Anders Claesson
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View article: Enumerative aspects of Caylerian polynomials
Enumerative aspects of Caylerian polynomials Open
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…
View article: Pattern-avoiding Cayley permutations via combinatorial species
Pattern-avoiding Cayley permutations via combinatorial species Open
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 ado…
View article: Restricted Permutations Enumerated by Inversions
Restricted Permutations Enumerated by Inversions Open
Permutations are usually enumerated by size, but new results can be found by\nenumerating them by inversions instead, in which case one must restrict one's\nattention to indecomposable permutations. In the style of the seminal paper by\nSi…
View article: Modified difference ascent sequences and Fishburn structures
Modified difference ascent sequences and Fishburn structures Open
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 bou…
View article: Permutations with Few Inversions
Permutations with Few Inversions Open
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…
View article: Caylerian polynomials
Caylerian polynomials Open
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 li…
View article: Permutations with few inversions
Permutations with few inversions Open
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…
View article: Turning cycle restrictions into mesh patterns via Foata's fundamental transformation
Turning cycle restrictions into mesh patterns via Foata's fundamental transformation Open
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. A…
View article: Counting tournament score sequences
Counting tournament score sequences Open
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…
View article: Fishburn trees
Fishburn trees Open
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 Fishb…
View article: Counting tournament score sequences
Counting tournament score sequences Open
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…
View article: Combinatorial Exploration: An algorithmic framework for enumeration
Combinatorial Exploration: An algorithmic framework for enumeration Open
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 an…
View article: Weak ascent sequences and related combinatorial structures
Weak ascent sequences and related combinatorial structures Open
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 leng…
View article: Counting Pop-Stacked Permutations in Polynomial Time
Counting Pop-Stacked Permutations in Polynomial Time Open
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 c…
View article: From Hertzsprung's problem to pattern-rewriting systems
From Hertzsprung's problem to pattern-rewriting systems Open
Drawing on a problem posed by Hertzsprung in 1887, we say that a given permutation $π\in\mathcal{S}_n$ contains the Hertzsprung pattern $σ\in\mathcal{S}_k$ if there is factor $π(d+1)π(d+2)\cdotsπ(d+k)$ of $π$ such that $π(d+1)-σ(1) =\cdots…
View article: Sorting with Pattern-Avoiding Stacks: The 132-Machine
Sorting with Pattern-Avoiding Stacks: The 132-Machine Open
This paper continues the analysis of the pattern-avoiding sorting machines recently introduced by Cerbai, Claesson and Ferrari (2020). These devices consist of two stacks, through which a permutation is passed in order to sort it, where th…
View article: Transport of patterns by Burge transpose
Transport of patterns by Burge transpose Open
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 …
View article: Number of pop-stacked permutations
Number of pop-stacked permutations Open
The number of pop-stacked permutations of [n] for n = 1 to 1000 (sequence A307030 in the OEIS) as well as a triangle of numbers giving the number of pop-stacked permutations of each length grouped by number of ascending runs up to n = 300.
View article: triangle.txt
triangle.txt Open
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View article: Sorting with pattern-avoiding stacks: The 132-machine
Sorting with pattern-avoiding stacks: The 132-machine Open
This paper continues the analysis of the pattern-avoiding sorting machines recently introduced by Cerbai, Claesson and Ferrari (2020). These devices consist of two stacks, through which a permutation is passed in order to sort it, where th…
View article: sequence.txt
sequence.txt Open
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View article: Stack sorting with restricted stacks
Stack sorting with restricted stacks Open
The (classical) problem of characterizing and enumerating permutations that can be sorted using two stacks connected in series is still largely open. In the present paper we address a related problem, in which we impose restrictions both o…
View article: Simultaneous Avoidance of a Vincular and a Covincular Pattern of Length 3
Simultaneous Avoidance of a Vincular and a Covincular Pattern of Length 3 Open
A pattern is said to be covincular if its inverse is vincular. In this paper we count the number of permutations simultaneously avoiding a vincular and a covincular pattern, both of length 3. We see familiar sequences, such as the Catalan …
View article: Enumerations of Permutations Simultaneously Avoiding a Vincular and a Covincular Pattern of Length 3
Enumerations of Permutations Simultaneously Avoiding a Vincular and a Covincular Pattern of Length 3 Open
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 per…
View article: Subword counting and the incidence algebra
Subword counting and the incidence algebra Open
The Pascal matrix, $P$, is an upper diagonal matrix whose entries are the binomial coefficients. In 1993 Call and Velleman demonstrated that it satisfies the beautiful relation $P=\exp(H)$ in which $H$ has the numbers 1, 2, 3, etc. on its …
View article: Isomorphisms between pattern classes
Isomorphisms between pattern classes Open
Isomorphisms p between pattern classes A and B are considered. It is shown that, if p is not a symmetry of the entire set of permutations, then, to within symmetry, A is a subset of one a small set of pattern classes whose structure, inclu…