arXiv subjects

Henning Ulfarsson

Publications and source records attributed to Henning Ulfarsson.

At least 19 recordsLinked to original sources

arXiv2026

Protected tails and polynomial-time enumeration of permutations avoiding a direct sum of an increasing pattern and 231

We give an exact algorithm counting the permutations that avoid a fixed pattern from the following family: the direct sum of an increasing pattern and the pattern 231. The first members of the family are 1342 and 12453. For each member, the algorithm computes the number of avoiding permutations of every length up to a given bound using polynomially many arithmetic operations and polynomially many stored integers, with degrees that grow linearly in the length of the pattern. We first obtain an exact recurrence by reading a permutation from left to right and recording, at each step, the constraints that the letters read so far impose on those still unread. Its state space grows exponentially, so evaluating it directly takes exponential time. We then show that part of the state is protected: later steps carry it along unchanged and do not depend on it. Factoring the protected part out turns the recurrence into a dynamic program with polynomially many stored transfer entries, and this gives the polynomial bounds for every member of the family. For the pattern 12453, a translation symmetry sharpens the bounds to degree seven for the operations and degree four for the storage. Separately written implementations and exact Chinese-remainder certification determine the number of 12453-avoiding permutations of every length up to 150. The previously published series reached length 38. The same tables also generate uniformly random avoiders in polynomial time. We illustrate this with a heatmap of one million 12453-avoiding permutations of length 300 sampled with floating-point tables. The counting recurrences for 1342 and 12453 are verified in the Lean 4 proof assistant.

math.CO↗

arXiv2026

Inversion monotonicity in subclasses of the 1324-avoiders

A collection $B$ of patterns is called inversion monotone if $\mathrm{av}_n^k(B)$, 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: $\mathrm{av}_n^k(1324)$ 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 $n \geq \frac{k+7}{2}$. The method yields an enumeration of $\mathrm{av}_n^k(1324, 1342)$ when $n \geq \frac{k+7}{2}$.

math.CO↗

arXiv2026

A Permutation Avoidance Game with Reverse Replies and Monotone Traps

We study the impartial game PAP (``permutations avoiding patterns''), in which players take turns choosing patterns to avoid. We define a set of length $k$ patterns, $B_k$, and show that it is the unique minimal monotone-forcing subset of $S_k$: every sufficiently long permutation that avoids $B_k$ is monotone, and every monotone-forcing subset of $S_k$ must contain $B_k$. We prove a quadratic upper bound for the monotone-forcing threshold, and determine the exact thresholds for $k=3,4,5,6$. We use properties of the sets $B_k$ to prove that a reverse-reply strategy wins PAP on $S_n$ when $k=4$ for all $n \geq 10$; for $k=3$, the same strategy can be analysed directly. We conjecture that it is a winning strategy for all $k$ and $n$ sufficiently large.

math.CO↗

arXiv2024

BiSC: An algorithm for discovering generalized permutation patterns

Theorems relating permutations with objects in other fields of mathematics are often stated in terms of avoided patterns. Examples include various classes of Schubert varieties from algebraic geometry (Billey and Abe 2013), commuting functions in analysis (Baxter 1964), beta-shifts in dynamical systems (Elizalde 2011) and homology of representations (Sundaram 1994). We present a new algorithm, BiSC, that, given any set of permutations, outputs a conjecture for describing the set in terms of avoided patterns. The algorithm automatically conjectures the statements of known theorems such as the descriptions of smooth (Lakshmibai and Sandhya 1990) and forest-like permutations (Bousquet-M{é}lou and Butler 2007), Baxter permutations (Chung et al. 1978), stack-sortable (Knuth 1975) and West-2-stack-sortable permutations (West 1990). The algorithm has also been used to discover new theorems and conjectures related to the dihedral and alternating subgroups of the symmetric group, Young tableaux, Wilf-equivalences, and sorting devices.

math.CO↗

arXiv2024

Combinatorial Exploration: An algorithmic framework for enumeration

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.

math.CO↗

arXiv2023

Permutations avoiding bipartite partially ordered patterns have a regular insertion encoding

We prove that any class of permutations defined by avoiding a partially ordered pattern (POP) with height at most two has a regular insertion encoding and thus has a rational generating function. Then, we use Combinatorial Exploration to find combinatorial specifications and generating functions for hundreds of other permutation classes defined by avoiding a size 5 POP, allowing us to resolve several conjectures of Gao and Kitaev and of Chen and Lin.

math.CO↗

arXiv2023

Turning cycle restrictions into mesh patterns via Foata's fundamental transformation

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.

math.CO↗

arXiv2021

Enumeration of Permutation Classes and Weighted Labelled Independent Sets

In this paper, we study the staircase encoding of permutations, which maps a permutation to a staircase grid with cells filled with permutations. We consider many cases, where restricted to a permutation class, the staircase encoding becomes a bijection to its image. We describe the image of those restrictions using independent sets of graphs weighted with permutations. We derive the generating function for the independent sets and then for their weighted counterparts. The bijections we establish provide the enumeration of permutation classes. We use our results to uncover some unbalanced Wilf-equivalences of permutation classes and outline how to do random sampling in the permutation classes. In particular, we cover the classes $\mathrm{Av}(2314,3124)$, $\mathrm{Av}(2413,3142)$, $\mathrm{Av}(2413,3124)$, $\mathrm{Av}(2413,2134)$ and $\mathrm{Av}(2314,2143)$, as well as many subclasses.

math.CO↗

arXiv2019

Pattern avoiding permutations and independent sets in graphs

We introduce a new method for encoding permutations as weighted independent sets in a family of graphs we call cores. The encoding allows us to enumerate (1324, 2143)-, (1234, 1324, 2143)-, (1234, 1324, 1432, 3214)-avoiding permutations with respect to their number of "boundary points" and the size of the independent set in the graph they correspond to.

math.CO↗

arXiv2019

Algorithmic coincidence classification of mesh patterns

We review and extend previous results on coincidence of mesh patterns. We introduce the notion of a force on a permutation pattern and apply it to the coincidence classification of mesh patterns, completing the classification up to size three. We also show that this concept can be used to enumerate classical permutation classes.

math.CO↗

arXiv2018

The Poset of Mesh Patterns

We introduce the poset of mesh patterns, which generalises the permutation pattern poset. We fully classify the mesh patterns for which the interval [1^\emptyset,m] is non-pure, where 1^\emptyset is the unshaded singleton mesh pattern. We present some results on the Möbius function of the poset, and show that μ(1^\emptyset,m) is almost always zero. Finally, we introduce a class of disconnected and non-shellable intervals by generalising the direct product operation from permutations to mesh patterns.

math.CO↗

arXiv2018

Equivalence classes of mesh patterns with a dominating pattern

Two mesh patterns are coincident if they are avoided by the same set of permutations, and are Wilf-equivalent if they have the same number of avoiders of each length. We provide sufficient conditions for coincidence of mesh patterns, when only permutations also avoiding a longer classical pattern are considered. Using these conditions we completely classify coincidences between families containing a mesh pattern of length 2 and a classical pattern of length 3. Furthermore, we completely Wilf-classify mesh patterns of length 2 inside the class of 231-avoiding permutations.

math.CO↗

arXiv2017

Enumerations of Permutations Simultaneously Avoiding a Vincular and a Covincular Pattern of Length 3

Vincular and covincular patterns are generalizations of classical patterns allowing restrictions on the indices and values of the occurrences in a permutation. In this paper we study the integer sequences arising as the enumerations of permutations simultaneously avoiding a vincular and a covincular pattern, both of length 3, with at most one restriction. We see familiar sequences, such as the Catalan and Motzkin numbers, but also some previously unknown sequences which have close links to other combinatorial objects such as lattice paths and integer partitions. Where possible we include a generating function for the enumeration. One of the cases considered settles a conjecture by Pudwell (2010) on the Wilf-equivalence of barred patterns. We also give an alternative proof of the classic result that permutations avoiding 123 are counted by the Catalan numbers.

math.CO↗

arXiv2017

Automatic discovery of structural rules of permutation classes

We introduce an algorithm that conjectures the structure of a permutation class in the form of a disjoint cover of "rules"; similar to generalized grid classes. The cover is usually easily verified by a human and translated into an enumeration. The algorithm is successful on different inputs than other algorithms and can succeed with any polynomial permutation class. We apply it to every non-polynomial permutation class avoiding a set of length four patterns. The structures found by the algorithm can sometimes allow an enumeration of the permutation class with respect to permutation statistics, as well as choosing a permutation uniformly at random from the permutation class. We sketch a new algorithm formalizing the human verification of the conjectured covers.

math.CO↗

arXiv2016

Occurrence graphs of patterns in permutations

We define the \emph{occurrence graph} $G_p(π$) of a pattern $p$ in a permutation $π$ as the graph with the occurrences of $p$ in $π$ as vertices and edges between the vertices if the occurrences differ by exactly one element. We then study properties of these graphs. The main theorem in this paper is that every \emph{hereditary property} of graphs gives rise to a \emph{permutation class}.

math.CO↗

arXiv2016

Coincidence among families of mesh patterns

Two mesh patterns are coincident if they are avoided by the same set of permutations. In this paper, we provide necessary conditions for this coincidence, which include having the same set of enclosed diagonals. This condition is sufficient to prove coincidence of vincular patterns, although it is not enough to guarantee coincidence of bivincular patterns. In addition, we provide a generalization of the Shading Lemma (Hilmarsson et al.), a result that examined when a square could be added to the mesh of a pattern.

math.CO↗

arXiv2015

Wilf-classification of mesh patterns of short length

This paper starts the Wilf-classification of mesh patterns of length 2. Although there are initially 1024 patterns to consider we introduce automatic methods to reduce the number of potentially different Wilf-classes to at most 65. By enumerating some of the remaining classes we bring that upper-bound further down to 56. Finally, we conjecture that the actual number of Wilf-classes of mesh patterns of length 2 is 46.

math.CO↗

arXiv2015

Collatz meets Fibonacci

The Collatz map is defined for a positive even integer as half that integer, and for a positive odd integer as that integer threefold, plus one. The Collatz conjecture states that when the map is iterated the number one is eventually reached. We study permutations that arise as sequences from this iteration. We show that permutations of this type of length up to 14 are enumerated by the Fibonacci numbers. Beyond that excess permutations appear. We will explain the appearance of these excess permutations and give an upper bound on the exact enumeration.

math.CO↗