DISCRETE CATS SEMINAR

discrete CATS seminar

Title: Lattice minors and Eulerian posets

 

Abstract: We define a notion of deletion and contraction for lattices. The result of a sequence of deletions and contractions is called a minor of the original lattice. The name minors is justified by the fact that the minors of the lattice of flats of a graph correspond to the simple minors of the graph when the vertices are labeled (and the edges unlabeled). For each finite lattice we define a poset of minors and show it is Eulerian and a PL sphere. We also obtain some inequalities for the cd-indices of these posets of minors.

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discrete CATS seminar

Title: Coloring (P5, gem)-free graphs with ∆ − 1 colors.

Abstract:   The Borodin–Kostochka Conjecture states that for a graph G, if ∆(G) ≥ 9 and ω(G) ≤ ∆(G)−1, then χ(G) ≤ ∆(G)−1. This conjecture is a strengthening of Brooks' Theorem and while known for certain graph classes it remains open for general graphs. In this talk we prove the Borodin– Kostochka Conjecture for (P5,gem)-free graphs, i.e. graphs with no induced P5 and no induced K1 ∨ P4.

 

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discrete CATS seminar

Title: Characterizing quotients of positroids

Abstract: We characterize quotients of specific families of positroids. Positroids are a special class of representable matroids introduced by Postnikov in the study of the nonnegative part of the Grassmannian. Postnikov defined several combinatorial objects that index positroids. In this talk, we make use of one of these objects called a decorated permutation to combinatorially characterize when certain positroids form quotients. Furthermore, we conjecture a general rule for quotients among arbitrary positroids on the same ground set. This is joint work with Carolina Benedetti and Daniel Tamayo.

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discrete CATS seminar

Title: Recurrence Relations of Linear Tilings
 
Abstract: Let U ⊆ ℕ and let T be a linear tiling of a 1 x n board using 1 x i polyomino tiles where u is in U. We define T_U(n) to be the set of linear tilings of length n using tile sizes in U. Classically, when U = { 1, 2} these tilings are known as the Lucanomial tilings and they follow the Lucas sequence. Now, let U and V be distinct subsets of the natural numbers. In this talk, we will provide some general results and supportive examples of a large family of linear recurrence relations between the sequences T_U(n) and T_V(n) for fixed sets U and V. This is joint work with Christopher O'Neill and Pamela E. Harris.
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discrete CATS seminar

Title: Log-Concavity of Littlewood-Richardson Coefficients
 
Abstract:  Schur polynomials and Littlewood-Richardson numbers are classical objects arising in symmetric function theory, representation theory, and the cohomology of the Grassmannian. I will give a quick introduction to them, and then describe a new log-concavity property of the Littlewood-Richardson numbers. I will then explain the mechanism behind the log-concavity, Brändén and Huh's theory of Lorentzian polynomials.
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discrete CATS seminar

Title: The Lalanne--Kreweras Involution, Rowmotion, and Birational Liftings
 
Abstract: Our work ties together a few different actions studied in combinatorics.  First, The Lalanne–Kreweras involution (LK) on Dyck paths yields a bijective proof of the symmetry of two statistics: the number of valleys and the major index.  Panyushev studied an equivalent involution can be considered on the set of antichains of the type A root poset.  Second, we will discuss the action of rowmotion on the set of antichains of a poset.  This action, which sends an antichain A to the minimal elements of the complement of the order ideal generated by A, has received significant attention recently in dynamical algebraic combinatorics due to various phenomena (e.g. periodicity, cyclic sieving, homomesy) on certain "nice" posets including root posets.  The LK involution and rowmotion are connected in that they generate a dihedral action on the set of antichains of the type A root poset.  Furthermore, the periodicity of rowmotion on the type A root poset lifts to a generalization called "birational rowmotion" first studied by David Einstein and James Propp.  This motivated us to search for a birational lifting of the LK involution, where we discovered that the key properties of the LK involution are also satisfied in this generalization.  This is joint work with Sam Hopkins.
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discrete CATS seminar

Title: Shelling the m=1 amplituhedron
 
Abstract: The amplituhedron is a topological space related to the totally nonnegative Grassmannian that was inspired by high energy physics. In this talk we will apply combinatorics and poset topology to analyze a special case known as the m=1 amplituhedron. There is a poset of sign vectors which is the closure poset of the m=1 amplituhedron, and we show this poset has an EL shelling. This implies the m=1 amplituhedron is a ball. We also show this poset is strongly Sperner.
 
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discrete CATS seminar

Title: Generalized permutahedra: Minkowski linear functionals and Ehrhart positivity
 
Abstract: Generalized permutahedra form a combinatorially rich class of polytopes that appear naturally in various areas of mathematics. They include many interesting and significant classes of polytopes such as matroid polytopes. We study functions on generalized permutahedra that behave linearly with respect to dilation and taking Minkowski sums. We present classification results and discuss how these can be applied to prove positivity of the linear coefficient of the Ehrhart polynomial of generalized permutahedra. This is joint work with Mohan Ravichandran.
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