scholarly journals Unsplittable Classes of Separable Permutations

10.37236/6115 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Michael Albert ◽  
Vít Jelínek
Keyword(s):  
Separable Permutations

A permutation class is splittable if it is contained in the merge of two of its proper subclasses. We characterise the unsplittable subclasses of the class of separable permutations both structurally and in terms of their bases.

scholarly journals Subclasses of the separable permutations

2011 ◽  
Vol 43 (5) ◽  
pp. 859-870 ◽  
Author(s):  
Michael H. Albert ◽  
M. D. Atkinson ◽  
Vincent Vatter
Keyword(s):  
Separable Permutations

scholarly journals Equipopularity Classes in the Separable Permutations

10.37236/4797 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Michael Albert ◽  
Cheyne Homberger ◽  
Jay Pantone
Keyword(s):  
Set Of Permutations ◽  
Separable Permutations

When two patterns occur equally often in a set of permutations, we say that these patterns are equipopular. Using both structural and analytic tools, we classify the equipopular patterns in the set of separable permutations. In particular, we show that the number of equipopularity classes for length $n$ patterns in the separable permutations is equal to the number of partitions of $n-1$.

scholarly journals A Combinatorial Bijection on di-sk Trees

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Shishuo Fu ◽  
Zhicong Lin ◽  
Yaling Wang
Keyword(s):  
Binary Tree ◽  
Number Of Components ◽  
Symmetric Extension ◽  
Alternative Approach ◽  
Tree Traversal ◽  
Separable Permutations ◽  
Rooted Binary Tree ◽  
Schur Positive

A di-sk tree is a rooted binary tree whose nodes are labeled by $\oplus$ or $\ominus$, and no node has the same label as its right child. The di-sk trees are in natural bijection with separable permutations. We construct a combinatorial bijection on di-sk trees proving  the two quintuples $(\mathrm{LMAX},\mathrm{LMIN},\mathrm{DESB},\mathsf{iar},\mathsf{comp})$ and $(\mathrm{LMAX},\mathrm{LMIN},\mathrm{DESB},\mathsf{comp},\mathsf{iar})$ have the same distribution over separable permutations. Here for a permutation $\pi$, $\mathrm{LMAX}(\pi)/\mathrm{LMIN}(\pi)$ is the set of values of the left-to-right maxima/minima of $\pi$ and $\mathrm{DESB}(\pi)$ is the set of descent bottoms of $\pi$, while $\mathsf{comp}(\pi)$ and $\mathsf{iar}(\pi)$ are respectively  the number of components of $\pi$ and the length of initial ascending run of $\pi$.  Interestingly, our bijection specializes to a bijection on $312$-avoiding permutations, which provides  (up to the classical Knuth–Richards bijection) an alternative approach to a result of Rubey (2016) that asserts the  two triples $(\mathrm{LMAX},\mathsf{iar},\mathsf{comp})$ and $(\mathrm{LMAX},\mathsf{comp},\mathsf{iar})$ are equidistributed  on $321$-avoiding permutations. Rubey's result is a symmetric extension of an equidistribution due to Adin–Bagno–Roichman, which implies the class of $321$-avoiding permutations with a prescribed number of components is Schur positive.  Some equidistribution results for various statistics concerning tree traversal are presented in the end.

scholarly journals Formal Basis of a Language Universal

2021 ◽  
Vol 47 (1) ◽  
pp. 9-42
Author(s):  
Miloš Stanojević ◽  
Mark Steedman
Keyword(s):  
Present Article ◽  
Machine Translation ◽  
Word Order ◽  
Practical Interest ◽  
Categorial Grammars ◽  
Computer Scientists ◽  
Formal Basis ◽  
Crosslinguistic Variation ◽  
Language Universal ◽  
Separable Permutations

Abstract Steedman (2020) proposes as a formal universal of natural language grammar that grammatical permutations of the kind that have given rise to transformational rules are limited to a class known to mathematicians and computer scientists as the “separable” permutations. This class of permutations is exactly the class that can be expressed in combinatory categorial grammars (CCGs). The excluded non-separable permutations do in fact seem to be absent in a number of studies of crosslinguistic variation in word order in nominal and verbal constructions. The number of permutations that are separable grows in the number n of lexical elements in the construction as the Large Schröder Number Sn−1. Because that number grows much more slowly than the n! number of all permutations, this generalization is also of considerable practical interest for computational applications such as parsing and machine translation. The present article examines the mathematical and computational origins of this restriction, and the reason it is exactly captured in CCG without the imposition of any further constraints.

scholarly journals The topology of the permutation pattern poset

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Peter McNamara ◽  
Einar Steingrımsson
Keyword(s):  
Homotopy Type ◽  
Recursive Formula ◽  
Large Classes ◽  
Permutation Pattern ◽  
International Audience ◽  
Rooted Forest ◽  
Carry Over ◽  
Almost All ◽  
Separable Permutations ◽  
Pattern Containment

International audience The set of all permutations, ordered by pattern containment, forms a poset. This extended abstract presents the first explicit major results on the topology of intervals in this poset. We show that almost all (open) intervals in this poset have a disconnected subinterval and are thus not shellable. Nevertheless, there seem to be large classes of intervals that are shellable and thus have the homotopy type of a wedge of spheres. We prove this to be the case for all intervals of layered permutations that have no disconnected subintervals of rank 3 or more. We also characterize in a simple way those intervals of layered permutations that are disconnected. These results carry over to the poset of generalized subword order when the ordering on the underlying alphabet is a rooted forest. We conjecture that the same applies to intervals of separable permutations, that is, that such an interval is shellable if and only if it has no disconnected subinterval of rank 3 or more. We also present a simplified version of the recursive formula for the Möbius function of decomposable permutations given by Burstein et al.

scholarly journals Parallel algorithms for separable permutations

2005 ◽  
Vol 146 (3) ◽  
pp. 343-364 ◽  
Author(s):  
V. Yugandhar ◽  
Sanjeev Saxena
Keyword(s):  
Parallel Algorithms ◽  
Separable Permutations

scholarly journals The Brownian limit of separable permutations

2018 ◽  
Vol 46 (4) ◽  
pp. 2134-2189 ◽  
Author(s):  
Frédérique Bassino ◽  
Mathilde Bouvel ◽  
Valentin Féray ◽  
Lucas Gerin ◽  
Adeline Pierrot
Keyword(s):  
Brownian Limit ◽  
Separable Permutations

scholarly journals The Möbius function of separable permutations (extended abstract)

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Vít Jelínek ◽  
Eva Jelínková ◽  
Einar Steingrímsson
Keyword(s):  
Möbius Function ◽  
Computationally Efficient ◽  
Mobius Function ◽  
Nous Montrons ◽  
International Audience ◽  
Separable Permutations ◽  
Pattern Containment

International audience A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142. Using the notion of separating tree, we give a computationally efficient formula for the Möbius function of an interval $(q,p)$ in the poset of separable permutations ordered by pattern containment. A consequence of the formula is that the Möbius function of such an interval $(q,p)$ is bounded by the number of occurrences of $q$ as a pattern in $p$. The formula also implies that for any separable permutation $p$ the Möbius function of $(1,p)$ is either 0, 1 or -1. Une permutation est séparable si elle peut être générée á partir de la permutation 1 par des sommes directes et des sommes indirectes, ou de façon équivalente, si elle évite les motifs 2413 et 3142. En utilisant le concept de l'arbre séparant, nous donnons une formule pour le calcul efficace de la fonction de Möbius d'un intervalle de $(q, p)$ dans l'ensemble partiellement ordonné des permutations séparables. Une conséquence est que la fonction de Möbius de $(q,p)$ pour $q$ et $p$ séparables est bornée par le nombre d'occurrences de $q$ comme un motif en $p$. Nous montrons aussi que pour une permutation $p$ séparable, la fonction de Möbius de $(1,p)$ est soit 0, 1 ou -1.

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