Some title

Some Person

USomething

Wednesday

Sept. 3, 2014

8:00-8:50am

Vincent Hall 313

Some Abstract. $e^x$

TBA

No seminar- 1st week of class

Friday

Sept. 7, 2018

3:35-4:25pm

Vincent Hall 570

No seminar- 1st week of class

Friday

Jan. 24, 2020

3:35-4:25pm

Vincent Hall 570

Weak order and descents for monotone triangles

Vic Reiner

Friday

Jan. 31, 2020

3:35-4:25pm

Vincent Hall 570

(joint work with Zach Hamaker; arXiv:1809.1057)
Monotone triangles are combinatorial objects in bijection with alternating sign matrices, a fascinating generalization of permutation matrices. We will review this connection, and the fact that strong Bruhat order on permutations has a natural extension to monotone triangles.
We will then explain an analogous extension of the weak Bruhat order on permutations to monotone triangles. This comes from extending the notions of descents in permutations and the "bubble-sorting" action of the 0-Hecke algebra on permutations to monotone triangles.
We will also explain one of our motivations: to give a natural family of shellings for Terwilliger's recently defined order on subsets.

Unconditional Reflexive Polytopes

McCabe Olsen

Ohio State

Friday

Feb. 7, 2020

3:35-4:25pm

Vincent Hall 570

A convex body is unconditional if it is symmetric with respect to reflections in all coordinate hyperplanes. In this paper, we investigate unconditional lattice polytopes with respect to geometric, combinatorial, and algebraic properties. In particular, we characterize unconditional reflexive polytopes in terms of perfect graphs. As a prime example, we study the signed Birkhoff polytope. Moreover, we derive constructions for Gale-dual pairs of polytopes and we explicitly describe Gröbner bases for unconditional reflexive polytopes coming from partially ordered sets. This is joint work with Florian Kohl (Aalto University) and Raman Sanyal (Goethe Universität Frankfurt).

Combinatorics of the double-dimer model

Helen Jenne

Oregon

Friday

Feb. 14, 2020

3:35-4:25pm

Vincent Hall 570

In this talk we will discuss a new result about the
double-dimer model: under certain conditions, the partition function for
double-dimer configurations of a planar bipartite graph satisfies an
elegant recurrence, related to the Desnanot-Jacobi identity from linear
algebra. A similar identity for the number of dimer configurations (or
perfect matchings) of a graph was established nearly 20 years ago by Kuo
and others. We will also explain one of the motivations for this work,
which is a problem in Donaldson-Thomas and Pandharipande-Thomas theory
that will be the subject of a forthcoming paper with Gautam Webb and Ben
Young.

Separable elements and splittings of Weyl groups

Yibo Gao

MIT

Friday

Feb. 21, 2020

3:35-4:25pm

Vincent Hall 570

We introduce separable elements in finite Weyl groups, generalizing the well-studied class of separable permutations. They enjoy nice properties in the weak Bruhat order, enumerate faces of the graph associahedron of the corresponding Dynkin diagrams, and can be characterized by pattern avoidance in the sense of Billey and Postnikov. We then prove that the multiplication map $W/V \times V \to W$ for a generalized quotient of the symmetric group is always surjective when V is a principal order ideal, providing the first combinatorial proof of an inequality due originally to Sidorenko in 1991, answering an open problem of Morales, Pak, and Panova. We show that this multiplication map is a bijection if and only if V is an order ideal in the right weak order generated by a separable element, answering an open question of Björner and Wachs in 1988. This is joint work with Christian Gaetz.

Generalized snake graphs from orbifolds

Elizabeth Kelley

UMN

Friday

Feb. 28, 2020

3:35-4:25pm

Vincent Hall 570

Cluster algebras, as originally defined by Fomin and Zelevinsky, are characterized by binomial exchange relations. A natural generalization of cluster algebras, due to Chekhov and Shapiro, allows the exchange relations to have arbitrarily many terms. A subset of these generalized cluster algebras can be associated with triangulations of orbifolds, analogous to the subset of ordinary cluster algebras associated with triangulated surfaces. We generalize Musiker-Schiffler-Williams' snake graph construction for this subset of generalized cluster algebras, yielding explicit combinatorial formulas for the cluster variables. We then show that our construction can be extended to give expansions for generalized arcs on triangulated orbifolds. This is joint work with Esther Banaian.

Grothendieck Polynomials from Chromatic Lattice Models

Katy Weber

UMN

Friday

Mar. 6, 2020

3:35-4:25pm

Vincent Hall 570

The $\beta$-Grothendieck polynomials are simultaneous generalizations of Schubert and Grothendieck polynomials that arise in the study of the connective K-theory of the flag variety. They can be calculated as a generating function of combinatorial objects known as pipe dreams, as well as recursively via geometrically-motivated divided difference operators. We combine these two points of view by defining a chromatic lattice model whose partition function is a $\beta$-Grothendieck polynomial. This is joint work-in-progress with Ben Brubaker, Claire Frechette, Andy Hardt, and Emily Tibor.

No seminar - Spring Break

Friday

Mar. 13, 2020

3:35-4:25pm

Vincent Hall 570

Talk cancelled

Friday

Mar. 20, 2020

3:35-4:25pm

Vincent Hall 570

Counting trees and nilpotent endomorphisms

Vic Reiner

UMN

Friday

Mar. 27, 2020

3:35-4:25pm

Zoom ID 731-678-3144

A formula of Cayley (1889) says that the number of trees on vertex set [n]:={1,2,...,n} is n^{n-2}. Among its many proofs, my favorite is a gorgeous bijection due to Andre Joyal in 1981. One can also view Cayley's formula as asserting that there are n^{n-1} vertex-rooted trees on [n], or equivalently n^{n-1} eventually constant self-maps on [n]. This talk will review Joyal's proof, and its recent revisitation by Tom Leinster in arXiv:1912.12562. Leinster gives a beautiful q-analogue of the proof, that proves a q-analogous theorem of Fine and Herstein (1958). The latter theorem counts those linear self-maps of an n-dimensional vector space over a finite field F_q which are eventually constant, that is, nilpotent as linear maps. Vic's slides are available here .

The box-ball system and cyclindric loop Schur functions

Gabriel Frieden

UQAM

Friday

Apr. 3, 2020

3:35-4:25pm

Zoom ID 391-941-053.

The box-ball system is a cellular automaton in which a sequence of balls moves along a row of boxes. An interesting feature of this automaton is its soliton behavior: regardless of the initial state, the balls in the system eventually form themselves into connected blocks (solitons) which remain together for the rest of time. In 2014, T. Lam, P. Pylyavskyy, and R. Sakamoto conjectured a formula which describes the solitons resulting from an initial state of the box-ball system in terms of the tropicalization of certain polynomials they called cylindric loop Schur functions. In this talk, I will describe the various ingredients of this conjecture and discuss its proof.

Vandermondes in superspace

Brendon Rhoades

UCSD

Friday

Apr. 10, 2020

3:35-4:25pm

Zoom ID 391-940-053

The *Vandermonde determinant* is ubiquitous in algebraic combinatorics and representation theory. One application of the Vandermonde is as a generator for a `harmonic' model of the coinvariant ring attached to the symmetric group which eschews the use of -- and computational issues involved with -- quotient rings. We present an extension of the Vandermonde determinant to * superspace* (a symmetric algebra tensor an exterior algebra) and use it to generate a variety of modules including the recently defined `Delta Conjecture coinvariant rings' of Haglund-Rhoades-Shimozono as well as (conjecturally) a trigraded module for the full Delta Conjecture. We use superspace Vandermondes to build bigraded superspace quotients tied to the geometry of * spanning configurations * studied by Pawlowski-Rhoades which satisfy a superspace version of Poincaré Duality and (conjecturally) exhibit unimodality properties which suggest a superspace version of Hard Lefschetz. Joint with Andy Wilson.

A combinatorial e-expansion of vertical strip LLT polynomials

Per Alexandersson

Stockholm

Friday

Apr. 17, 2020

3:35-4:25pm

Zoom ID 391-940-053

In 2019, D'Adderio proved that if G(x;q) is a
vertical-strip LLT polynomial, then G(x;q+1) is positive in the
elementary symmetric functions basis. A conjectured formula
for the coefficients in this basis was given earlier in 2019 by
Alexandersson. We give a new proof of D'Adderio's result which
also proves the conjectured formula.
The problem of finding such an e-expansion is surprisingly similar to
the still open problem of Shareshian-Wachs, regarding the e-expansion
of chromatic polynomials associated with unit-interval graphs.
We shall discuss this connection as well.
Here is a video recording of Per's talk.

Coxeter factorizations and the Matrix Tree theorem with generalized Jucys-Murphy weights. (Click for video recording).

Theo Douvropoulos

IRIF

Friday

Apr. 24, 2020

3:35-4:25pm

Zoom ID 391-940-053

One of the most far reaching proofs of Cayley's formula, that the number $n^{n-2}$ counts trees on n labeled vertices, is via Kirchhoff's Matrix Tree theorem. After Denes, Schaeffer, and many others, there is a well-exploited correspondence between trees and transitive factorizations in the symmetric group; in particular, the number $n^{n-2}$ counts shortest factorizations of the long cycle (12..n) in transpositions. Furthermore, Burman and Zvonkine (and independently Alon and Kozma) have given a "higher-genus" formula that enumerates arbitrary length factorizations of long cycles, where each transposition (ij) is weighted by its own variable $w_{ij}$, and which has a product form involving the eigenvalues of the Laplacian L(K$_n$) of the complete graph.
In joint work with Guillaume Chapuy, we consider a (partial) analog of the weighted Laplacian for complex reflection groups W. The weights are specified via any given flag of parabolic subgroups, generalizing the definition of Jucys-Murphy elements. We prove a product formula for the enumeration of weighted reflection factorizations of Coxeter elements, that subsumes the Chapuy-Stump formula and in part the Burman-Zvonkine formula. Its proof is based on an interesting fact that relates the exterior powers of the reflection representation with those W-characters that are non-zero on the Coxeter class. We present some further applications of these techniques, in particular, a uniform simple(r) way to produce the chain number $\frac{h^n n!}{|W|}$ of the noncrossing lattice NC(W). An extended abstract for this work was accepted for FPSAC 2020 and is available at my website website). A video recording of the talk is available here .

Associahedra, Cyclohedra and inversion of power series

José Bastidas

Cornell

Friday

May 1, 2020

3:35-4:25pm

Zoom ID 391-940-053

Species and Hopf monoids are powerful algebraic tools to study families of combinatorial structures. Aguiar and Ardila introduced the Hopf monoid of generalized permutahedra and realized many combinatorial Hopf monoids as submonoids of generalized permutahedra. They solved the antipode problem for the Hopf monoid of associahedra and explained how the classical Lagrange inversion formula for power series follows from this. In this talk, we bring cyclohedra into the picture. We solve the antipode problem for this new Hopf monoid and use this result to describe inversion in a group of pairs of power series using the face structure of associahedra and cyclohedra. The talk is based on joint work with Marcelo Aguiar (Cornell University).

Troupes and cumulants. (Click for video recording.)

Colin Defant

Princeton

Friday

May 8, 2020

3:35-4:25 pm

Zoom ID 391-940-053

Cumulants are the fundamental combinatorial tools used in noncommutative probability theory. Sequences of free cumulants and sequences of classical cumulants are paired with each other via summation formulas involving partition lattices and noncrossing partition lattices. In several cases, a sequence of free cumulants that counts a set of colored binary plane trees happens to correspond, somewhat miraculously, to a sequence of classical cumulants that counts the decreasing labeled versions of the same trees. We will see that this strange phenomenon holds for families of trees that we call troupes, which are defined using two new operations on colored binary plane trees that we call insertion and decomposition. Troupes also provide a broad framework for generalizing several of the results that are known about West's stack-sorting map. We will hint at just a couple of the many ways in which the investigation of troupes could be extended further. The video recording of Colin's talk is here.