Ascent sequences and their modified version play a central role in the bijective framework relating several combinatorial structures counted by the Fishburn numbers. Ascent sequences are positive integer sequences defined by imposing a bound on the growth of their entries in terms of the number of ascents contained in the corresponding prefix, while modified ascent sequences are the image of ascent sequences under the so-called hat map. By relaxing the notion of ascent, Dukes and Sagan have recently introduced difference ascent sequences. Here we define modified difference ascent sequences and study their combinatorial properties. Inversion sequences are a superset of the difference ascent sequences and we extend the hat map to this domain. Our extension depends on a parameter which we specialize to obtain a new set of permutations counted by the Fishburn numbers and characterized by a subdiagonality property.

We show that, in the ring of virtual species, ℐ=∑_m≥ 0(-1)^m∏_i=1^m((E^-1)^i-1), where ℐ is the species of interval orders and E^-1 is the multiplicative inverse of the species E of sets. The right-hand side is the virtual species of signed ballot matrices introduced by Claesson and Hannah. They showed that its signed cardinality counts labeled interval orders. We strengthen this to a species identity, which we prove twice: first algebraically and then bijectively, using a natural sign-reversing involution. The cycle index series of ℐ specializes to the generating series for labeled and unlabeled interval orders. We describe the automorphism group of an interval order as a Young subgroup and prove the identity ℐ=ℛ∘ E_+, where ℛ is the species of rigid interval orders. We also show that Glaisher's T-number T_n counts the 24-colored interval orders on [n] in which no isolated element has color 24.

A Cayley permutation is a word of positive integers such that if a letter appears in this word, then all positive integers smaller than that letter also appear. We initiate a systematic study of pattern avoidance on Cayley permutations adopting a combinatorial species approach. Our methods lead to species equations, generating series, and counting formulas for Cayley permutations avoiding any pattern of length at most three. We also introduce the species of primitive structures as a generalization of Cayley permutations with no "flat steps". Finally, we explore various notions of Wilf equivalence arising in this context.

Two-sort species yield differential equations for functional digraphs of Cayley permutations. From these, we obtain an explicit formula for fixed-point-free Cayley permutations and prove that their proportion tends to 1 slash e $1/e$1/e, as for permutations and endofunctions. Our approach also yields counting formulas when the functional digraph is a tree, forest, or connected.

We define an involution h on Catalan structures through an abstract framework, prove an equidistribution theorem for four canonical statistics and present a generating function carrying these. This framework encompasses all combinatorial structures with a decomposition mirroring the first-return decomposition of Dyck paths. The fixed points of h are counted by Catalan numbers. Canonical bijections transport the equidistribution to eight well known concrete families, identifying the canonical statistics with native ones on each. In addition to its primary structure, each Catalan structure has a derived secondary structure, and h interchanges primary and secondary structure. The involution factors as h = ∘∘, where and are two simpler involutions, and the composition M = h ∘ coincides with Donaghey's automorphism on plane trees. This yields M^-1 = ∘ M ∘ and a period theorem: Iterating the secondary structure construction produces a sequence that repeats with period equal to the order of M. It is an open problem to describe h and the canonical statistics explicitly on most of the more than two hundred known families of Catalan structures.

A collection B of patterns is called inversion monotone if av_n^k(B), the number of B-avoiding permutations of length n with k inversions, is weakly increasing in n for any fixed k. In 2012, Claesson, Jelínek and Steingrímsson posed the inversion monotonicity conjecture, which states that the pattern 1324 is inversion monotone and implies a new upper bound for its Stanley–Wilf limit. We prove that the collections {1324, 231} and {1324, 2314, 3214, 4213} are inversion monotone via explicit injections. The latter follows from a general procedure for constructing inversion-monotone sets. Our results constitute the first known nontrivial examples of inversion-monotone sets. A key feature of the inversion monotonicity conjecture is that 1324 has a limit sequence: av_n^k(1324) is constant in n when n is large. We characterize the sets of patterns that have limit sequences, and determine the limit sequences of all pairs {1324, p}, where p is a pattern of length four. Connections to various families of integer partitions arise. Finally, we expand on work by Linusson and Verkama (2025) on almost decomposable permutations to determine a broad family of sets containing 1324 that are inversion monotone under the assumption n ≥k+7/2. The method yields an enumeration of av_n^k(1324, 1342) when n ≥k+7/2.

Combinatorial Exploration is a new domain-agnostic algorithmic framework to automatically and rigorously study the structure of combinatorial objects and derive their counting sequences and generating functions. We describe how it works and provide an open-source Python implementation. As a prerequisite, we build up a new theoretical foundation for combinatorial decomposition strategies and combinatorial specifications. We then apply Combinatorial Exploration to the domain of permutation patterns, to great effect. We rederive hundreds of results in the literature in a uniform manner and prove many new ones. These results can be found in a new public database, the Permutation Pattern Avoidance Library (PermPAL) at https://permpal.com. Finally, we give three additional proofs-of-concept, showing examples of how Combinatorial Exploration can prove results in the domains of alternating sign matrices, polyominoes, and set partitions.

Cayley permutations generalize permutations by allowing repeated values. We use two-sort species to derive differential equations that capture the recursive structure of their functional digraphs. These equations enable us to enumerate fixed-point-free Cayley permutations. Our approach also yields combinatorial identities and counting formulas for Cayley permutations whose functional digraphs have specific graph-theoretical properties, such as being a tree, being a forest, or being connected.

Ascent sequences play a key role in the combinatorics of Fishburn structures. Difference ascent sequences are a natural generalization obtained by replacing ascents with d-ascents. We have recently extended the so-called hat map to difference ascent sequences, and self-modified difference ascent sequences are the fixed points under this map. We characterize self-modified difference ascent sequences and enumerate them in terms of certain generalized Fibonacci polynomials. Furthermore, we describe the corresponding subset of d-Fishburn permutations.

Eulerian polynomials record the distribution of descents over permutations. Caylerian polynomials likewise record the distribution of descents over Cayley permutations, where a Cayley permutation is a word of positive integers such that if a number appears in the word then all positive integers less than that number also appear in the word. Using combinatorial species and sign-reversing involutions we derive counting formulas and generating functions for the Caylerian polynomials as well as for related refined polynomials.

An adjacent q-cycle is a natural generalization of an adjacent transposition. We show that the number of adjacent q-cycles in a permutation maps to the sum of occurrences of two mesh patterns under Foata's fundamental transformation. As a corollary we resolve Conjecture 3.14 in the paper "From Hertzprung's problem to pattern-rewriting systems" by the first author.

The Eulerian polynomials enumerate permutations according to their number of descents. We initiate the study of descent polynomials over Cayley permutations, which we call Caylerian polynomials. Some classical results are generalized by linking Caylerian polynomials to Burge words and Burge matrices. The $\gamma$-nonegativity of the two-sided Eulerian polynomials is reformulated in terms of Burge structures. Finally, Cayley permutations with a prescribed ascent set are shown to be counted by Burge matrices with fixed row sums.

Permutations are usually enumerated by size, but new results can be found by enumerating them by inversions instead, in which case one must restrict one's attention to indecomposable permutations.In the style of the seminal paper by Simion and Schmidt [6], we investigate all combinations of permutation patterns of length at most 3.

The score sequence of a tournament is the sequence of the out-degrees of its vertices arranged in nondecreasing order. The problem of counting score sequences of a tournament with $n$ vertices is more than 100 years old (MacMahon 1920). In 2013 Hanna conjectured a surprising and elegant recursion for these numbers. We settle this conjecture in the affirmative by showing that it is a corollary to our main theorem, which is a factorization of the generating function for score sequences with a distinguished index. We also derive a closed formula and a quadratic time algorithm for counting score sequences.

The in-order traversal provides a natural correspondence between binary trees with a decreasing vertex labeling and endofunctions on a finite set. By suitably restricting the vertex labeling we arrive at a class of trees that we call Fishburn trees. We give bijections between Fishburn trees and other well-known combinatorial structures that are counted by the Fishburn numbers, and by composing these new maps we obtain simplified versions of some of the known maps. Finally, we apply this new machinery to the so called flip and sum problems on modified ascent sequences.

A curious generating function $S_0(x)$ for permutations of $[n]$ with exactly $n$ inversions is presented. Moreover, $(xC(x))^iS_0(x)$ is shown to be the generating function for permutations of $[n]$ with exactly $n-i$ inversions, where $C(x)$ is the generating function for the Catalan numbers.

Permutations in the image of the pop-stack operator are said to be pop-stacked. We give a polynomial-time algorithm to count pop-stacked permutations up to a fixed length and we use it to compute the first 1000 terms of the corresponding counting sequence. Only the first 16 terms had previously been computed. With the 1000 terms we prove some negative results concerning the nature of the generating function for pop-stacked permutations. We also predict the asymptotic behavior of the counting sequence using differential approximation.

In this paper we introduce 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 new sequences and give a closed form expression for the number of weak ascent sequences having a prescribed length and number of weak ascents.

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 on Cayley permutations, we present an alternative view of the transport of patterns as a Wilf-equivalence between subsets of Cayley permutations. We also highlight a connection with primitive ascent sequences.

Drawing on a problem posed by Hertzsprung in 1887, we say that a given permutation π ∈ Sn contains the Hertzsprung pattern σ ∈ Sk if there is factor π(d+1)π(d+2) · · · π(d+k) of π such that π(d+1)−σ(1) = · · · = π(d+k)−σ(k). Using a combination of the Goulden-Jackson cluster method and the transfermatrix method we determine the joint distribution of occurrences of any set of (incomparable) Hertzsprung patterns, thus substantially generalizing earlier results by Jackson et al. on the distribution of ascending and descending runs in permutations. We apply our results to the problem of counting permutations up to pattern-replacement equivalences, and using pattern-rewriting systems— a new formalism similar to the much studied string-rewriting systems—we solve a couple of open problems raised by Linton et al. in 2012.