A Major-Index Preserving Map on Fillings

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.

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

The Electronic Journal of Combinatorics ◽

10.37236/5485 ◽

2016 ◽

Vol 23 (1) ◽

Cited By ~ 10

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.

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

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2405 ◽

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.

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Pathways for penetration of iron and negative stain across the protein shell of ferritin: Ultrastructural perspectives

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100129693 ◽

1992 ◽

Vol 50 (2) ◽

pp. 1012-1013

Author(s):

William H. Massover

Keyword(s):

Outer Shell ◽

X Ray Diffraction ◽

Central Cavity ◽

X Ray ◽

Subunit Dissociation ◽

Negative Stain ◽

Analogous Question ◽

Protein Shell

Molecules of the metalloprotein, ferritin, have an outer shell comprised of a polymeric assembly of 24 polypeptide subunits (apoferritin). This protein shell encloses a hydrated space, the central cavity, within which up to several thousand iron atoms can be deposited as the biomineral, ferrihydrite. The actual pathway taken by iron moving across the protein shell is not known; an analogous question exists for the demonstrated entrance of negative stains into the central cavity. Intersubunit interstices at the 4-fold and 3-fold symmetry axes have been defined with x-ray diffraction, and were hypothesized to provide a pathway for penetration through the outer shell; however, since these channels are only 4Å in width, they are much too small to allow simple permeation of either hydrated iron or stain ions. A different hypothesis, based on studies of subunit dissociation from highly diluted ferritin, proposes that transient gaps in the protein shell are created by a rapid reversible subunit release and permit the direct passage of large ions into the central cavity.

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