Browsing by Subject "combinatorics"

Now showing 1 - 5 of 5
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    Combinatorics of Cluster Algebras from Surfaces
    (2016-08) Gunawan, Emily
    We construct a periodic infinite frieze using a class of peripheral elements of a cluster algebra of type D or affine A􏰕. We discover new symmetries and formulas relating the entries of this frieze and bracelet elements. We also present a correspondence between Broline, Crowe and Isaacs’s classical matching tuples and various recent interpretations of elements of cluster algebras from surfaces. We extend a T-path expansion formula for arcs on an unpunctured surface to the case of arcs on a once-punctured polygon and use this formula to give a combinatorial proof that cluster monomials form the atomic basis of a cluster algebra of type D. We further generalize our work and present T-path formulas for tagged arcs with one or two notchings on a marked surface with punctures.
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    Conference Athletic Schedules: An Application of Projective Geometry, Finite Fields, and Graph Theory
    (2016-05) Nelson, Alexandra
    This paper solves a problem faced by the Suburban East Conference of the Minnesota State High School League in 2009, of designing a consistent wrestling schedule to accommodate a new school. We adapt the application of projective geometry of finite fields to general scheduling problems, and develop an algorithm for determining a schedule. We prove that this algorithm can be completed, will yield the desired schedule, and can yield all possible schedules in the desired format. We also model the schedule with bipartite graphs, and use edge colorings to complete the schedule with home and away assignments.
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    Cyclic Actions in Combinatorial Invariant Theory
    (2021-07) Stucky, Eric
    The major original contributions of this thesis are as follows: Theorem 3.3.1 and Proposition 3.3.3 together show that a natural q-analogue of the rational Schr\"oder polynomial is (separately) unimodal in both its even and odd coefficient sequences. Theorem 4.1.2 which, for certain parameters, defines an elementary (WxC)-action on the classical parking space for a Weyl group. When this action is defined, it agrees with the more technical algebraic construction of Armstrong, Reiner, and Rhoades. Theorem 5.1.3 is a general cyclic sieving result which in particular recovers the q=-1 phenomenon for Catalan necklaces, as well as higher-order sieving for a more general family of necklaces.
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    Generalizations of Total Positivity
    (2020-07) Chepuri, Sunita
    The theory of total positivity was classically concerned with totally nonnegative matrices (matrices with all nonnegative minors). These matrices appear in many varied areas of mathematics including probability, asymptotic representation theory, algebraic and enumerative combinatorics, and linear algebra. However, motivated by surprising positivity properties of Lusztig's canonical bases for quantum groups, the field of total positivity has more recently grown to include other totally nonnegative varieties. We first discuss results regarding immanants on the space of k-positive matrices (matrices where all minors of size up to k are positive). Immanants are functions on square matrices generalizing the determinant and permanent. Kazhdan--Lustzig immanants, defined by Rhoades and Skandera, are of particular interest, as elements of the dual canonical basis of the coordinate ring of GL_n(C) can be expressed as Kazhdan--Lustzig immanants. Results of Stembridge, Haiman, and Rhoades--Skandera show that Kazhdan--Lustzig immanants are nonnegative on totally nonnegative matrices. Here, we give conditions on v in S_n so that the Kazhdan-Lusztig immanant corresponding to v is positive on k-positive matrices. We then consider a space that arises from the study of totally nonnegative Grassmannians. Postnikov's plabic graphs in a disk are used to parametrize these spaces. In recent years plabic graphs have found numerous applications in math and physics. One of the key features of the theory is the fact that if a plabic graph is reduced, the face weights can be uniquely recovered from boundary measurements. On surfaces more complicated than a disk this property is lost. In this thesis, we undertake a comprehensive study of a certain semi-local transformation of weights for plabic networks on a cylinder that preserve boundary measurements. We call this a plabic R-matrix. We show that plabic R-matrices have underlying cluster algebra structure, generalizing work of Inoue--Lam--Pylyavskyy. Special cases of transformations we consider include geometric R-matrices appearing in Berenstein--Kazhdan theory of geometric crystals, and also certain transformations appearing in a recent work of Goncharov--Shen.
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    Involutions on Baxter Objects and q-Gamma Nonnegativity
    (2015-08) Dilks, Kevin
    Baxter numbers are known to count several families of combinatorial objects, all of which come equipped with a natural involution. In this paper, we add a combinatorial family to the list, and show that the known bijections between these objects respect these involutions. We also give a formula for the number of objects fixed under this involution, showing that it is an instance of Stembridge's ``$q=-1$ phenomenon''. A polynomial $\sum_{i=0}^{n} a_i t^i$ with symmetric coefficients ($a_{n-i}=a_i$) has a unique expansion $\sum_{k=0}^{\lfloor n/2 \rfloor} \gamma_k t^k(1+t)^{n-2k}$, and is said to be \emph{gamma-nonnegative} if $\gamma_k\geq 0$ for all $k$. We either prove or conjecture a stronger $q$-analogue of this property for several polynomials in two variables $t$,$q$, whose $q=1$ specializations are known to be gamma-nonnegative.