scholarly journals An Extension of MacMahon's Equidistribution Theorem to Ordered Multiset Partitions

10.37236/5485 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Andrew Timothy Wilson
Keyword(s):  
Classical Result ◽  
Macdonald Polynomials ◽  
Major Index ◽  
Insertion Method ◽  
Inversion Number

A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work, we prove a strengthening of MacMahon's theorem originally conjectured by Haglund. Our result can be seen as an equidistribution theorem over the ordered partitions of a multiset into sets, which we call ordered multiset partitions. Our proof is bijective and involves a new generalization of Carlitz's insertion method. This generalization leads to a new extension of Macdonald polynomials for hook shapes. We use our main theorem to show that these polynomials are symmetric and we give their Schur expansion.   A corrigendum was added 17 September 2019.

scholarly journals An extension of MacMahon's Equidistribution Theorem to ordered multiset partitions

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Andrew Timothy Wilson
Keyword(s):  
Classical Result ◽  
Macdonald Polynomials ◽  
Major Index ◽  
Insertion Method ◽  
Inversion Number ◽  
Nous Montrons ◽  
International Audience ◽  
Nous Appellerons ◽  
Comme Application

International audience A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work we prove a strengthening of this theorem originally conjectured by Haglund. Our result can be seen as an equidistribution theorem over the ordered partitions of a multiset into sets, which we call ordered multiset partitions. Our proof is bijective and involves a new generalization of Carlitz's insertion method. As an application, we develop refined Macdonald polynomials for hook shapes. We show that these polynomials are symmetric and give their Schur expansion. Un résultat classique de MacMahon affirme que nombre d’inversion et l’indice majeur ont la même distribution sur permutations d’un multi-ensemble donné. Dans ce travail, nous démontrons un renforcement de ce théorème origine conjecturé par Haglund. Notre résultat peut être considéré comme un théorème d’équirépartition sur les partitions ordonnées d’un multi-ensemble en ensembles, que nous appellerons partitions de multiset commandés. Notre preuve est bijective et implique une nouvelle généralisation de la méthode d’insertion de Carlitz. Comme application, nous développons des polynômes de Macdonald raffinés pour formes d’hameçons. Nous montrons que ces polynômes sont symétriques et donnent leur expansion Schur.

scholarly journals 0-Hecke algebra actions on coinvariants and flags

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Jia Huang
Keyword(s):  
Generating Functions ◽  
Hecke Algebra ◽  
Flag Variety ◽  
Major Index ◽  
Inversion Number ◽  
International Audience ◽  
Descent Set ◽  
Complete Flag ◽  
Coinvariant Algebra

International audience By investigating the action of the 0-Hecke algebra on the coinvariant algebra and the complete flag variety, we interpret generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. En étudiant l'action de l'algèbre de 0-Hecke sur l'algèbre coinvariante et la variété de drapeaux complète, nous interprétons les fonctions génératrices qui comptent les permutations avec un ensemble inverse de descentes fixé, selon leur nombre d'inversions et leur "major index''.

scholarly journals Signed Words and Permutations II; The Euler-Mahonian Polynomials

10.37236/1879 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
Dominique Foata ◽  
Guo-Niu Han
Keyword(s):  
Symmetric Group ◽  
Generating Functions ◽  
Statistical Study ◽  
Record Values ◽  
The Other ◽  
Length Function ◽  
Major Index ◽  
Signed Permutations ◽  
Inversion Number ◽  
The One

As for the symmetric group of ordinary permutations there is also a statistical study of the group of signed permutations, that consists of calculating multivariable generating functions for this group by statistics involving record values and the length function. Two approaches are here systematically explored, using the flag-major index on the one hand, and the flag-inversion number on the other hand. The MacMahon Verfahren appears as a powerful tool throughout.

scholarly journals The Inversion Number and the Major Index are Asymptotically Jointly Normally Distributed on Words

2016 ◽  
Vol 25 (3) ◽  
pp. 470-483
Author(s):  
MARKO THIEL
Keyword(s):  
Normal Distribution ◽  
Joint Distribution ◽  
Bivariate Normal Distribution ◽  
Bivariate Normal ◽  
Asymptotically Normal ◽  
Major Index ◽  
Inversion Number ◽  
Mahonian Statistics ◽  
Normally Distributed

In a recent paper, Baxter and Zeilberger showed that the two most important Mahonian statistics, the inversion number and the major index, are asymptotically independently normally distributed on permutations. In another recent paper, Canfield, Janson and Zeilberger proved the result, already known to statisticians, that the Mahonian distribution is asymptotically normal on words. This leaves one question unanswered: What, asymptotically, is the joint distribution of the inversion number and the major index on words? We answer this question by establishing convergence to a bivariate normal distribution.

scholarly journals A Bijective Proof of a Major Index Theorem of Garsia and Gessel

10.37236/336 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Mordechai Novick
Keyword(s):  
Generating Function ◽  
Index Theorem ◽  
Bijective Proof ◽  
Major Index ◽  
Inversion Number ◽  
And Inversion ◽  
Special Case

In this paper we provide a bijective proof of a theorem of Garsia and Gessel describing the generating function of the major index over the set of all permutations of $[n]=\{1,...,n\}$ which are shuffles of given disjoint ordered sequences $\pi_1,...,\pi_k$ whose union is $[n]$. The proof is based on a result (an "insertion lemma") of Haglund, Loehr, and Remmel which describes the change in major index resulting from the insertion of a given new element in any place in a given permutation. Using this lemma we prove the theorem by establishing a bijection between shuffles of ordered sequences and a certain set of partitions. A special case of Garsia and Gessel's theorem provides a proof of the equidistribution of major index and inversion number over inverse descent classes, a result first proved bijectively by Foata and Schutzenberger in 1978. We provide, based on the method of our first proof, another bijective proof of this result.

scholarly journals A Major-Index Preserving Map on Fillings

10.37236/6893 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Per Alexandersson ◽  
Mehtaab Sawhney
Keyword(s):  
Macdonald Polynomials ◽  
Uniqueness Property ◽  
Alternative Models ◽  
Major Index ◽  
Analogous Question ◽  
Key Polynomials

We generalize a map by S. Mason regarding two combinatorial models for key polynomials, in a way that accounts for the major index. Furthermore we define a similar variant of this map, that regards alternative models for the modified Macdonald polynomials at t=0, and thus partially answers a question by J. Haglund. These maps together imply a certain uniqueness property regarding inversion–and coinversion-free fillings. These uniqueness properties allow us to generalize the notion of charge to a non-symmetric setting, thus answering a question by A. Lascoux and the analogous question in the symmetric setting proves a conjecture by K. Nelson.

Major Index and Inversion Number of Permutations

1978 ◽  
Vol 83 (1) ◽  
pp. 143-159 ◽  
Author(s):  
Dominique Foata ◽  
Marcel-Paul Schützenberger
Keyword(s):  
Major Index ◽  
Inversion Number ◽  
And Inversion

Augmented reality–assisted pedicle screw insertion: a cadaveric proof-of-concept study

2019 ◽  
Vol 31 (1) ◽  
pp. 139-146 ◽  
Author(s):  
Camilo A. Molina ◽  
Nicholas Theodore ◽  
A. Karim Ahmed ◽  
Erick M. Westbroek ◽  
Yigal Mirovsky ◽  
...  
Keyword(s):  
Augmented Reality ◽  
Pedicle Screw ◽  
User Experience ◽  
Pedicle Screws ◽  
Screw Insertion ◽  
Pedicle Screw Insertion ◽  
Insertion Method ◽  
Comparative Accuracy ◽  
Placement Accuracy ◽  
Surgical Field

OBJECTIVEAugmented reality (AR) is a novel technology that has the potential to increase the technical feasibility, accuracy, and safety of conventional manual and robotic computer-navigated pedicle insertion methods. Visual data are directly projected to the operator’s retina and overlaid onto the surgical field, thereby removing the requirement to shift attention to a remote display. The objective of this study was to assess the comparative accuracy of AR-assisted pedicle screw insertion in comparison to conventional pedicle screw insertion methods.METHODSFive cadaveric male torsos were instrumented bilaterally from T6 to L5 for a total of 120 inserted pedicle screws. Postprocedural CT scans were obtained, and screw insertion accuracy was graded by 2 independent neuroradiologists using both the Gertzbein scale (GS) and a combination of that scale and the Heary classification, referred to in this paper as the Heary-Gertzbein scale (HGS). Non-inferiority analysis was performed, comparing the accuracy to freehand, manual computer-navigated, and robotics-assisted computer-navigated insertion accuracy rates reported in the literature. User experience analysis was conducted via a user experience questionnaire filled out by operators after the procedures.RESULTSThe overall screw placement accuracy achieved with the AR system was 96.7% based on the HGS and 94.6% based on the GS. Insertion accuracy was non-inferior to accuracy reported for manual computer-navigated pedicle insertion based on both the GS and the HGS scores. When compared to accuracy reported for robotics-assisted computer-navigated insertion, accuracy achieved with the AR system was found to be non-inferior when assessed with the GS, but superior when assessed with the HGS. Last, accuracy results achieved with the AR system were found to be superior to results obtained with freehand insertion based on both the HGS and the GS scores. Accuracy results were not found to be inferior in any comparison. User experience analysis yielded “excellent” usability classification.CONCLUSIONSAR-assisted pedicle screw insertion is a technically feasible and accurate insertion method.

scholarly journals Evaluation of Nonsymmetric Macdonald Superpolynomials at Special Points

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 779
Author(s):  
Charles F. Dunkl
Keyword(s):  
Preceding Paper ◽  
Hecke Algebra ◽  
Type A ◽  
Macdonald Polynomials ◽  
Special Points ◽  
Two Parameters

In a preceding paper the theory of nonsymmetric Macdonald polynomials taking values in modules of the Hecke algebra of type A (Dunkl and Luque SLC 2012) was applied to such modules consisting of polynomials in anti-commuting variables, to define nonsymmetric Macdonald superpolynomials. These polynomials depend on two parameters q,t and are defined by means of a Yang–Baxter graph. The present paper determines the values of a subclass of the polynomials at the special points 1,t,t2,… or 1,t−1,t−2,…. The arguments use induction on the degree and computations with products of generators of the Hecke algebra. The resulting formulas involve q,t-hook products. Evaluations are also found for Macdonald superpolynomials having restricted symmetry and antisymmetry properties.

Versal topological stratification and the bifurcation geometry of map-germs of the plane

1990 ◽  
Vol 107 (1) ◽  
pp. 127-147 ◽  
Author(s):  
J. H. Rieger
Keyword(s):  
Differential Geometry ◽  
Classical Result ◽  
Smooth Surface ◽  
Critical Values ◽  
Central Projections ◽  
Apparent Contour ◽  
Smooth Map

The set of critical values of a map of the plane (of corank 1) can be regarded as the apparent contour of a smooth surface. It is a classical result of Whitney[13] that generically the apparent contour is a smooth fold curve with isolated cusps and transverse fold crossings. More recent classifications of smooth map-germs of the plane of low codimension (occurring in generic 2- or 3-parameter families), e.g. in [1, 5, 7], were motivated by a question in differential geometry: given any smooth surface which is generically embedded in ℝ3, produce a list of all possible (orthogonal or central) projections of such a surface.

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