Spring 2022

- The Combinatorics Seminar meets on
**Fridays, 3-4pm**, in**Snow 306**. (We may sometimes use Zoom instead.) - Organizers: Mark Denker and Jeremy Martin.
- Good general seminar-attending advice, especially for graduate students: The "Three Things" Exercise for getting things out of talks by Ravi Vakil
- Also consider attending the Graduate Online Combinatorics Colloquium.

**Friday 1/21**

Organizational meeting

**Friday 1/28**

Dania Morales

*The character group of a Hopf monoid, Part II*

**Friday 2/4**

Mark Denker

*A Hopf monoid on posets*

**Tuesday 2/8: Kevin Marshall
A Hopf Monoid on Set Families (Ph.D. Oral Comprehensive Exam)**

Zoom link (passcode: 2022)

**Friday 2/11**

Eleonore Bach (FU Berlin)

*A concrete construction of the cographic hyperplane arrangement*

**Zoom link (passcode: 1430)**

Zoom recording

__Abstract:__
Geometrically carrying a trove of information about the underlying
simple graph, the graphic hyperplane arrangement \(H_G\) yields an
interesting mathematical object to study a simple graph \(G\). For
example, one proves that the normal vectors of \(H_G\) are linearly
independent if and only if they induce forests on \(G\) and the regions of
\(H_G\) are in a one-to-one correspondence to the acyclic orientations of
\(G\). With the graphic hyperplane arrangement we can associate the
graphic matroid whose bases are spanning forests of \(G\). What
information do we obtain if we apply duality, i.e., if we start with the
dual of the graphic matroid, called the cographic matroid whose bases
are complements of spanning forests of \(G\)? In this talk we are going to
start answering the above question by constructing the normal vectors of
the cographic hyperplane arrangement associated with the cographic
matroid for simple, connected and bridgeless graphs.

**Friday 2/18**

Jeremy Martin

*Simplicial Effective Resistance and Tree Enumeration*

__Abstract:__
A graph \(G\) can be viewed as an electrical network, whose parameters
are the resistances along its edges. Remarkably, this model leads to a
way to calculate combinatorial information about \(G\), including
generating functions for spanning trees. Also remarkably, my coauthors
Woong Kook and Kang-Ju Lee have shown that the same method can be used
more generally to enumerate spanning trees of simplicial complexes
(which I will explain). In particular, we can recover explicit formulas
for *shifted* complexes (first obtained by Art Duval, Caroline
Klivans and myself) and prove a new formula for *color-shifted*
complexes (conjectured by Ghodrat Aalipour and Duval). This is joint
work with Duval, Kook, and Lee.

**Friday 2/25**

Dylan Beck

*Some new invariants of the edge ring of a finite simple graph*

__Abstract:__
Given any positive integer \(n\), there is a one-to-one correspondence
between the collection of finite simple graphs on \(n\) vertices and the
collection of squarefree monomial ideals of a polynomial ring in \(n\)
indeterminates over a specified field; this bijection is induced by the
assignment of an edge \(\{i, j\}\) to the squarefree monomial \(x_i x_j\).
Consequently, the quotient of a polynomial ring over a field by a
squarefree monomial ideal can be understood by studying the associated
finite simple graph and vice-versa. One of the most celebrated examples
of this relationship is due to Fröberg, who showed in 1990 that the
complement of a finite simple graph is chordal if and only if the
corresponding squarefree monomial ideal admits a linear resolution. In
joint work with Souvik Dey, we introduce two new invariants of a finite
simple graph via the quotient of the complex polynomial ring by the
corresponding squarefree monomial ideal. We demonstrate that all though
these invariants are often quite subtle, they can be computed or bounded
explicitly in several familiar cases. Even more, we show that these
invariants behave well with respect to operations such as the join of
two finite simple graphs or the wedge of complete graphs. We conclude by
stating some questions and possible directions for future work.

**Friday 3/4**

Sheila Sundaram (Pierrepont School)

*Subword Order: Topology, Homology Representations, and Enumeration*

Zoom recording

__Abstract:__
We show that the homology module for words of bounded length, over an
alphabet of size \(n\), decomposes into a sum of tensor powers of the
\(S_n\)-irreducible \(S_{(n-1,1)}\) indexed by the partition
\((n-1,1)\), recovering, as a special case, a result of Bj\"orner and
Stanley for words of length at most \(k\). For arbitrary ranks we show
that the homology is an integer combination of positive tensor powers of
the reflection representation \(S_{(n-1,1)}\), and conjecture that this
combination is nonnegative. We uncover a curious duality in homology in
the case when one rank is deleted.

Our most definitive result describes the homology representation for an arbitrary set of ranks, as being almost a permutation module, modulo one copy of the reflection representation.

This talk will be self-contained. I will describe the techniques used to determine the topology and the homology representations via the Hopf trace formula. I will also present some open enumerative conjectures involving Stirling numbers of both kinds.

**Friday 3/11**

No seminar - Spring Break

**Friday 3/18**

No seminar - Spring Break

**Friday 3/25**

Derek Hanely (University of Kentucky)

*Triangulations of Flow Polytopes, Ample Framings, and Path Algebras*

Zoom recording

__Abstract:__
The cone of nonnegative flows for a directed acyclic graph (DAG) is
known to admit regular unimodular triangulations induced by framings of
the DAG. These triangulations restrict to triangulations of the flow
polytope for strength one flows, which are called DKK triangulations.
For a special class of framings called ample framings, these
triangulations of the flow cone project to a complete fan. In this talk,
we will characterize the DAGs that admit ample framings and enumerate
the number of ample framings for a fixed DAG. Moreover, we will
establish a connection between maximal simplices in DKK triangulations
and \(\tau\)-tilting posets for certain gentle algebras, which allows us
to impose a poset structure on the dual graph of any DKK triangulation
for an amply framed DAG. Using this connection, we are able to prove
that for full DAGs, i.e., those DAGs with inner vertices having
in-degree and out-degree equal to two, the flow polytopes are Gorenstein
and have unimodal Ehrhart \(h^\ast\)-polynomials. This is joint work with
Matias von Bell, Benjamin Braun, Kaitlin Bruegge, Zachery Peterson,
Khrystyna Serhiyenko, and Martha Yip.

**Friday 4/1**

Ritika Nair

*Weak Lefschetz Property for quotients by monomial ideals containing squares of variables*

__Abstract:__
Let \(\Delta\) be a simplicial complex on \(n\) vertices. The corresponding
Stanley-Reisner ring is obtained from \(S = \Bbbk[x_1, \ldots, x_n]\) by
quotienting out \(I_{\Delta} = \langle x_{i_1} \cdots x_{i_m} : \{i_1,
\ldots, i_m \} \notin \Delta \rangle\), where \(\Bbbk\) is a field of
characteristic \(0\). Let \(J_{\Delta}\) be the ideal generated by
\(I_{\Delta}\) and the squares of the variables. \(A(\Delta) =
S/J_{\Delta}\) is a graded artinian algebra. For a general linear form
\(\ell\), \(A(\Delta)\) has the *Weak Lefschetz Property* (WLP) if
the homomorphism induced by multiplication by \(\ell\), \(\mu_i :
A(\Delta)_i \to A(\Delta)_{i+1}\) has maximal rank for all \(i\). In
joint work with Hailong Dao, we characterize the Weak Lefschetz Property
of \(A(\Delta)\) in terms of the simplicial complex \(\Delta\). In
particular, we see a complete characterization of WLP in degree \(1\) in
terms of the edge graph of \(\Delta\). We shall also look at
pseudomanifolds of small dimension and if time permits, some interesting
corollaries including relating WLP with face 2-colorability of certain
triangulations.

**Friday 4/8**

Geoffrey Critzer

*Enumeration of Finite Semigroups via the Symbolic Method*

__Abstract:__
We introduce the symbolic method as developed in the seminal work
*Analytic Combinatorics* by Phillipe Flajolet and Robert Sedgewick.
We employ the method to derive generating functions that enumerate some
interesting statistics relating properties of the full transformation
semigroup \(\mathcal{T}_n\), the semi-group \(\mathcal{PT}_n\) of
partial transformations, the symmetric inverse semigroup
\(\mathcal{IS}_n\), and the semigroup \(\mathcal{B}_n\) of all relations
on \([n]\). In particular, we count the number of nilpotent, idempotent,
and group elements in each semigroup. We classify these semigroups
according to various properties, including but not limited to: index,
connected components, number of recurrent, fixed, defective, and
undefined points. We derive probability generating functions for some
distributions associated with the non-functional points of a random
partial function.

**Friday 4/15**

Kyle Maddox

*Semigroups and commutative algebra*

__Abstract:__
Semigroups are simple combinatorial objects which provide a wealth of
interesting and beautiful examples in commutative algebra by forming
what's known as an affine semigroup ring. In this talk, we will recap
some of the basic definitions surrounding semigroups and discuss recent
joint work with Vaibhav Pandey about affine semigroup rings.

**Friday 4/22**

Kyle Maddox

*Commutative algebra and semigroups*

__Abstract:__
In this follow-up talk from last week, I will focus a bit more on the
algebraic side. We will cover a tiny bit of necessary homological
algebra, then demonstrate how the simple idea of "\(p\)-nilpotent
embeddings" of affine semigroups can help give interesting and
powerful information about otherwise mysterious local cohomology
modules of the affine semigroup ring.
Kevin Marshall

Title TBA

**Friday 4/29**

No seminar; attend the Great Plains Combinatorics Conference instead!

**Friday 5/6**

No seminar (Stop Day)

For seminars from previous semesters, please see the KU Combinatorics Group page.

Last updated Mon 5/9/22