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View article: Mini-Workshop: Permutation Patterns
Mini-Workshop: Permutation Patterns Open
The study of permutation patterns has recently seen several surprising results, and the purpose of this mini-workshop was to bring together researchers from across the field to focus on four hot topics related to these recent developments.…
View article: Permutations Avoiding Bipartite Partially Ordered Patterns Have a Regular Insertion Encoding
Permutations Avoiding Bipartite Partially Ordered Patterns Have a Regular Insertion Encoding Open
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 f…
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: The enumeration of inversion sequences avoiding the patterns 201 and 210
The enumeration of inversion sequences avoiding the patterns 201 and 210 Open
We derive the algebraic generating function for inversion sequences avoiding the patterns 201 and 210 by describing a set of succession rules, converting them to a system of generating function equations with one catalytic variable, and th…
View article: Permutations avoiding bipartite partially ordered patterns have a regular insertion encoding
Permutations avoiding bipartite partially ordered patterns have a regular insertion encoding Open
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 f…
View article: The enumeration of inversion sequences avoiding the patterns 201 and 210
The enumeration of inversion sequences avoiding the patterns 201 and 210 Open
We derive the algebraic generating function for inversion sequences avoiding the patterns $201$ and $210$ by describing a set of succession rules, converting them to a system of generating function equations with one catalytic variable, an…
View article: Using large random permutations to partition permutation classes
Using large random permutations to partition permutation classes Open
Permutation classes are sets of permutations defined by the absence of certain substructures. In some cases permutation classes can be decomposed as unions of subclasses. We use combinatorial specifications automatically discovered by Comb…
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: 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: Permutations avoiding sets of patterns with long monotone subsequences
Permutations avoiding sets of patterns with long monotone subsequences Open
We enumerate permutations that avoid all but one of the $k$ patterns of length $k$ starting with a monotone increasing subsequence of length $k-1$. We compare the size of such permutation classes to the size of the class of permutations av…
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: sequence.txt
sequence.txt Open
:unav
View article: Colored Multipermutations and a Combinatorial Generalization of Worpitzky’s Identity
Colored Multipermutations and a Combinatorial Generalization of Worpitzky’s Identity Open
Worpitzky's identity expresses $n^p$ in terms of the Eulerian numbers and binomial coefficients: $$n^p = \sum_{i=0}^{p-1} \genfrac {0pt}{}{p}{i} \binom{n+i}{p}.$$
Pita-Ruiz recently defined numbers $A_{a,b,r}(p,i)$ implicitly to satisfy …
View article: Colored Multipermutations and a Combinatorial Generalization of Worpitzky's Identity
Colored Multipermutations and a Combinatorial Generalization of Worpitzky's Identity Open
Worpitzky's identity expresses $n^p$ in terms of the Eulerian numbers and binomial coefficients: $$n^p = \sum_{i=0}^{p-1} \genfrac<>{0pt}{}{p}{i} \binom{n+i}{p}.$$ Pita-Ruiz recently defined numbers $A_{a,b,r}(p,i)$ implicitly to satisfy a…
View article: On the growth of merges and staircases of permutation classes
On the growth of merges and staircases of permutation classes Open
There is a well-known upper bound on the growth rate of the merge of two permutation classes. Curiously, there is no known merge for which this bound is not achieved. Using staircases of permutation classes, we provide sufficient condition…
View article: Universal Layered Permutations
Universal Layered Permutations Open
We establish an exact formula for the length of the shortest permutation containing all layered permutations of length $n$, proving a conjecture of Gray.
View article: Completing the Structural Analysis of the 2x4 Permutation Classes
Completing the Structural Analysis of the 2x4 Permutation Classes Open
We study the structure and enumeration of the final two 2x4 permutation classes, completing a research program that has spanned almost two decades. For both classes, careful structural analysis produces a complicated functional equation. O…
View article: The asymptotic number of simple singular vector tuples of a cubical tensor
The asymptotic number of simple singular vector tuples of a cubical tensor Open
S. Ekhad and D. Zeilberger recently proved that the multivariate generating function for the number of simple singular vector tuples of a generic \(m_1 \times · · · \times m_d\) tensor has an elegant rational form involving elementary symm…
View article: New bounds on the growth rate of 1324-avoiders
New bounds on the growth rate of 1324-avoiders Open
We establish an improved lower bound of 10.271 for the exponential growth rate of the class of permutations avoiding the pattern 1324, and an improved upper bound of 13.5. These results depend on a new exact structural characterisation of …
View article: A structural characterisation of Av(1324) and new bounds on its growth\n rate
A structural characterisation of Av(1324) and new bounds on its growth\n rate Open
We establish an improved lower bound of 10.271 for the exponential growth\nrate of the class of permutations avoiding the pattern 1324, and an improved\nupper bound of 13.5. These results depend on a new exact structural\ncharacterisation …
View article: Shift equivalence in the generalized factor order
Shift equivalence in the generalized factor order Open
We provide a geometric condition that guarantees strong Wilf equivalence in the generalized factor order. This provides a powerful tool for proving specific and general Wilf equivalence results, and several such examples are given.
View article: Is the full susceptibility of the square-lattice Ising model a differentially algebraic function?
Is the full susceptibility of the square-lattice Ising model a differentially algebraic function? Open
We study the class of non-holonomic power series with integer coefficients\nthat reduce, modulo primes, or powers of primes, to algebraic functions. In\nparticular we try to determine whether the susceptibility of the square-lattice\nIsing…
View article: Pattern avoidance in forests of binary shrubs
Pattern avoidance in forests of binary shrubs Open
We investigate pattern avoidance in permutations satisfying some additional restrictions. These are naturally considered in terms of avoiding patterns in linear extensions of certain forest-like partially ordered sets, which we call binary…
View article: The Asymptotic Number of Simple Singular Vector Tuples of a Cubical Tensor
The Asymptotic Number of Simple Singular Vector Tuples of a Cubical Tensor Open
S. Ekhad and D. Zeilberger recently proved that the multivariate generating function for the number of simple singular vector tuples of a generic $m_1 \times \cdots \times m_d$ tensor has an elegant rational form involving elementary symme…
View article: Growth rates of permutation classes: categorization up to the\n uncountability threshold
Growth rates of permutation classes: categorization up to the\n uncountability threshold Open
In the antecedent paper to this it was established that there is an algebraic\nnumber $\\xi\\approx 2.30522$ such that while there are uncountably many growth\nrates of permutation classes arbitrarily close to $\\xi$, there are only\ncount…