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Quasisymmetric Functions from Combinatorial Hopf Monoids And
Discrete Mathematics and Theoretical Computer Science DMTCS vol. (subm.), by the authors, 1–1 Quasisymmetric functions from combinatorial Hopf monoids and Ehrhart Theory Jacob A. White School of Mathematical and Statistical Sciences University of Texas - Rio Grande Valley Edinburg, TX 78539 Abstract. We investigate quasisymmetric functions coming from combinatorial Hopf monoids. We show that these invariants arise naturally in Ehrhart theory, and that some of their specializations are Hilbert functions for relative simplicial complexes. This class of complexes, called forbidden composition complexes, also forms a Hopf monoid, thus demonstrating a link between Hopf algebras, Ehrhart theory, and commutative algebra. We also study various specializations of quasisymmetric functions. Resume.´ Nous ´etudions les fonctions quasisym´etriques associ´ees aux mono¨ıdes de Hopf combinatoriaux. Nous d´emontrons que ces invariants sont des objets naturels `ala th´eorie de Ehrhart. De plus, certains correspondent `ades fonctions de Hilbert associ´ees `ades complexes simpliciaux relatifs. Cette classe de complexes, constitue un monode¨ de Hopf, r´ev´elant ainsi un lien entre les alg`ebres de Hopf, la th´eorie de Ehrhart, et l’alg`ebre commutative. Nous ´etudions ´egalement diverses cat´egories de fonctions quasisym´etriques. Keywords: Chromatic Polynomials, Symmetric Functions, Combinatorial Species, Combinatorial Hopf Algebras, Ehrhart Theory, Hilbert functions 1 Introduction Chromatic polynomials of graphs, introduced by Birkhoff and Lewis (1946) are wonderful polynomials. Their properties can be understood as coming from three different theories: 1. Chromatic polynomials were shown by Beck and Zaslavsky (2006) to be Ehrhart functions for inside-out polytopes. arXiv:1604.00076v1 [math.CO] 31 Mar 2016 2. They arise as the Hilbert polynomial for the coloring ideal introduced by Steingr´ımsson (2001). -
Bijective Proofs for Schur Function Identities Which Imply Dodgson's Condensation Formula and Plücker Relations
Bijective proofs for Schur function identities which imply Dodgson’s condensation formula and Pl¨ucker relations Markus Fulmek Institut f¨urMathematik der Universit¨atWien Strudlhofgasse 4, A-1090 Wien, Austria [email protected] Michael Kleber Massachusetts Institute of Technology 77 Massachusetts Avenue, Cambridge, MA 02139, USA [email protected] Submitted: July 3, 2000; Accepted: March 7, 2001. MR Subject Classifications: 05E05 05E15 Abstract We present a “method” for bijective proofs for determinant identities, which is based on translating determinants to Schur functions by the Jacobi–Trudi identity. We illustrate this “method” by generalizing a bijective construction (which was first used by Goulden) to a class of Schur function identities, from which we shall obtain bijective proofs for Dodgson’s condensation formula, Pl¨ucker relations and a recent identity of the second author. 1 Introduction Usually, bijective proofs of determinant identities involve the following steps (cf., e.g, [19, Chapter 4] or [23, 24]): Expansion of the determinant as sum over the symmetric group, • Interpretation of this sum as the generating function of some set of combinatorial • objects which are equipped with some signed weight, Construction of an explicit weight– and sign–preserving bijection between the re- • spective combinatorial objects, maybe supported by the construction of a sign– reversing involution for certain objects. the electronic journal of combinatorics 8 (2001), #R16 1 Here, we will present another “method” of bijective proofs for determinant identitities, which involves the following steps: First, we replace the entries ai;j of the determinants by hλ i+j (where hm denotes i− • the m–th complete homogeneous function), Second, by the Jacobi–Trudi identity we transform the original determinant identity • into an equivalent identity for Schur functions, Third, we obtain a bijective proof for this equivalent identity by using the interpre- • tation of Schur functions in terms of nonintersecting lattice paths. -
Covariances of Symmetric Statistics
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector JOURNAL OF MULTIVARIATE ANALYSIS 41, 14-26 (1992) Covariances of Symmetric Statistics RICHARD A. VITALE Universily of Connecticut Communicared by C. R. Rao We examine the second-order structure arising when a symmetric function is evaluated over intersecting subsets of random variables. The original work of Hoeffding is updated with reference to later results. New representations and inequalities are presented for covariances and applied to U-statistics. 0 1992 Academic Press. Inc. 1. INTRODUCTION The importance of the symmetric use of sample observations was apparently first observed by Halmos [6] and, following the pioneering work of Hoeffding [7], has been reflected in the active study of U-statistics (e.g., Hoeffding [8], Rubin and Vitale [ 111, Dynkin and Mandelbaum [2], Mandelbaum and Taqqu [lo] and, for a lucid survey of the classical theory Serfling [12]). An important feature is the ANOVA-type expansion introduced by Hoeffding and its application to finite-sample inequalities, Efron and Stein [3], Karlin and Rinnott [9], Bhargava Cl], Takemura [ 143, Vitale [16], Steele [ 131). The purpose here is to survey work on this latter topic and to present new results. After some preliminaries, Section 3 organizes for comparison three approaches to the ANOVA-type expansion. The next section takes up characterizations and representations of covariances. These are then used to tighten an important inequality of Hoeffding and, in the last section, to study the structure of U-statistics. 2. NOTATION AND PRELIMINARIES The general setup is a supply X,, X,, . -
Pre-Publication Accepted Manuscript
Sami H. Assaf, David E. Speyer Specht modules decompose as alternating sums of restrictions of Schur modules Proceedings of the American Mathematical Society DOI: 10.1090/proc/14815 Accepted Manuscript This is a preliminary PDF of the author-produced manuscript that has been peer-reviewed and accepted for publication. It has not been copyedited, proofread, or finalized by AMS Production staff. Once the accepted manuscript has been copyedited, proofread, and finalized by AMS Production staff, the article will be published in electronic form as a \Recently Published Article" before being placed in an issue. That electronically published article will become the Version of Record. This preliminary version is available to AMS members prior to publication of the Version of Record, and in limited cases it is also made accessible to everyone one year after the publication date of the Version of Record. The Version of Record is accessible to everyone five years after publication in an issue. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000{000 S 0002-9939(XX)0000-0 SPECHT MODULES DECOMPOSE AS ALTERNATING SUMS OF RESTRICTIONS OF SCHUR MODULES SAMI H. ASSAF AND DAVID E. SPEYER (Communicated by Benjamin Brubaker) Abstract. Schur modules give the irreducible polynomial representations of the general linear group GLt. Viewing the symmetric group St as a subgroup of GLt, we may restrict Schur modules to St and decompose the result into a direct sum of Specht modules, the irreducible representations of St. We give an equivariant M¨obiusinversion formula that we use to invert this expansion in the representation ring for St for t large. -
Random Walks on Quasisymmetric Functions
RANDOM WALKS ON QUASISYMMETRIC FUNCTIONS PATRICIA HERSH AND SAMUEL K. HSIAO Abstract. Conditions are provided under which an endomorphism on quasisymmetric functions gives rise to a left random walk on the descent algebra which is also a lumping of a left random walk on permutations. Spectral results are also obtained. Several important random walks are now realized this way: Stanley’s QS-distribution results from endomor- phisms given by evaluation maps, a-shuffles result from the a-th convolution power of the universal character, and the Tchebyshev operator of the second kind introduced recently by Ehrenborg and Readdy yields traditional riffle shuffles. A conjecture of Ehrenborg regarding the spectra for a family of random walks on ab-words is proven. A theorem of Stembridge from the theory of enriched P -partitions is also recovered as a special case. 1. Introduction Quasisymmetric functions have long been used for encoding and manipulating enumerative ∞ combinatorial data. They admit a natural graded Hopf algebra structure Q = n=0 Qn central to the study of combinatorial Hopf algebras [2]. Their dual relationship to Solomon’s descent algebras [31, 20, 27], as well as to noncommutative symmetric functions [19],L are an important part of the story and have inspired a broad literature. A major goal of this paper is to identify and develop bridges between some of this lit- erature and the body of work surrounding Bidigare, Hanlon, and Rockmore’s far-reaching generalizations of Markov chains for various common shuffling and sorting schemes, the main references being [4, 5, 10, 11, 12]. Stanley first recognized and established a connec- tion between quasisymmetric functions and this work in [34]. -
Generating Functions from Symmetric Functions Anthony Mendes And
Generating functions from symmetric functions Anthony Mendes and Jeffrey Remmel Abstract. This monograph introduces a method of finding and refining gen- erating functions. By manipulating combinatorial objects known as brick tabloids, we will show how many well known generating functions may be found and subsequently generalized. New results are given as well. The techniques described in this monograph originate from a thorough understanding of a connection between symmetric functions and the permu- tation enumeration of the symmetric group. Define a homomorphism ξ on the ring of symmetric functions by defining it on the elementary symmetric n−1 function en such that ξ(en) = (1 − x) /n!. Brenti showed that applying ξ to the homogeneous symmetric function gave a generating function for the Eulerian polynomials [14, 13]. Beck and Remmel reproved the results of Brenti combinatorially [6]. A handful of authors have tinkered with their proof to discover results about the permutation enumeration for signed permutations and multiples of permuta- tions [4, 5, 51, 52, 53, 58, 70, 71]. However, this monograph records the true power and adaptability of this relationship between symmetric functions and permutation enumeration. We will give versatile methods unifying a large number of results in the theory of permutation enumeration for the symmet- ric group, subsets of the symmetric group, and assorted Coxeter groups, and many other objects. Contents Chapter 1. Brick tabloids in permutation enumeration 1 1.1. The ring of formal power series 1 1.2. The ring of symmetric functions 7 1.3. Brenti’s homomorphism 21 1.4. Published uses of brick tabloids in permutation enumeration 30 1.5. -
Newton Polytopes in Algebraic Combinatorics
Newton Polytopes in Algebraic Combinatorics Neriman Tokcan University of Michigan, Ann Arbor April 6, 2018 Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Outline Newton Polytopes Geometric Operations on Polytopes Saturated Newton Polytope (SNP) The ring of symmetric polynomials SNP of symmetric functions Partitions - Dominance Order Newton polytope of monomial symmetric function SNP and Schur Positivity Some other symmetric polynomials Additional results Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Newton Polytopes d A polytope is a subset of R ; d ≥ 1 that is a convex hull of finite d set of points in R : Convex polytopes are very useful for analyzing and solving polynomial equations. The interplay between polytopes and polynomials can be traced back to the work of Isaac Newton on plane curve singularities. P α The Newton polytope of a polynomial f = n c x α2Z ≥0 α 2 C[x1; :::; xn] is the convex hull of its exponent vectors, i.e., n Newton(f ) = conv(fα : cα 6= 0g) ⊆ R : Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Geometric operations on Polytopes d Let P and Q be two polytopes in R , then their Minkowski sum is given as P + Q = fp + q : p 2 P; q 2 Qg: The geometric operation of taking Minkowski sum of polytopes mirrors the algebraic operation of multiplying polynomials. Properties: Pn Pn n conv( i=1 Si ) = i=1 conv(Si ); for Si ⊆ R : Newton(fg) = Newton(f ) + Newton(g); for f ; g 2 C[x1; :::; xn]: Newton(f + g) = conv(Newton(f ) [ Newton(g)): Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Saturated Newton Polytope{SNP A polynomial has saturated Newton polytope (SNP) if every lattice point of the convex hull of its exponent vectors corresponds to a monomial. -
Math 263A Notes: Algebraic Combinatorics and Symmetric Functions
MATH 263A NOTES: ALGEBRAIC COMBINATORICS AND SYMMETRIC FUNCTIONS AARON LANDESMAN CONTENTS 1. Introduction 4 2. 10/26/16 5 2.1. Logistics 5 2.2. Overview 5 2.3. Down to Math 5 2.4. Partitions 6 2.5. Partial Orders 7 2.6. Monomial Symmetric Functions 7 2.7. Elementary symmetric functions 8 2.8. Course Outline 8 3. 9/28/16 9 3.1. Elementary symmetric functions eλ 9 3.2. Homogeneous symmetric functions, hλ 10 3.3. Power sums pλ 12 4. 9/30/16 14 5. 10/3/16 20 5.1. Expected Number of Fixed Points 20 5.2. Random Matrix Groups 22 5.3. Schur Functions 23 6. 10/5/16 24 6.1. Review 24 6.2. Schur Basis 24 6.3. Hall Inner product 27 7. 10/7/16 29 7.1. Basic properties of the Cauchy product 29 7.2. Discussion of the Cauchy product and related formulas 30 8. 10/10/16 32 8.1. Finishing up last class 32 8.2. Skew-Schur Functions 33 8.3. Jacobi-Trudi 36 9. 10/12/16 37 1 2 AARON LANDESMAN 9.1. Eigenvalues of unitary matrices 37 9.2. Application 39 9.3. Strong Szego limit theorem 40 10. 10/14/16 41 10.1. Background on Tableau 43 10.2. KOSKA Numbers 44 11. 10/17/16 45 11.1. Relations of skew-Schur functions to other fields 45 11.2. Characters of the symmetric group 46 12. 10/19/16 49 13. 10/21/16 55 13.1. -
Math 958–Topics in Discrete Mathematics Spring Semester 2018 TR 09:30–10:45 in Avery Hall (AVH) 351 1 Instructor
Math 958{Topics in Discrete Mathematics Spring Semester 2018 TR 09:30{10:45 in Avery Hall (AVH) 351 1 Instructor Dr. Tri Lai Assistant Professor Department of Mathematics University of Nebraska - Lincoln Lincoln, NE 68588-0130, USA Email: [email protected] Website: http://www.math.unl.edu/$\sim$tlai3/ Office: Avery Hall 339 Office Hours: By Appointment 2 Prerequisites Math 450. 3 Contacting me The best way to contact with me is by email, [email protected]. Please put \[MATH 958]" at the beginning of the title and make sure to include your whole name in your email. Using your official UNL email to contact me is strongly recommended. My office is Avery Hall 339. My office hours are by appoinment. 4 Course Description Bijective Combinatorics is a branch of combinatorics focusing on bijections between mathematical objects. The fact is that there many totally different objects in mathematics that are actually equinumerous, in this case, we always want to seek for a simple bijection between them. For example, the number of ways to divide that a convex (n + 2)-gon into triangles by non-intersecting diagonals is equal to the number of full binary trees with n + 1 leaves. Both objects are counted 1 2n by the well known Catalan number Cn = n+1 n . Do you know that there are more than two hundred (!!) mathematical objects are counted by the Catalan number? In this course, we will go over a number of such \Catalan objects" and investigate the beautiful bijections between them. One more example is the well-known theorem by Euler about integer partitions stating that the number of ways to write a positive integer n as the sum of distinct positive integers is equal to the number of ways to write n as the sum of odd positive integers. -
Lecture 21: Symmetric Products and Algebras
LECTURE 21: SYMMETRIC PRODUCTS AND ALGEBRAS Symmetric Products The last few lectures have focused on alternating multilinear functions. This one will focus on symmetric multilinear functions. Recall that a multilinear function f : U ×m ! V is symmetric if f(~v1; : : : ;~vi; : : : ;~vj; : : : ;~vm) = f(~v1; : : : ;~vj; : : : ;~vi; : : : ;~vm) for any i and j, and for any vectors ~vk. Just as with the exterior product, we can get the universal object associated to symmetric multilinear functions by forming various quotients of the tensor powers. Definition 1. The mth symmetric power of V , denoted Sm(V ), is the quotient of V ⊗m by the subspace generated by ~v1 ⊗ · · · ⊗ ~vi ⊗ · · · ⊗ ~vj ⊗ · · · ⊗ ~vm − ~v1 ⊗ · · · ⊗ ~vj ⊗ · · · ⊗ ~vi ⊗ · · · ⊗ ~vm where i and j and the vectors ~vk are arbitrary. Let Sym(U ×m;V ) denote the set of symmetric multilinear functions U ×m to V . The following is immediate from our construction. Lemma 1. We have an natural bijection Sym(U ×m;V ) =∼ L(Sm(U);V ): n We will denote the image of ~v1 ⊗: : :~vm in S (V ) by ~v1 ·····~vm, the usual notation for multiplication to remind us that the terms here can be rearranged. Unlike with the exterior product, it is easy to determine a basis for the symmetric powers. Theorem 1. Let f~v1; : : : ;~vmg be a basis for V . Then _ f~vi1 · :::~vin ji1 ≤ · · · ≤ ing is a basis for Sn(V ). Before we prove this, we note a key difference between this and the exterior product. For the exterior product, we have strict inequalities. For the symmetric product, we have non-strict inequalities. -
Finding Direct Partition Bijections by Two-Directional Rewriting Techniques
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 285 (2004) 151–166 www.elsevier.com/locate/disc Finding direct partition bijections by two-directional rewriting techniques Max Kanovich Department of Computer and Information Science, University of Pennsylvania, 3330 Walnut Street, Philadelphia, PA 19104, USA Received 7 May 2003; received in revised form 27 October 2003; accepted 21 January 2004 Abstract One basic activity in combinatorics is to establish combinatorial identities by so-called ‘bijective proofs,’ which consists in constructing explicit bijections between two types of the combinatorial objects under consideration. We show how such bijective proofs can be established in a systematic way from the ‘lattice properties’ of partition ideals, and how the desired bijections are computed by means of multiset rewriting, for a variety of combinatorial problems involving partitions. In particular, we fully characterizes all equinumerous partition ideals with ‘disjointly supported’ complements. This geometrical characterization is proved to automatically provide the desired bijection between partition ideals but in terms of the minimal elements of the order ÿlters, their complements. As a corollary, a new transparent proof, the ‘bijective’ one, is given for all equinumerous classes of the partition ideals of order 1 from the classical book “The Theory of Partitions” by G.Andrews. Establishing the required bijections involves two-directional reductions technique novel in the sense that forward and backward application of rewrite rules heads, respectively, for two di?erent normal forms (representing the two combinatorial types). It is well-known that non-overlapping multiset rules are con@uent. -
Chromatic Quasisymmetric Functions of Directed Graphs
Séminaire Lotharingien de Combinatoire 78B (2017) Proceedings of the 29th Conference on Formal Power Article #74, 12 pp. Series and Algebraic Combinatorics (London) Chromatic Quasisymmetric Functions of Directed Graphs Brittney Ellzey∗ Department of Mathematics, University of Miami, Miami, FL Abstract. Chromatic quasisymmetric functions of labeled graphs were defined by Shareshian and Wachs as a refinement of Stanley’s chromatic symmetric functions. In this extended abstract, we consider an extension of their definition from labeled graphs to directed graphs, suggested by Richard Stanley. We obtain an F-basis expan- sion of the chromatic quasisymmetric functions of all digraphs and a p-basis expan- sion for all symmetric chromatic quasisymmetric functions of digraphs, extending work of Shareshian-Wachs and Athanasiadis. We show that the chromatic quasisymmetric functions of proper circular arc digraphs are symmetric functions, which generalizes a result of Shareshian and Wachs on natural unit interval graphs. The directed cycle on n vertices is contained in the class of proper circular arc digraphs, and we give a generat- ing function for the e-basis expansion of the chromatic quasisymmetric function of the directed cycle, refining a result of Stanley for the undirected cycle. We present a gen- eralization of the Shareshian-Wachs refinement of the Stanley-Stembridge e-positivity conjecture. Keywords: quasisymmetric functions, directed graphs, graph colorings 1 Introduction Let G = (V, E) be a (simple) graph. A proper coloring, k : V ! P, of G is an assignment of positive integers, which we can think of as colors, to the vertices of G such that adjacent vertices have different colors; in other words, if fi, jg 2 E, then k(i) 6= k(j).