Discrete CATS Seminar

Master Exam

Speaker:  Maxwell Hosler, University of Kentucky

Title:  An order on circular permutations

Abstract:

Discussing a paper by Abram, Chapelier-Laget, and Reutenauer, we examine a family a lattices with three isomorphic expressions; first, as a lattice of circular permutations, second, as a lattice of natural-valued functions called 'admitted vectors,' and third, as an interval in the weak order on the affine symmetric group. This family turns out to have strong analogies with the weak order on the symmetric group, despite not being a weak order. Amongst other things, admitted vectors act as 'inversion sets with multiplicity' for  these permutations, and the Hasse diagram can be labelled by transpositions in a way reminiscent of how the same can be done for the weak order. We end by proving the fact that, in some sense, the 'limit' of this family of posets is Young's lattice.

https://sites.google.com/view/discretecatsseminar

Discrete CATS Seminar

Master Exam

Speaker:  Maxwell Hosler, University of Kentucky

Title:  An order on circular permutations

Abstract:

Discussing a paper by Abram, Chapelier-Laget, and Reutenauer, we examine a family a lattices with three isomorphic expressions; first, as a lattice of circular permutations, second, as a lattice of natural-valued functions called 'admitted vectors,' and third, as an interval in the weak order on the affine symmetric group. This family turns out to have strong analogies with the weak order on the symmetric group, despite not being a weak order. Amongst other things, admitted vectors act as 'inversion sets with multiplicity' for  these permutations, and the Hasse diagram can be labelled by transpositions in a way reminiscent of how the same can be done for the weak order. We end by proving the fact that, in some sense, the 'limit' of this family of posets is Young's lattice.

https://sites.google.com/view/discretecatsseminar

Discrete CATS Seminar

Master Talk

Speaker:  Evan Henning, University of Kentucky

Title: The incidence Hopf algebra of the non-crossing partition lattice

Abstract:

First studied in the literature by H.W Becker as planar rhyme schemes, non-crossing partitions have long been a combinatorial object of interest. It is well known that the set of non-crossing partitions of [n] inherit a lattice structure as a sublattice of the partition lattice ordered by refinement. Simion and Ullman showed that this lattice is self-dual. Moreover, intervals in the non-crossing partition lattice factor nicely hence the incidence Hopf algebra on the family of intervals of the non-crossing partition lattice has a nice structure. In this talk we will discuss Hillary Einziger's contributions to the study of the structure of this incidence Hopf algebra. In doing so we will find multiple bases, various formulas for the antipode, and show a bijection between this Hopf algebra and that of the symmetric functions.

https://sites.google.com/view/discretecatsseminar

Discrete CATS Seminar

Master Talk

Speaker:  Evan Henning, University of Kentucky

Title: The incidence Hopf algebra of the non-crossing partition lattice

Abstract:

First studied in the literature by H.W Becker as planar rhyme schemes, non-crossing partitions have long been a combinatorial object of interest. It is well known that the set of non-crossing partitions of [n] inherit a lattice structure as a sublattice of the partition lattice ordered by refinement. Simion and Ullman showed that this lattice is self-dual. Moreover, intervals in the non-crossing partition lattice factor nicely hence the incidence Hopf algebra on the family of intervals of the non-crossing partition lattice has a nice structure. In this talk we will discuss Hillary Einziger's contributions to the study of the structure of this incidence Hopf algebra. In doing so we will find multiple bases, various formulas for the antipode, and show a bijection between this Hopf algebra and that of the symmetric functions.

https://sites.google.com/view/discretecatsseminar

Speaker:  Evan Henning

Title:  The incidence Hopf algebra of the non-crossing partition lattice

Abstract:

First studied in the literature by H.W Becker as planar rhyme schemes, non-crossing partitions have long been a combinatorial object of interest. It is well known that the set of non-crossing partitions of [n] inherit a lattice structure as a sublattice of the partition lattice ordered by refinement. Simion and Ullman showed that this lattice is self-dual. Moreover, intervals in the non-crossing partition lattice factor nicely hence the incidence Hopf algebra on the family of intervals of the non-crossing partition lattice has a nice structure. In this talk we will discuss Hillary Einziger's contributions to the study of the structure of this incidence Hopf algebra. In doing so we will find multiple bases, various formulas for the antipode, and show a bijection between this Hopf algebra and that of the symmetric functions.

https://sites.google.com/view/discretecatsseminar/

Speaker:  Evan Henning

Title:  The incidence Hopf algebra of the non-crossing partition lattice

Abstract:

First studied in the literature by H.W Becker as planar rhyme schemes, non-crossing partitions have long been a combinatorial object of interest. It is well known that the set of non-crossing partitions of [n] inherit a lattice structure as a sublattice of the partition lattice ordered by refinement. Simion and Ullman showed that this lattice is self-dual. Moreover, intervals in the non-crossing partition lattice factor nicely hence the incidence Hopf algebra on the family of intervals of the non-crossing partition lattice has a nice structure. In this talk we will discuss Hillary Einziger's contributions to the study of the structure of this incidence Hopf algebra. In doing so we will find multiple bases, various formulas for the antipode, and show a bijection between this Hopf algebra and that of the symmetric functions.

https://sites.google.com/view/discretecatsseminar/

Speaker:  Maxwell Hosler

Title:  TBA

Abstract:

Speaker:  Maxwell Hosler

Title:  TBA

Abstract:

Speaker:  Pablo Castilla, University of Kentucky

Title:  TBA

Date:  April 21, 2025

Abstract:

This is a Masters Exam talk.

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Speaker:  Pablo Castilla, University of Kentucky

Title:  TBA

Date:  April 21, 2025

Abstract:

This is a Masters Exam talk.

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