Discrete Math Seminar

Discrete CATS Seminar

Title: Mutation of friezes

 

Abstract: Frieze is an array of positive integers satisfying certain rules.  Friezes of type A were first studied by Conway and Coxeter in 1970's, but they gained fresh interest in the last decade in relation to cluster algebras.   Moreover, the categorification of cluster algebras developed in 2006 yields a new realization of friezes in terms of representation theory of Jacobian algebras.   In this talk, we will discuss friezes of types A and D and their mutations.

 

Date:
-
Location:
POT 745

Discrete CATS Seminar

Title: Mutation of friezes

 

Abstract: Frieze is an array of positive integers satisfying certain rules.  Friezes of type A were first studied by Conway and Coxeter in 1970's, but they gained fresh interest in the last decade in relation to cluster algebras.   Moreover, the categorification of cluster algebras developed in 2006 yields a new realization of friezes in terms of representation theory of Jacobian algebras.   In this talk, we will discuss friezes of types A and D and their mutations.

 

Date:
-
Location:
POT 745

Discrete CATS Seminar

Title: The $gamma$-coefficients of the tree Eulerian polynomials.

Abstract: We consider the generating polynomial $T_n(t)$ of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. In particular it has palindromic coefficients and hence it can be expressed in the  the basis $\left \{ t^i(1+t)^{n-1-2i}\,\mid\, 0\le i \le \lfloor \frac{n-1}{2}\rfloor\right \}$, known as the $\gamma$-basis. We show that $\T_n(t)$ has nonnegative $\gamma$-coefficients and we present various combinatorial interpretations for them.
 
Date:
-
Location:
POT 745

Discrete CATS Seminar

Title: The $gamma$-coefficients of the tree Eulerian polynomials.

Abstract: We consider the generating polynomial $T_n(t)$ of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. In particular it has palindromic coefficients and hence it can be expressed in the  the basis $\left \{ t^i(1+t)^{n-1-2i}\,\mid\, 0\le i \le \lfloor \frac{n-1}{2}\rfloor\right \}$, known as the $\gamma$-basis. We show that $\T_n(t)$ has nonnegative $\gamma$-coefficients and we present various combinatorial interpretations for them.
 
Date:
-
Location:
POT 745

Discrete CATS Seminar

Speaker: Pamela Harris, Williams College

Title: A proof of the peak polynomial positivity conjecture

 

Abstract: We say that a permutation $\pi=\pi_1\pi_2\cdots \pi_n \in \mathfrak{S}_n$ has a peak at index $i$ if $\pi_{i-1} < \pi_i > \pi_{i+1}$. Let $P(\pi)$ denote the set of indices where $\pi$ has a peak. Given a set $S$ of positive integers, we define $P(S;n)=\{\pi\in\mathfrak{S}_n:P(\pi)=S\}$. In 2013 Billey, Burdzy, and Sagan showed that for subsets of positive integers $S$ and sufficiently large $n$, $| P(S;n)|=p_S(n)2^{n-|S|-1}$ where $p_S(x)$ is a polynomial depending on $S$. They gave a recursive formula for $p_S(x)$ involving an alternating sum, and they conjectured that the coefficients of $p_S(x)$ expanded in a binomial coefficient basis centered at $\max(S)$ are all nonnegative. In this talk we introduce a new recursive formula for $|P(S;n)|$ without alternating sums and we use this recursion to prove that their conjecture is true.

Date:
-
Location:
POT 745

Discrete CATS Seminar

Speaker: Pamela Harris, Williams College

Title: A proof of the peak polynomial positivity conjecture

 

Abstract: We say that a permutation $\pi=\pi_1\pi_2\cdots \pi_n \in \mathfrak{S}_n$ has a peak at index $i$ if $\pi_{i-1} < \pi_i > \pi_{i+1}$. Let $P(\pi)$ denote the set of indices where $\pi$ has a peak. Given a set $S$ of positive integers, we define $P(S;n)=\{\pi\in\mathfrak{S}_n:P(\pi)=S\}$. In 2013 Billey, Burdzy, and Sagan showed that for subsets of positive integers $S$ and sufficiently large $n$, $| P(S;n)|=p_S(n)2^{n-|S|-1}$ where $p_S(x)$ is a polynomial depending on $S$. They gave a recursive formula for $p_S(x)$ involving an alternating sum, and they conjectured that the coefficients of $p_S(x)$ expanded in a binomial coefficient basis centered at $\max(S)$ are all nonnegative. In this talk we introduce a new recursive formula for $|P(S;n)|$ without alternating sums and we use this recursion to prove that their conjecture is true.

Date:
-
Location:
POT 745

Discrete CATS Seminar

Speaker: Luis David Garcia Puente, Sam Houston State University
 
Title: What is a sandpile group?
 
Abstract: The theory of sandpile groups started in 1987, when physicists Bak, Tang, and Wiesenfeld created an idealized version of a sandpile in which sand is stacked on the vertices of a (combinatorial) graph and is subjected to certain avalanching rules. The long-term dynamics of this system is encoded by the set of recurrent sandpiles. This set has the structure of a finite

abelian group. This group has been discovered in different contexts and received many names: the sandpile group (statistical physics), the critical group (algebraic combinatorics), the group of

components (arithmetic geometry), and the jacobian of a graph (algebraic geometry).  Algebraically, the sandpile group is isomorphic to the cokernel of the (reduced) Laplacian matrix of the underlying graph. Among many beautiful properties, the order of the sandpile group equals the number of spanning trees of the underlying graph. In this sense, the sandpile group is a more subtle isomorphism invariant of a graph.  In this talk, I will provide an introduction to the subject and showcase a few of my favorite results. Some of these results were obtained in collaboration with students in many undergraduate research projects over the last few years.
Date:
-
Location:
POT 745

Discrete CATS Seminar

Speaker: Luis David Garcia Puente, Sam Houston State University
 
Title: What is a sandpile group?
 
Abstract: The theory of sandpile groups started in 1987, when physicists Bak, Tang, and Wiesenfeld created an idealized version of a sandpile in which sand is stacked on the vertices of a (combinatorial) graph and is subjected to certain avalanching rules. The long-term dynamics of this system is encoded by the set of recurrent sandpiles. This set has the structure of a finite

abelian group. This group has been discovered in different contexts and received many names: the sandpile group (statistical physics), the critical group (algebraic combinatorics), the group of

components (arithmetic geometry), and the jacobian of a graph (algebraic geometry).  Algebraically, the sandpile group is isomorphic to the cokernel of the (reduced) Laplacian matrix of the underlying graph. Among many beautiful properties, the order of the sandpile group equals the number of spanning trees of the underlying graph. In this sense, the sandpile group is a more subtle isomorphism invariant of a graph.  In this talk, I will provide an introduction to the subject and showcase a few of my favorite results. Some of these results were obtained in collaboration with students in many undergraduate research projects over the last few years.
Date:
-
Location:
POT 745