DOCSLIB.ORG

On the Product of Generating Functions for Domino and Bi-Tableaux

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Product of Generating Functions for Domino and Bi-Tableaux On the product of generating functions for domino and bi-tableaux Ekaterina A. Vassilieva November 26, 2019 Abstract. The connection between the generating functions of various sets of tableaux and the appropriate families of quasisymmetric functions is a signifi- cant tool to give a direct analytical proof of some advanced bijective results and provide new combinatorial formulas. In this paper we focus on two kinds of type B Littlewood-Richardson coefficients and derive new formulas using weak com- position quasisymmetric functions and Chow’s quasisymmetric functions. In the type A case these coefficients give the multiplication table for Schur functions, i.e. the generating functions for classical Young tableaux. We look at their generalisations involving a set of bi-tableaux and domino tableaux. 1 Type A quasisymmetric functions 1.1 Permutations, Young tableaux and descent sets For any positive integer n write [n] = {1, ··· ,n}, Sn the symmetric group on [n]. A partition λ of n, denoted λ ⊢ n is a sequence λ = (λ1, λ2, ··· , λp) of ℓ(λ) = p parts sorted in decreasing order such that |λ| = i λi = n. A partition λ is usually represented as a Young diagram of n = |λ| boxes arranged P in ℓ(λ) left justified rows so that the i-th row from the top contains λi boxes. A Young diagram whose boxes are filled with positive integers such that the entries are increasing along the rows and strictly increasing down the columns is called a semistandard Young tableau. If the entries of a semistandard arXiv:1911.10430v1 [math.CO] 23 Nov 2019 Young tableau are restricted to the elements of [n] and strictly increasing along the rows, we call it a standard Young tableau. The partition λ is the shape of the tableau and we denote SYT (λ) (resp. SSYT (λ)) the set of standard (resp. semistandard) Young tableaux of shape λ. One important feature of a permutation π in Sn is its descent set, i.e. the subset of [n−1] defined as Des(π)= {1 ≤ i ≤ n−1 | π(i) > π(i+1)}. Similarly, define the descent set of a standard Young tableau T as Des(T )= {1 ≤ i ≤ n − 1 | i is in a strictly higher row than i +1}. 1 1.2 Quasisymmetric expansion of Schur polynomials Let X = {x1, x2, ···} and Y = {y1,y2, ···} be two alphabets of commutative indeterminates. Denote also XY the product alphabet {xiyj }i,j ordered by the lexicographical order. Let FI (X) be the Gessel’s fundamental quasisym- metric function on X indexed by the subset I ⊂ [n − 1]: FI (X)= xi1 xi2 ··· xin . (1) i1≤···≤in k∈I⇒Xik <ik+1 The power series FI (X) is not symmetric in X but verifies the property that for any strictly increasing sequence of indices i1 < i2 < ··· < ip the coefficient k k xk1 xk2 x p xk1 xk2 x p s X of 1 2 ··· p is equal to the coefficient of i1 i2 ··· ip . Let λ( ) be the Schur polynomial indexed by partition λ on alphabet X. Schur polynomials are the generating functions for semistandard Young tableaux. As a result, T sλ(X)= X = FDes(T )(X). (2) T ∈SSYX T (λ) T ∈SYX T (λ) Using for any π ∈ Sn Gessel’s formula for the coproduct of fundamental qua- sisymmetric functions in [4] FDes(π)(XY )= FDes(σ)(X)FDes(ρ)(Y ), (3) σ,ρ S σρ π ∈ Xn; = and the Cauchy formula for Schur polynomials sλ(X)sλ(Y )= s(n)(XY ), (4) λ n X⊢ one can derive FDes(T )(X)FDes(U)(Y )= F∅(XY )= FDes(π)(X)FDes(π−1 )(Y ). λ⊢n π∈Sn T,U∈XSY T (λ) X (5) This gives a direct analytical proof of one important property usually proved using the Robinson-Schensted correspondence, i.e. that permutations π in Sn are in bijection with pairs of standard Young tableaux T,U of the same shape such that Des(T )= Des(π) and Des(U)= Des(π−1). 1.3 Product of Schur polynomials One may write a permutation α ∈ Sn as a word on the letters in [n], α = α1α2 ··· αn where αi = α(i). Given α ∈ Sn and β ∈ Sm, let α β be the set of permutations γ ∈ Sm+n obtained by shuffling the letters α and β = n + β1 n + β2 ··· n + βm such that the initial order of the letters of α (resp. of β) is preserved in γ. By extension if A and B are two sets of permutations, we 2 denote A B the set of all the shuffles of a permutation of A and a permutation of B. We have the product formula FDes(α)(X)FDes(β)(X)= FDes(γ)(X). (6) γ α β ∈X Given a standard Young tableau T of shape λ, Equation (5) proves the existence −1 of classes of permutations CT such that for any π ∈ CT , Des(π ) = Des(T ) and there is a descent preserving bijection between CT and SYT (λ). Remark 1. Classically the CT are the Knuth classes of permutations but here we derive these results directly without using plactic relations and/or the Robinson- Schensted correspondence. Equation (6) directly implies for integer partitions λ and µ and T,U ∈ SYT (λ)× SYT (µ): sλ(X)sµ(X)= FDes(α)(X) FDes(β)(X)= FDes(γ)(X). (7) α C β C γ C C X∈ T X∈ U ∈ XT U ν For any ν ⊢ |λ| + |µ| define formally the coefficients cλµ as the coefficients of the expansion of sλ(X)sµ(X) in the Schur basis (sν )ν . Equation (7) shows for some Vν ∈ SYT (ν) that ν ν cλµsν (X)= cλµFDes(γ)(X)= FDes(γ)(X). (8) γ C C ν⊢|Xλ|+|µ| ν⊢|Xλ|+|µ| ∈ XT U γ∈CVν As a result, we have the following proposition. Proposition 1. For I ⊂ [|λ| + |µ|− 1] denote DI the set of permutations π such that Des(π)= I. Using the notations above for integer partitions λ, µ and for T,U in SYT (λ) × SYT (µ), there is a set of tableaux Vν ∈ SYT (ν) indexed by the set of integer partitions ν ⊢ |λ| + |µ| such that ν cλµ|CVν ∩ DI | = | (CT CU ) ∩ DI |. (9) ν⊢|Xλ|+|µ| The following sections are dedicated to the extension of this method to weak composition quasisymmetric functions and Chow’s type B quasisymmetric functions. 2 Type B extension to weak composition qua- sisymmetric functions 2.1 Weak composition quasisymmetric functions and coloured permutations Weak composition quasisymmetric functions are a generalisation of Gessel’s qua- sisymmetric functions introduced in [5] and [12] in order to provide a framework 3 for the study of a question a G.-C. Rota relating Rota-Baxter algebras and sym- metric type functions. We consider a finite alphabet X = {x1, x2, ...,xM }. For any non-negative integer n, let WC(n) be the set of weak compositions of n, i.e. the set of ordered tuples of non-negative integers that sum up to n. A weak composition α ∈ WC(n) is of the form i1 i2 ik ik+1 α = (0 ,s1, 0 ,s2,..., 0 ,sk, 0 ) with ij ≥ 0, 1 ≤ j ≤ k + 1, si > 0, 1 ≤ j ≤ k and si = n. We further define ℓ(α)= k + i1 + ··· + ik+1 the length of α, ℓ0(α)= i1 + ··· + ik+1 as its P 0-length, |α| = n the weight of α and ||α|| = |α| + ℓ0(α) the total weight of α. The descent set of α is D(α)= {a1,...,ak|aj = i1 + s1 + ··· + ij + sj , j =1,...,k}. j1 j2 jk jk Definition 1. Given a weak composition α = (0 ,s1, 0 ,s2,..., 0 ,sk, 0 +1 ) with D(α) = {a1,a2,...,ak} define the weak composition fundamental quasisymmetric function indexed by α the formal power series Fα(X) = e 0 0 0 0 X xi ··· xi xij +1 ··· xia ··· xia +j +1 ··· xia xi ··· xi (10) 1 j1 1 1 k−1 k k ak+1 ak+jk+1 1≤i ≤···≤i 1 ak+jk+1 j∈D(α)⇒ij <ij+1 Weak composition fundamental quasisymmetric functions may also be de- scribed in terms of P -partitions indexed by coloured permutations. Define the total order 1 < 2 < 3 < ··· < 1 < 2 < 3 ··· and denote |i| = i for i > 0. We look at chain posets Pπ = {π1 <Pπ π2 <Pπ ··· <Pπ πl} indexed by coloured permutations π for any non-negative integer l where the πi are distinct letters in [l] ∪ [l] and such that |π1||π2| · · · |πl| has no repeated letters. A Pπ-partition is a map f : Pπ → [M] such that for x<Pπ y, f(x) ≤ f(y) with x>y ⇒ f(x) <f(y). The weight of f, w(f) is the monomial δ 0 if π is an overlined letter, w(f)= x πi , where δ = i f(πi) πi π P (1 otherwise. iY∈ π Denote A(π) the set of Pπ-partitions. We are interested in the generating func- tions Γ indexed by coloured permutations π Γ(π)= w(f). (11) f∈AX(π) Li shows in [6] that the Γ(π) can be expressed as a Z-linear combination of the Fα but not always positive. In particular if the indices j1 < j2 < ··· < jp of the 4 overlined letters in a coloured permutation π are such that πj1 < πj2 < ··· < πjp , denote the weak composition i1 i2 ik ik+1 comp(π)=(0 ,s1, 0 ,s2,..., 0 ,sk, 0 ) where (s1,s2,...,sk) is the tuple of lengths of the sequences of consecutive increasing non-overlined letters in π and ir is the quantity of overlined letters (p) between sequences r − 1 and r.
Load more

Recommended publications
  • 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).
    [Show full text]
  • 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].
    [Show full text]
  • 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).
    [Show full text]
  • The Contributions of Stanley to the Fabric of Symmetric and Quasisymmetric Functions
    The contributions of Stanley to the fabric of symmetric and quasisymmetric functions Sara C. Billey and Peter R. W. McNamara A E S A D I Dedicated to L N T C H R on the occasion of his 70th birthday. Y R Abstract. We weave together a tale of two rings, SYM and QSYM, following one gold thread spun by Richard Stanley. The lesson we learn from this tale is that \Combinatorial objects like to be counted by quasisymmetric functions." 1. Introduction The twentieth century was a remarkable era for the theory of symmetric func- tions. Schur expanded the range of applications far beyond roots of polynomials to the representation theory of GLn and Sn and beyond. Specht, Hall and Mac- donald unified the algebraic theory making it far more accessible. Lesieur recog- nized the connection between Schur functions and the topology of Grassmannian manifolds spurring interest and further developments by Borel, Bott, Bernstein{ Gelfand{Gelfand, Demazure and many others. Now, symmetric functions routinely appear in many aspects of mathematics and theoretical physics, and have significant importance in quantum computation. In that era of mathematical giants, Richard Stanley's contributions to symmet- ric functions are shining examples of how enumerative combinatorics has inspired and influenced some of the best work of the century. In this article, we focus on a few of the gems that continue to grow in importance over time. Specifically, we survey some results and applications for Stanley symmetric functions, chromatic symmetric functions, P -partitions, generalized Robinson{Schensted{Knuth corre- spondence, and flag symmetry of posets.
    [Show full text]
  • Arxiv:1810.07148V1 [Math.CO]
    RECENT TRENDS IN QUASISYMMETRIC FUNCTIONS SARAH K. MASON Abstract. This article serves as an introduction to several recent developments in the study of quasisymmetric functions. The focus of this survey is on connections between quasisymmetric functions and the combinatorial Hopf algebra of noncom- mutative symmetric functions, appearances of quasisymmetric functions within the theory of Macdonald polynomials, and analogues of symmetric functions. Topics include the significance of quasisymmetric functions in representation theory (such as representations of the 0-Hecke algebra), recently discovered bases (including ana- logues of well-studied symmetric function bases), and applications to open problems in symmetric function theory. Contents 1. Introduction 1 1.1. Basic definitions and background on symmetric functions 2 1.2. Schur functions 4 1.3. Quasisymmetric functions 5 2. Algebra and representation theory 8 2.1. Combinatorial Hopf algebras 9 2.2. Solomon’s descent algebra 11 2.3. Representationsofthe0-Heckealgebra 12 3. Macdonald polynomials 13 3.1. Quasisymmetric decomposition of Macdonald polynomials 15 3.2. Quasisymmetric Schur functions 16 4. Quasisymmetricanaloguesofsymmetricfunctionbases 18 4.1. Dual immaculate quasisymmetric functions 18 4.2. Quasisymmetric analogues of the power sum basis 20 4.3. The shuffle algebra 22 5. Connections to symmetric functions and the polynomial ring 24 arXiv:1810.07148v1 [math.CO] 16 Oct 2018 5.1. Chromatic quasisymmetric functions 25 5.2. Quasisymmetric expansions of symmetric functions 27 5.3. Slide polynomials and the quasi-key basis 29 6. Acknowledgements 31 References 31 1. Introduction Quasisymmetric functions first appeared in the work of Stanley [117] and were formally developed in Gessel’s seminal article on multipartite P -partitions [44].
    [Show full text]
  • Skew Row-Strict Quasisymmetric Schur Functions Sarah K
    Marshall University Marshall Digital Scholar Mathematics Faculty Research Mathematics 11-2015 Skew row-strict quasisymmetric Schur functions Sarah K. Mason Elizabeth Niese Marshall University, [email protected] Follow this and additional works at: https://mds.marshall.edu/mathematics_faculty Part of the Discrete Mathematics and Combinatorics Commons Recommended Citation Mason, S.K. & Niese, E. Skew row-strict quasisymmetric Schur functions. Journal of Algebraic Combinatorics (2015) 42:763-791. This Article is brought to you for free and open access by the Mathematics at Marshall Digital Scholar. It has been accepted for inclusion in Mathematics Faculty Research by an authorized administrator of Marshall Digital Scholar. For more information, please contact [email protected], [email protected]. Skew row-strict quasisymmetric Schur functions S. K. Mason E. Niese Wake Forest University Marshall University Department of Mathematics Department of Mathematics [email protected] [email protected] April 2015 Abstract Mason and Remmel introduced a basis for quasisymmetric func- tions known as the row-strict quasisymmetric Schur functions. This basis is generated combinatorially by fillings of composition diagrams that are analogous to the row-strict tableaux that generate Schur func- tions. We introduce a modification known as Young row-strict qua- sisymmetric Schur functions, which are generated by row-strict Young composition fillings. After discussing basic combinatorial properties of these functions, we define a skew Young row-strict quasisymmetric Schur function using the Hopf algebra of quasisymmetric functions and then prove this is equivalent to a combinatorial description. We also provide a decomposition of the skew Young row-strict quasisymmetric Schur functions into a sum of Gessel's fundamental quasisymmetric functions and prove a multiplication rule for the product of a Young row-strict quasisymmetric Schur function and a Schur function.
    [Show full text]
  • Divided Symmetrization and Quasisymmetric Functions (Extended Abstract) Philippe Nadeau, Vasu Tewari
    Divided symmetrization and quasisymmetric functions (extended abstract) Philippe Nadeau, Vasu Tewari To cite this version: Philippe Nadeau, Vasu Tewari. Divided symmetrization and quasisymmetric functions (extended abstract). Seminaire Lotharingien de Combinatoire, Université Louis Pasteur, 2020. hal-03099482 HAL Id: hal-03099482 https://hal.archives-ouvertes.fr/hal-03099482 Submitted on 13 Jan 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Séminaire Lotharingien de Combinatoire XX (2020) Proceedings of the 32nd Conference on Formal Power Article #YY, 12 pp. Series and Algebraic Combinatorics (Ramat Gan) Divided symmetrization and quasisymmetric functions Philippe Nadeau∗1, and Vasu Tewariy2 1 Univ Lyon, CNRS, Université Claude Bernard Lyon 1, UMR 5208, Institut Camille Jordan, France 2 Department of Mathematics, University of Pennsylvania, USA Abstract. We study various aspects of the divided symmetrization operator, which was introduced by Postnikov in the context of volume polynomials of permutahe- dra. Divided symmetrization is a linear form which acts on the space of polynomials in n indeterminates of degree n − 1. Our main results are related to quasisymmet- ric polynomials. We show that divided symmetrization applied to a quasisymmetric polynomial in m ≤ n indeterminates has a natural interpretation.
    [Show full text]
  • Quasisymmetric Functions and Kazhdan-Lusztig Polynomials
    QUASISYMMETRIC FUNCTIONS AND KAZHDAN-LUSZTIG POLYNOMIALS LOUIS J. BILLERA AND FRANCESCO BRENTI Abstract. We associate a quasisymmetric function to any Bruhat in- terval in a general Coxeter group. This association can be seen to be a morphism of Hopf algebras to the subalgebra of all peak functions, lead- ing to an extension of the cd-index of convex polytopes. We show how the Kazhdan-Lusztig polynomial of the Bruhat interval can be expressed in terms of this complete cd-index and otherwise explicit combinatori- ally defined polynomials. In particular, we obtain the simplest closed formula for the Kazhdan-Lusztig polynomials that holds in complete generality. Contents 1. Introduction 1 2. The R-quasisymmetric function of a Bruhat interval and the complete cd-index 9 3. Kazhdan-Lusztig polynomials and the complete cd-index 15 4. Kazhdan-Lusztig polynomials and ballot polynomials 18 5. Other representations of the complete cd-index 22 6. A sign conjecture for the complete cd-index 24 References 25 1. Introduction The Kazhdan-Lusztig polynomials of a Coxeter group are of fundamen- tal importance in representation theory and in the geometry and topology of Schubert varieties. Defined by means of two separate recursions, they have proved difficult to unravel in any straightforward manner. Here, we reduce their computation to the computation of another invariant of Cox- eter groups. This new invariant, a quasisymmetric function that can be 2000 Mathematics Subject Classification. Primary 20F55, 05E99; Secondary 05E15. Key words and phrases. Coxeter group, Kazhdan-Lusztig polynomial, cd-index, qua- sisymmetric function. The first author was supported in part by NSF grants DMS-0100323 and DMS-0555268.
    [Show full text]
  • Chromatic Quasisymmetric Functions
    CHROMATIC QUASISYMMETRIC FUNCTIONS JOHN SHARESHIAN1 AND MICHELLE L. WACHS2 Abstract. We introduce a quasisymmetric refinement of Stan- ley's chromatic symmetric function. We derive refinements of both Gasharov's Schur-basis expansion of the chromatic symmetric func- tion and Chow's expansion in Gessel's basis of fundamental qua- sisymmetric functions. We present a conjectural refinement of Stanley's power sum basis expansion, which we prove in special cases. We describe connections between the chromatic quasisym- metric function and both the q-Eulerian polynomials introduced in our earlier work and, conjecturally, representations of symmetric groups on cohomology of regular semisimple Hessenberg varieties, which have been studied by Tymoczko and others. We discuss an approach, using the results and conjectures herein, to the e- positivity conjecture of Stanley and Stembridge for incomparabil- ity graphs of (3 + 1)-free posets. Contents 1. Introduction 2 2. Basic properties and examples 8 3. Expansion in the fundamental basis 10 4. Natural unit interval orders 14 5. Expansion in the elementary basis 18 6. Expansion in the Schur basis 20 7. Expansion in the power sum basis 27 8. Some calculations 34 9. Specialization 38 10. Hessenberg varieties 44 11. Recent developments 48 Appendix A. Permutation statistics 48 Appendix B. Palindromicity and unimodality 49 Appendix C. Unimodality of q-Eulerian polynomials and Smirnov word enumerators 52 Acknowledgments 54 References 54 Date: June 30, 2014; Revised March 18, 2016. 1Supported in part by NSF Grants DMS 0902142 and DMS 1202337. 2Supported in part by NSF Grants DMS 0902323 and DMS 1202755 and by Simons Foundation Grant #267236.
    [Show full text]
  • A Note on Three Types of Quasisymmetric Functions
    A note on three types of quasisymmetric functions T. Kyle Petersen Department of Mathematics Brandeis University, Waltham, MA, USA [email protected] Submitted: Aug 8, 2005; Accepted: Nov 14, 2005; Published: Nov 22, 2005 Mathematics Subject Classifications: 05E99, 16S34 Abstract In the context of generating functions for P -partitions, we revisit three flavors of quasisymmetric functions: Gessel’s quasisymmetric functions, Chow’s type B quasisymmetric functions, and Poirier’s signed quasisymmetric functions. In each case we use the inner coproduct to give a combinatorial description (counting pairs of permutations) to the multiplication in: Solomon’s type A descent algebra, Solomon’s type B descent algebra, and the Mantaci-Reutenauer algebra, respectively. The presentation is brief and elementary, our main results coming as consequences of P -partition theorems already in the literature. 1 Quasisymmetric functions and Solomon's descent algebra The ring of quasisymmetric functions is well-known (see [12], ch. 7.19). Recall that a quasisymmetric function is a formal series Q(x1,x2,...) ∈ Z[[x1,x2,...]] xα1 xα2 ···xαk i < of bounded degree such that the coefficient of i1 i2 ik is the same for all 1 i2 < ··· <ik and all compositions α =(α1,α2,...,αk). Recall that a composition of n, written α |= n, is an ordered tuple of positive integers α =(α1,α2,...,αk) such that |α| = α1 + α2 + ···+ αk = n. In this case we say that α has k parts, or #α = k.Wecan put a partial order on the set of all compositions of n by reverse refinement. The covering relations are of the form (α1,...,αi + αi+1,...,αk) ≺ (α1,...,αi,αi+1,...,αk).
    [Show full text]
  • The Probability of Positivity in Symmetric and Quasisymmetric
    THE PROBABILITY OF POSITIVITY IN SYMMETRIC AND QUASISYMMETRIC FUNCTIONS REBECCA PATRIAS AND STEPHANIE VAN WILLIGENBURG Abstract. Given an element in a finite-dimensional real vector space, V , that is a nonneg- ative linear combination of basis vectors for some basis B, we compute the probability that it is furthermore a nonnegative linear combination of basis vectors for a second basis, A. We then apply this general result to combinatorially compute the probability that a sym- metric function is Schur-positive (recovering the recent result of Bergeron–Patrias–Reiner), e-positive or h-positive. Similarly we compute the probability that a quasisymmetric func- tion is quasisymmetric Schur-positive or fundamental-positive. In every case we conclude that the probability tends to zero as the degree of a function tends to infinity. Contents 1. Introduction 1 2. The probability of vector positivity 2 3. Probabilities of symmetric function positivity 5 4. Probabilities of quasisymmetric function positivity 9 Acknowledgements 14 References 14 1. Introduction The subject of when a symmetric function is Schur-positive, that is, a nonnegative linear arXiv:1810.11038v3 [math.CO] 27 May 2019 combination of Schur functions, is an active area of research. If a homogeneous symmetric function of degree n is Schur-positive, then it is the image of some representation of the symmetric group Sn under the Frobenius characteristic map. Furthermore, if it is a poly- nomial, then it is the character of a polynomial representation of the general linear group GL(n, C). Consequently, much effort has been devoted to determining when the difference of two symmetric functions is Schur-positive, for example [2, 8, 12, 13, 14, 16, 18, 19].
    [Show full text]
  • The Kernel of Chromatic Quasisymmetric Functions on Graphs and Hypergraphic Polytopes
    The kernel of chromatic quasisymmetric functions on graphs and hypergraphic polytopes Raul Penaguiao∗ March 27, 2020 Abstract The chromatic symmetric function on graphs is a celebrated graph invariant. Analogous chromatic maps can be defined on other objects, as presented by Aguiar, Bergeron and Sottile. The problem of identifying the kernel of some of these maps was addressed by F´eray, for the Gessel quasisymmetric function on posets. On graphs, we show that the modular relations and isomorphism relations span the kernel of the chromatic symmetric function. This helps us to construct a new invariant on graphs, which may be helpful in the context of the tree conjecture. We also address the kernel problem in the Hopf algebra of generalized permutahedra, introduced by Aguiar and Ardila. We present a solution to the kernel problem on the Hopf algebra spanned by hypergraphic polytopes, which is a subfamily of generalized permutahedra that contains a number of polytope families. Finally, we consider the non-commutative analogues of these quasisymmetric invariants, and establish that the word quasisymmetric functions, also called non- commutative quasisymmetric functions, form the terminal object in the category of combinatorial Hopf monoids. As a corollary, we show that there is no combinatorial Hopf monoid morphism between the combinatorial Hopf monoid of posets and that of hypergraphic polytopes. 1 Introduction Chromatic function on graphs For a graph G with vertex set V (G), a coloring f of the graph G is a function f : V (G) ! N. A coloring is proper in G if no edge is monochromatic. arXiv:1803.08824v4 [math.CO] 27 Mar 2020 We denote by G the graph Hopf algebra, which is a vector space freely generated by the graphs whose vertex sets are of the form [n] for some n ≥ 0.
    [Show full text]