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 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 Emulation of Utility Functions Over a Set of Permutations: Sequencing Reliability Growth Tasks

Technometrics ◽  
2018 ◽  
Vol 60 (3) ◽  
pp. 273-285 ◽  
Author(s):  
Kevin J. Wilson ◽  
Daniel A. Henderson ◽  
John Quigley
Keyword(s):  
Utility Functions ◽  
Reliability Growth ◽  
Set Of Permutations

scholarly journals INTERVAL STRUCTURE OF THE PIERI FORMULA FOR GROTHENDIECK POLYNOMIALS

2013 ◽  
Vol 23 (01) ◽  
pp. 123-146 ◽  
Author(s):  
VIVIANE PONS
Keyword(s):  
Direct Proof ◽  
Hecke Algebra ◽  
Bruhat Order ◽  
Combinatorial Interpretation ◽  
Pieri Formula ◽  
Set Of Permutations ◽  
Grothendieck Polynomials

We give a combinatorial interpretation of a Pieri formula for double Grothendieck polynomials in terms of an interval of the Bruhat order. Another description had been given by Lenart and Postnikov in terms of chain enumerations. We use Lascoux's interpretation of a product of Grothendieck polynomials as a product of two kinds of generators of the 0-Hecke algebra, or sorting operators. In this way, we obtain a direct proof of the result of Lenart and Postnikov and then prove that the set of permutations occurring in the result is actually an interval of the Bruhat order.

Discretely Deranged Squares

1990 ◽  
Vol 83 (4) ◽  
pp. 318-320
Author(s):  
Paul O. Eckhardt
Keyword(s):  
Secondary School ◽  
School Level ◽  
Discrete Mathematics ◽  
College Courses ◽  
College Mathematics ◽  
Entry Level ◽  
Secondary School Level ◽  
Set Of Permutations ◽  
The Waves

The waves of change in entry-level college mathematics should not go unnoticed by those of us who teach at the secondary school level. One such wave, discrete mathematics, is lapping on the shore of the calculus. Many of our students will elect or be required to take college courses that include discrete combinatoric concepts. The focus of this article is to examine a special set of permutations and some of their extensions.

Minimization of additive functionals on a set of permutations

Cybernetics ◽  
1974 ◽  
Vol 8 (6) ◽  
pp. 963-965
Author(s):  
�. I. Volynskii
Keyword(s):  
Additive Functionals ◽  
Set Of Permutations

scholarly journals Mixed volumes of hypersimplices

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Gaku Liu
Keyword(s):  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Type B ◽  
Combinatorial Interpretation ◽  
Mixed Volumes ◽  
Eulerian Numbers ◽  
Eulerian Number ◽  
Nous Montrons ◽  
Set Of Permutations ◽  
International Audience

International audience In this extended abstract we consider mixed volumes of combinations of hypersimplices. These numbers, called mixed Eulerian numbers, were first considered by A. Postnikov and were shown to satisfy many properties related to Eulerian numbers, Catalan numbers, binomial coefficients, etc. We give a general combinatorial interpretation for mixed Eulerian numbers and prove the above properties combinatorially. In particular, we show that each mixed Eulerian number enumerates a certain set of permutations in $S_n$. We also prove several new properties of mixed Eulerian numbers using our methods. Finally, we consider a type $B$ analogue of mixed Eulerian numbers and give an analogous combinatorial interpretation for these numbers. Dans ce résumé étendu nous considérons les volumes mixtes de combinaisons d’hyper-simplexes. Ces nombres, appelés les nombres Eulériens mixtes, ont été pour la première fois étudiés par A. Postnikov, et il a été montré qu’ils satisfont à de nombreuses propriétés reliées aux nombres Eulériens, au nombres de Catalan, aux coefficients binomiaux, etc. Nous donnons une interprétation combinatoire générale des nombres Eulériens mixtes, et nous prouvons combinatoirement les propriétés mentionnées ci-dessus. En particulier, nous montrons que chaque nombre Eulérien mixte compte les éléments d’un certain sous-ensemble de l’ensemble des permutations $S_n$. Nous établissons également plusieurs nouvelles propriétés des nombres Eulériens mixtes grâce à notre méthode. Pour finir, nous introduisons une généralisation en type $B$ des nombres Eulériens mixtes, et nous en donnons une interprétation combinatoire analogue.

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.

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