dyck paths in which repetitions correspond to valleys. For 021-avoiding descent sequences, the number with $j$3 descents is

$j$4

and the total number by length is equinumerous with Dyck paths of the same semilength avoiding the contiguous pattern $j$5 (Callan, 2019).

Semiorder theory supplies another important subclass. Hereditary semiorders are those semiorders whose entire prefix history under the ascent-sequence bijection remains inside the class of semiorders. They admit a block decomposition in their minimal endpoint representations, have a rational ordinary generating function, and coincide exactly with restricted ascent sequences in the sense of Kitaev and Remmel (Keller et al., 2018).

Modified ascent sequences arise from the hat map, which repeatedly increments earlier entries relative to ascent tops. In the classical case, the image consists of Cayley permutations satisfying $j$6, where $j$7 is the set of positions of first occurrences of values. This construction extends to $j$8-ascent sequences, where ordinary ascents are replaced by $j$9-ascents, and produces modified $x_j<x_{j+1}$0-ascent sequences together with a factorization

$x_j<x_{j+1}$1

to $x_j<x_{j+1}$2-Fishburn permutations (Cerbai et al., 2024).

A further specialization is the class of self-modified difference ascent sequences, the fixed points of the $x_j<x_{j+1}$3-hat map. These are characterized by a factorization

$x_j<x_{j+1}$4

in which each block $x_j<x_{j+1}$5 is decreasing with pace $x_j<x_{j+1}$6. Their generating functions are expressed in terms of generalized Fibonacci polynomials, and the corresponding permutations form a subclass of $x_j<x_{j+1}$7-Fishburn permutations avoiding an additional pattern $x_j<x_{j+1}$8 (Cerbai et al., 2024).

6. Classification programs, recent developments, and open problems

The interaction between ascent sequences and Fishburn permutations remains active. The BMCDK bijection $x_j<x_{j+1}$9 restricts to a bijection

$i$0

settling a conjecture of Gil and Weiner. The same paper shows that 123-avoiding Fishburn permutations correspond to binary ascent sequences avoiding 012, and formulates the open problem of characterizing those permutations $i$1 for which $i$2 (Egge, 2022).

Simultaneous pattern avoidance is now developed well beyond the original single-pattern results. For pairs of patterns of length $i$3, exact enumerations are known for 16 classes, using simple recurrences, Dyck-path bijections, and generating trees; the resulting sequences include powers of $i$4, Catalan, Fibonacci, Motzkin, and binomial Catalan convolutions (Baxter et al., 2014). For selected length-$i$5 patterns $i$6, recent work determines the enumeration for every subset except $i$7, $i$8, and $i$9, and shows that the remaining solved classes fall into 16 Wilf equivalence classes (Liu et al., 8 Apr 2026).

Several open directions recur across the literature. One is the search for direct bijections where only equinumerosity is known: for instance, 021-avoiding descent sequences are counted by $x_1=0$00-avoiding Dyck paths, but no direct bijection is currently given (Callan, 2019). Another is the extension of growth-diagram and Ferrers-filling methods beyond the 210 case to other patterns and to higher $x_1=0$01-nonnesting analogues (Yan, 2012). A broader implication is that ascent sequences continue to function as a transfer language: new structural results on one Fishburn family typically propagate, by bijection or by shared generating trees, to interval orders, set partitions, Dyck-path classes, and pattern-avoiding permutations.