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16
Quasisymmetric Schur functions
 J. COMBIN. THEORY SER. A
, 2008
"... We introduce a new basis for quasisymmetric functions, which arise from a specialization of nonsymmetric Macdonald polynomials to Demazure atoms. This basis is called the basis of quasisymmetric Schur functions, since the basis elements refine Schur functions in a natural way. We derive expansions ..."
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We introduce a new basis for quasisymmetric functions, which arise from a specialization of nonsymmetric Macdonald polynomials to Demazure atoms. This basis is called the basis of quasisymmetric Schur functions, since the basis elements refine Schur functions in a natural way. We derive expansions for quasisymmetric Schur functions in terms of monomial and fundamental quasisymmetric functions, which give rise to quasisymmetric refinements of Kostka numbers and standard (reverse) tableaux. We conclude by deriving a Pieri rule for quasisymmetric Schur functions that naturally refines the Pieri rule for Schur functions.
A matroidfriendly basis for quasisymmetric functions
 J. OF COMB. THEORY
, 2008
"... A new Zbasis for the space of quasisymmetric functions (QSym, for short) is presented. It is shown to have nonnegative structure constants, and several interesting properties relative to the space of quasisymmetric functions associated to matroids by the Hopf algebra morphism (F) of Billera, Jia, a ..."
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Cited by 9 (1 self)
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A new Zbasis for the space of quasisymmetric functions (QSym, for short) is presented. It is shown to have nonnegative structure constants, and several interesting properties relative to the space of quasisymmetric functions associated to matroids by the Hopf algebra morphism (F) of Billera, Jia, and Reiner [3]. In particular, for loopless matroids, this basis reflects the grading by matroid rank, as well as by the size of the ground set. It is shown that the morphism F is injective on the set of rank two matroids, and that decomposability of the quasisymmetric function of a rank two matroid mirrors the decomposability of its base polytope. An affirmative answer to the Hilbert basis question raised in [3] is given.
MATROID BASE POLYTOPE DECOMPOSITION
"... Let P(M) be the matroid base polytope of a matroid M. A matroid base polytope decomposition of P(M) is a decomposition of the form P(M) = tS P(Mi) where each P(Mi) is also a matroid base polytope for some matroid Mi, and for each 1 ≤ i ̸ = j ≤ t, the intersection P(Mi)∩P(Mj) is a face of both P(Mi) ..."
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Cited by 2 (0 self)
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Let P(M) be the matroid base polytope of a matroid M. A matroid base polytope decomposition of P(M) is a decomposition of the form P(M) = tS P(Mi) where each P(Mi) is also a matroid base polytope for some matroid Mi, and for each 1 ≤ i ̸ = j ≤ t, the intersection P(Mi)∩P(Mj) is a face of both P(Mi) and P(Mj). In this paper, we investigate hyperplane splits, that is, polytope decompositions when t = 2. We give sufficient conditions for M so P(M) has a hyperplane split and characterize when P(M1 ⊕ M2) has a hyperplane split where M1 ⊕ M2 denote the direct sum of matroids M1 and M2. We also prove that P(M) has not a hyperplane split if M is binary. Finally, we show that P(M) has not a decomposition if its 1skeleton is the hypercube.
HOPF MONOIDS IN THE CATEGORY OF SPECIES
"... Abstract. A Hopf monoid (in Joyal’s category of species) is an algebraic structure akin to that of a Hopf algebra. We provide a selfcontained introduction to the theory of Hopf monoids in the category of species. Combinatorial structures which compose and decompose give rise to Hopf monoids. We stu ..."
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Abstract. A Hopf monoid (in Joyal’s category of species) is an algebraic structure akin to that of a Hopf algebra. We provide a selfcontained introduction to the theory of Hopf monoids in the category of species. Combinatorial structures which compose and decompose give rise to Hopf monoids. We study several examples of this nature. We emphasize the central role played in the theory by the Tits algebra of set compositions. Its product is tightly knit with the Hopf monoid axioms, and its elements constitute universal operations on connected Hopf monoids. We study analogues of the classical Eulerian and Dynkin idempotents and discuss the PoincaréBirkhoffWitt and CartierMilnorMoore theorems for Hopf monoids.
Hopf algebras in combinatorics
, 2013
"... Certain Hopf algebras arise in combinatorics because they have bases naturally parametrized by combinatorial objects (partitions, compositions, permutations, tableaux, graphs, trees, posets, polytopes, etc). The rigidity in the structure of a Hopf algebra can lead to enlightening proofs, and many i ..."
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Certain Hopf algebras arise in combinatorics because they have bases naturally parametrized by combinatorial objects (partitions, compositions, permutations, tableaux, graphs, trees, posets, polytopes, etc). The rigidity in the structure of a Hopf algebra can lead to enlightening proofs, and many interesting invariants of combinatorial objects turn out to be evaluations of Hopf morphisms. These are lecture notes for Fall 2012 Math 8680 Topics in Combinatorics at the University of Minnesota. The course is an attempt to focus on examples that I find interesting, but which are hard to find fully explained currently in books or in one paper. Be warned that these notes are highly idiosyncratic in choice
Scheduling problems
, 2014
"... We introduce the notion of a scheduling problem which is a boolean function S over atomic formulas of the form xi ≤ xj. Considering the xi as jobs to be performed, an integer assignment satisfying S schedules the jobs subject to the constraints of the atomic formulas. The scheduling counting funct ..."
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We introduce the notion of a scheduling problem which is a boolean function S over atomic formulas of the form xi ≤ xj. Considering the xi as jobs to be performed, an integer assignment satisfying S schedules the jobs subject to the constraints of the atomic formulas. The scheduling counting function counts the number of solutions to S. We prove that this counting function is a polynomial in the number of time slots allowed. Scheduling polynomials include the chromatic polynomial of a graph, the zeta polynomial of a lattice, the BilleraJiaReiner polynomial of a matroid and the newly defined arboricity polynomial of a matroid. To any scheduling problem, we associate not only a counting function for solutions, but also a quasisymmetric function and a quasisymmetric function in noncommuting variables. These scheduling functions include the chromatic symmetric functions of Sagan, Gebhard, and Stanley, and a close variant of Ehrenborg’s quasisymmetric function for posets. Geometrically, we consider the space of all solutions to a given scheduling problem. We extend a result of Steingŕımmson by proving that the hvector of the space of solutions is given by a shift of the scheduling polynomial. Furthermore, under certain niceness conditions on the defining boolean function, we prove partitionability of the space of solutions and positivity of fundamental expansions of the scheduling quasisymmetric functions and of the hvector of the scheduling polynomial.