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  • 1. Borcea, J.
    et al.
    Brändén, Petter
    Department of Mathematics, Stockholm University.
    Hyperbolicity preservers and majorization2010In: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, ISSN 1631-073X, Vol. 348, no 15-16, p. 843-846Article in journal (Refereed)
    Abstract [en]

    The majorization order on R(n) induces a natural partial ordering on the space of univariate hyperbolic polynomials of degree n. We characterize all linear operators on polynomials that preserve majorization, and show that it is sufficient (modulo obvious degree constraints) to preserve hyperbolicity.

  • 2. Borcea, J.
    et al.
    Brändén, Petter
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Multivariate Polya-Schur classification problems in the Weyl algebra2010In: Proceedings of the London Mathematical Society, ISSN 0024-6115, E-ISSN 1460-244X, Vol. 101, p. 73-104Article in journal (Refereed)
    Abstract [en]

    A multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts. We classify all linear partial differential operators in the Weyl algebra A(n) that preserve stability. An important tool that we develop in the process is the higher-dimensional generalization of Polya-Schur's notion of multiplier sequence. We characterize all multivariate multiplier sequences as well as those of finite order. Next, we establish a multivariate extension of the Cauchy-Poincare interlacing theorem and prove a natural analog of the Lax conjecture for real stable polynomials in two variables. Using the latter we describe all operators in A(1) that preserve univariate hyperbolic polynomials by means of determinants and homogenized symbols. Our methods also yield homotopical properties for symbols of linear stability preservers and a duality theorem showing that an operator in A(n) preserves stability if and only if its Fischer-Fock adjoint does. These are powerful multivariate extensions of the classical Hermite-Poulain-Jensen theorem, Polya's curve theorem and Schur-Malo-Szegocomposition theorems. Examples and applications to strict stability preservers are also discussed.

  • 3. Borcea, Julius
    et al.
    Bränden, Petter
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products2008In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 143, no 2, p. 205-223Article in journal (Refereed)
    Abstract [en]

    For (n x n)-matrices A and B, define eta(A, B) = Sigma(S)det(A[S])det(B[S']), where the summation is over all subsets of {1,..., n}, S' is the complement of S', and A [S] is the principal submatrix of A with rows and columns indexed by S. We prove that if A >= 0 and B is Hermitian, then (1) the polynomial eta(zA, -B) has all real roots; (2) the latter polynomial has as many positive, negative, and zero roots (counting multiplicities) as suggested by the inertia of B if A > 0; and (3) for 1 <= i <= n, the roots of eta(zA[{1}'], -B[{i}']) interlace those of eta(zA, -B). Assertions (1) - (3) solve three important conjectures proposed by C. R. Johnson in the mid-1980s in [20, pp. 169, 170], [21]. Moreover, we substantially extend these results to tuples of matrix pencils and real stable polynomials. In the process, we establish unimodality properties in the sense of majorization for the coefficients of homogeneous real stable polynomials, and as an application, we derive similar properties for symmetrized Fischer products of positive-definite matrices. We also obtain Laguerre-type inequalities for characteristic polynomials of principal submatrices of arbitrary Hermitian matrices which considerably generalize a certain subset of the Hadamard., Fischer, and Koteljanskii inequalities for principal minors of positive-definite matrices. Finally, we propose Lax-type problems for real stable polynomials and mixed determinants.

  • 4. Borcea, Julius
    et al.
    Bränden, Petter
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lee-Yang Problems and the Geometry of Multivariate Polynomials2008In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 86, no 1, p. 53-61Article in journal (Refereed)
    Abstract [en]

    We describe all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing in open circular domains. This completes the multivariate generalization of the classification program initiated by Polya-Schur for univariate real polynomials and provides a natural framework for dealing in a uniform way with Lee-Yang type problems in statistical mechanics, combinatorics, and geometric function theory.

  • 5. Borcea, Julius
    et al.
    Bränden, Petter
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Polya-Schur master theorems for circular domains and their boundaries2009In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 170, no 1, p. 465-492Article in journal (Refereed)
    Abstract [en]

    We characterize all linear operators on finite or infinite-dimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region Omega subset of C for arbitrary closed circular domains Omega (i.e., images of the closed unit disk under a Mobius transformation) and their boundaries. This provides a natural framework for dealing with several long-standing fundamental problems, which we solve in a unified way. In particular, for Omega = R our results settle open questions that go back to Laguerre and Polya-Schur.

  • 6. Borcea, Julius
    et al.
    Bränden, Petter
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The Lee-Yang and Polya-Schur programs. I. Linear operators preserving stability2009In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 177, no 3, p. 541-569Article in journal (Refereed)
    Abstract [en]

    In 1952 Lee and Yang proposed the program of analyzing phase transitions in terms of zeros of partition functions. Linear operators preserving non-vanishing properties are essential in this program and various contexts in complex analysis, probability theory, combinatorics, and matrix theory. We characterize all linear operators on finite or infinite-dimensional spaces of multivariate polynomials preserving the property of being non-vanishing whenever the variables are in prescribed open circular domains. In particular, this solves the higher dimensional counterpart of a long-standing classification problem originating from classical works of Hermite, Laguerre, Hurwitz and Polya-Schur on univariate polynomials with such properties.

  • 7. Borcea, Julius
    et al.
    Bränden, Petter
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The Lee-Yang and Polya-Schur Programs. II. Theory of Stable Polynomials and Applications2009In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 62, no 12, p. 1595-1631Article in journal (Refereed)
    Abstract [en]

    In the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains. For such sets this completes the multivariate generalization of the classification program initiated by Polya and Schur for univariate real polynomials. We build on these classification theorems to develop here a theory of multivariate stable polynomials. Applications and examples show that this theory provides a natural framework for dealing in a uniform way with Lee-Yang type problems in statistical mechanics, combinatorics, and geometric function theory in one or several variables. In particular, we answer a question of Hinkkanen on multivariate apolarity.

  • 8. Borcea, Julius
    et al.
    Bränden, Petter
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Liggett, Thomas M.
    NEGATIVE DEPENDENCE AND THE GEOMETRY OF POLYNOMIALS2009In: Journal of The American Mathematical Society, ISSN 0894-0347, E-ISSN 1088-6834, Vol. 22, no 2, p. 521-567Article in journal (Refereed)
  • 9. Bränden, Petter
    Actions on permutations and unimodality of descent polynomials2008In: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 29, no 2, p. 514-531Article in journal (Refereed)
    Abstract [en]

    We study a group action on permutations due to Foata and Strehl and use it to prove that the descent generating polynomial of certain sets of permutations has a non-negative expansion in the basis {t(i) (1 + t)(n-1-2i)}(i=0)(m), m = [(n - 1)/2]. This property implies symmetry and unimodality. We prove that the action is invariant under stack sorting which strengthens recent unimodality results of Bona. We prove that the generalized permutation patterns (13-2) and (2-31) are invariant under the action and use this to prove unimodality properties for a q-analog of the Eulerian numbers recently studied by Corteel, Postnikov, Steingrimsson and Williams. We also extend the action to linear extensions of sign-graded posets to give a new proof of the unimodality of the (P, omega)-Eulerian polynomials of sign-graded posets and a combinatorial interpretations (in terms of Stembridge's peak polynomials) of the corresponding coefficients when expanded in the above basis. Finally, we prove that the statistic defined as the number of vertices of even height in the unordered decreasing tree of a permutation has the same distribution as the number of descents on any set of permutations invariant under the action. On restricting to the set of stack sortable permutations we recover a result of Kreweras.

  • 10. Bränden, Petter
    Counterexamples to the Neggers-Stanley conjecture2004In: Electronic research announcements of the American Mathematical Society, ISSN 1079-6762, Vol. 10, p. 155-158Article in journal (Refereed)
    Abstract [en]

    The Neggers-Stanley conjecture asserts that the polynomial counting the linear extensions of a labeled finite partially ordered set by the number of descents has real zeros only. We provide counterexamples to this conjecture.

  • 11. Bränden, Petter
    On linear transformations preserving the Polya frequency property2006In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 358, no 8, p. 3697-3716Article in journal (Refereed)
    Abstract [en]

    We prove that certain linear operators preserve the Polya frequency property and real-rootedness, and apply our results to settle some conjectures and open problems in combinatorics proposed by Bona, Brenti and Reiner-Welker.

  • 12. Bränden, Petter
    On operators on polynomials preserving real-rootedness and the Neggers-Stanley conjecture2004In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 20, no 2, p. 119-130Article in journal (Refereed)
    Abstract [en]

    We refine a technique used in a paper by Schur on real-rooted polynomials. This amounts to an extension of a theorem of Wagner on Hadamard products of Polya frequency sequences. We also apply our results to polynomials for which the Neggers-Stanley Conjecture is known to hold. More precisely, we settle interlacing properties for E-polynomials of series-parallel posets and column-strict labelled Ferrers posets.

  • 13. Bränden, Petter
    Polynomials with the half-plane property and matroid theory2007In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 216, no 1, p. 302-320Article in journal (Refereed)
    Abstract [en]

    A polynomial f is said to have the half-plane property if there is an open half-plane H subset of C, whose boundary contains the origin, such that f is non-zero whenever all the variables are in H. This paper answers several open questions relating multivariate polynomials with the half-plane property to matroid theory. (1) We prove that the support of a multivariate polynomial with the half-plane property is a jump system. This answers an open question posed by Choe, Oxley, Sokal and Wagner and generalizes their recent result claiming that the same is true whenever the polynomial is also homogeneous. (2) We prove that a multivariate multi-affine polynomial f is an element of R[z(1),..., z(n)] has the half-plane property (with respect to the upper half-plane) if and only if partial derivative f/partial derivative(zi)(x)center dot partial derivative f/partial derivative(zj)(x)-partial derivative(2)f/partial derivative(zi)partial derivative(zj)(x)center dot f(x)>= 0 for all x is an element of R-n and 1 <= i, j <= n. This is used to answer two open questions posed by Choe and Wagner regarding strongly Rayleigh matroids. (3) We prove that the Fano matroid is not the support of a polynomial with the half-plane property. This is the first instance of a matroid which does not appear as the support of a polynomial with the half-plane property and answers a question posed by Choe et al. We also discuss further directions and open problems.

  • 14. Bränden, Petter
    q-Narayana numbers and the flag h-vector2004In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 281, no 03-jan, p. 67-81Article in journal (Refereed)
    Abstract [en]

    The Narayana numbers are N(n,k) = (1/n)((n)(k))((n)(k+1)). There are several natural statistics on Dyck paths with a distribution given by N(n, k). We show the equidistribution of Narayana statistics by computing the flag h-vector of J(2 x n) in different ways. In the process we discover new Narayana statistics and provide co-statistics for the Narayana statistics so that the bi-statistics have a distribution given by Furlinger and Hofbauer's q-Narayana numbers. We interpret the flag h-vector in terms of semi-standard Young tableaux, which enables us to express the q-Narayana numbers in terms of Schur functions. We also introduce what we call pre-shellings of simplicial complexes.

  • 15. Bränden, Petter
    Sign-graded posets, unimodality of W-polynomials and the Charney-Davis conjecture2004In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 11, no 2Article in journal (Refereed)
    Abstract [en]

    We generalize the notion of graded posets to what we call sign-graded (labeled) posets. We prove that the W-polynomial of a sign-graded poset is symmetric and unimodal. This extends a recent result of Reiner and Welker who proved it for graded posets by associating a simplicial polytopal sphere to each graded poset. By proving that the W-polynomials of sign-graded posets has the right sign at -1, we are able to prove the Charney-Davis Conjecture for these spheres (whenever they are flag).

  • 16. Bränden, Petter
    et al.
    Claesson, A.
    Steingrimsson, E.
    Catalan continued fractions, and increasing subsequences in permutations2002In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 258, no 03-jan, p. 275-287Article in journal (Refereed)
    Abstract [en]

    We call a Stieltjes continued fraction with monic monomial numerators a Catalan continued fraction. Let e(k)(pi) be the number of increasing subsequences of length k + 1 in the permutation pi. We prove that any Catalan continued fraction is the multivariate generating function of a family of statistics on the 132-avoiding permutations, each consisting of a (possibly infinite) linear combination of the e(k)S. Moreover, there is an invertible linear transformation that translates between linear combinations of ekS and the corresponding continued fractions. Some applications are given, one of which relates fountains of coins to 132-avoiding permutations according to number of inversions. Another relates ballot numbers to such permutations according to number of right-to-left maxima.

  • 17.
    Bränden, Petter
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Moci, Luca
    The multivariate arithmetic Tutte polynomial2014In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 366, no 10, p. 5523-5540Article in journal (Refereed)
    Abstract [en]

    We introduce an arithmetic version of the multivariate Tutte polynomial and a quasi-polynomial that interpolates between the two. A generalized Fortuin-Kasteleyn representation with applications to arithmetic colorings and flows is obtained. We give a new and more general proof of the positivity of the coefficients of the arithmetic Tutte polynomial and (in the representable case) a geometrical interpretation of them.

  • 18.
    Bränden, Petter
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Wagner, David G.
    A converse to the Grace-Walsh-Szego theorem2009In: Mathematical proceedings of the Cambridge Philosophical Society (Print), ISSN 0305-0041, E-ISSN 1469-8064, Vol. 147, p. 447-453Article in journal (Refereed)
    Abstract [en]

    We prove that the symmetrizer of a permutation group preserves stability if and only if the group is orbit homogeneous. A consequence is that the hypothesis of permutation invariance in the Grace-Walsh-Szego Coincidence Theorem cannot be relaxed. In the process we obtain a new characterization of the Grace-like polynomials, introduced by D. Ruelle, and prove that the class of such polynomials can be endowed with a natural multiplication.

  • 19.
    Brändén, Petter
    Department of Mathematics, Stockholm University.
    A generalization of the Heine-Stieltjes theorem2011In: Constructive approximation, ISSN 0176-4276, E-ISSN 1432-0940, Vol. 34, no 1, p. 135-148Article in journal (Refereed)
    Abstract [en]

    The Heine-Stieltjes theorem describes the polynomial solutions, (v,f) such that T(f)=vf, to specific second-order differential operators, T, with polynomial coefficients. We extend the theorem to concern all (nondegenerate) differential operators preserving the property of having only real zeros, thus solving a conjecture of B. Shapiro. The new methods developed are used to describe intricate interlacing relations between the zeros of different pairs of solutions. This extends recent results of Bourget, McMillen and Vargas for the Heun equation and answers their question of how to generalize their results to higher degrees. Many of the results are new even for the classical case.

  • 20.
    Brändén, Petter
    Department of Mathematics, Stockholm University.
    Discrete concavity and the half-plane property2010In: SIAM Journal on Discrete Mathematics, ISSN 0895-4801, E-ISSN 1095-7146, Vol. 24, no 3, p. 921-933Article in journal (Refereed)
    Abstract [en]

    Murota et al. have recently developed a theory of discrete convex analysis which concerns M-convex functions on jump systems. We introduce here a family of M-concave functions arising naturally from polynomials (over a field of generalized Puiseux series) with prescribed non-anishing properties. This family contains several of the most well studied M-concave functions in the literature. In the language of tropical geometry, we study the tropicalization of the space of polynomials with the half-plane property and show that it is strictly contained in the space of M-concave functions. We also provide a short proof of Speyer's "hive theorem" which he used to give a new proof of Horn's conjecture on eigenvalues of sums of Hermitian matrices.

  • 21.
    Brändén, Petter
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Discrete Concavity and Zeros of Polynomials2009In: Electronic Notes in Discrete Mathematics, ISSN 1571-0653, E-ISSN 1571-0653, Vol. 34, p. 531-535Article in journal (Refereed)
    Abstract [en]

    Murota et al. have recently developed a theory of discrete convex analysis as a framework to solve combinatorial optimization problems using ideas from continuous optimization. This theory concerns M-convex functions on jump systems. We introduce here a family of M-concave functions arising naturally from polynomials (over the field of Puiseux series) with prescribed non-vanishing properties. We also provide a short proof of Speyer's "hive theorem" which he used to give a new proof of Horn's conjecture on eigenvalues of sums of Hermitian matrices. Due to limited space a more coherent treatment and proofs will appear elsewhere.

  • 22.
    Brändén, Petter
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Hyperbolicity cones of elementary symmetric polynomials are spectrahedral2014In: Optimization Letters, ISSN 1862-4472, E-ISSN 1862-4480, Vol. 8, no 5, p. 1773-1782Article in journal (Refereed)
    Abstract [en]

    We prove that the hyperbolicity cones of elementary symmetric polynomials are spectrahedral, i.e., they are slices of the cone of positive semidefinite matrices. The proof uses the matrix-tree theorem, an idea already present in Choe et al.

  • 23.
    Brändén, Petter
    Department of Mathematics, Stockholm University.
    Iterated sequences and the geometry of zeros2011In: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, ISSN 0075-4102, Vol. 658, p. 115-131Article in journal (Refereed)
    Abstract [en]

    We study the effect on the zeros of generating functions of sequences under certain non-linear transformations. Characterizations of Polya-Schur type are given of the transformations that preserve the property of having only real and non-positive zeros. In particular, if a polynomial a(0) + a(1)z + ... + a(n)z(n) has only real and non-positive zeros, then so does the polynomial a(0)(2) + (a(1)(2) - a(0)a(2))z + ... + (a(n-1)(2) - a(n-2)a(n))z(n-1) + a(n)(2)z(n). This confirms a conjecture of Fisk, McNamara-Sagan and Stanley, respectively. A consequence is that if a polynomial has only real and non-positive zeros, then its Taylor coefficients form an infinitely log-concave sequence. We extend the results to transcendental entire functions in the Laguerre-Polya class, and discuss the consequences to problems on iterated Turan inequalities, studied by Craven and Csordas. Finally, we propose a new approach to a conjecture of Boros and Moll.

  • 24.
    Brändén, Petter
    Department of Mathematics, Stockholm University.
    Obstructions to determinantal representability2011In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 226, no 2, p. 1202-1212Article in journal (Refereed)
    Abstract [en]

    There has recently been ample interest in the question of which sets can be represented by linear matrix inequalities (LMIs). A necessary condition is that the set is rigidly convex, and it has been conjectured that rigid convexity is also sufficient. To this end Helton and Vinnikov conjectured that any real zero polynomial admits a determinantal representation with symmetric matrices. We disprove this conjecture. By relating the question of finding LMI representations to the problem of determining whether a polymatroid is representable over the complex numbers, we find a real zero polynomial such that no power of it admits a determinantal representation. The proof uses recent results of Wagner and Wei on matroids with the half-plane property, and the polymatroids associated to hyperbolic polynomials introduced by Gurvits.

  • 25.
    Brändén, Petter
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Solutions to two problems on permanents2012In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 436, no 1, p. 53-58Article in journal (Refereed)
    Abstract [en]

    In this note we settle two open problems in the theory of permanents by using recent results from other areas of mathematics. Both problems were recently discussed in Bapat's survey [2]. Bapat conjectured that certain quotients of permanents, which generalize symmetric function means, are concave. We prove this conjecture by using concavity properties of hyperbolic polynomials. Motivated by problems on random point processes, Shirai and Takahashi raised the problem: Determine all real numbers a for which the alpha-permanent (or alpha-determinant) is nonnegative for all positive semidefinite matrices. We give a complete solution to this problem by using recent results of Scott and Sokal on completely monotone functions. It turns out that the conjectured answer to the problem is false.

  • 26.
    Brändén, Petter
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    The Lee-Yang and Pólya-Schur programs. III. Zero-preservers on Bargmann-Fock spaces2014In: American Journal of Mathematics, ISSN 0002-9327, E-ISSN 1080-6377, Vol. 136, no 1, p. 241-253Article in journal (Refereed)
    Abstract [en]

    We characterize linear operators preserving zero-restrictions on entire functions in weighted Bargmann-Fock spaces. This extends the characterization of linear operators on polynomials preserving stability (due to Borcea and the author) to the realm of entire functions, and translates into an optimal, albeit formal, Lee-Yang theorem.

  • 27.
    Brändén, Petter
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Chasse, Matthew
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Infinite log-concavity for polynomial pólya frequency sequences2015In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 143, no 12, p. 5147-5158Article in journal (Refereed)
    Abstract [en]

    McNamara and Sagan conjectured that if a0, a1, a2, . . . is a Pólya frequency (PF) sequence, then so is (formula presented), . . .. We prove this conjecture for a natural class of PF-sequences which are interpolated by polynomials. In particular, this proves that the columns of Pascal’s triangle are infinitely log-concave, as conjectured by McNamara and Sagan. We also give counterexamples to the first mentioned conjecture. Our methods provide families of nonlinear operators that preserve the property of having only real and nonpositive zeros.

  • 28.
    Brändén, Petter
    et al.
    Department of Mathematics, Stockholm University.
    Claesson, A.
    Mesh patterns and the expansion of permutation statistics as sums of permutation patterns2011In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, ISSN 1077-8926, Vol. 18, no 2, p. Paper 5-14Article in journal (Refereed)
    Abstract [en]

    Any permutation statistic f : G -> C may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: f = Sigma(tau)lambda(f)(tau)tau . To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern p = (pi, R) is an occurrence of the permutation pattern pi with additional restrictions specified by R on the relative position of the entries of the occurrence. We show that, for any mesh pattern p = (pi, R), wehave lambda(p)(tau) = (-1)(vertical bar tau vertical bar-vertical bar pi vertical bar)p*(tau) where p* = (pi, R(c)) is the mesh pattern with the same underlying permutation as p but with complementary restrictions. We use this result to expand some well known permutation statistics, such as the number of left-to-right maxima, descents, excedances, fixed points, strong fixed points, and the major index. We also show that alternating permutations, Andre permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns. Finally, we provide new natural Mahonian statistics.

  • 29.
    Brändén, Petter
    et al.
    Department of Mathematics, Stockholm University.
    González D'Leon, Rafael
    Department of Mathematics, University of Miami, Coral Gables.
    On the half-plane property and the {T}utte group of a matroid2010In: Journal of combinatorial theory. Series B (Print), ISSN 0095-8956, E-ISSN 1096-0902, Vol. 100, no 5, p. 485-492Article in journal (Refereed)
    Abstract [en]

    A multivariate polynomial is stable if it is non-vanishing whenever all variables have positive imaginary parts. A matroid has the weak half-plane property (WHPP) if there exists a stable polynomial with support equal to the set of bases of the matroid. If the polynomial can be chosen with all of its non-zero coefficients equal to one then the matroid has the half-plane property (HPP). We describe a systematic method that allows us to reduce the WHPP to the HPP for large families of matroids. This method makes use of the Tutte group of a matroid. We prove that no projective geometry has the WHPP and that a binary matroid has the WHPP if and only if it is regular. We also prove that T(8) and R(9) fail to have the WHPP.

  • 30.
    Brändén, Petter
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Haglund, James
    Visontai, Mirko
    Wagner, David G.
    Proof of the Monotone Column Permanent Conjecture2011In: Notions of Positivity and the Geometry of Polynomials / [ed] Petter Brändén, Mikael Passare, Mihai Putinar, Birkhäuser Verlag, 2011, 1, p. 63-78Chapter in book (Refereed)
  • 31.
    Brändén, Petter
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Jonasson, Johan
    Negative Dependence in Sampling2012In: Scandinavian Journal of Statistics, ISSN 0303-6898, E-ISSN 1467-9469, Vol. 39, no 4, p. 830-838Article in journal (Refereed)
    Abstract [en]

    . The strong Rayleigh property is a new and robust negative dependence property that implies negative association; in fact it implies conditional negative association closed under external fields (CNA+). Suppose that and are two families of 0-1 random variables that satisfy the strong Rayleigh property and let . We show that {Zi} conditioned on is also strongly Rayleigh; this turns out to be an easy consequence of the results on preservation of stability of polynomials of Borcea & Branden (Invent. Math., 177, 2009, 521569). This entails that a number of important pps sampling algorithms, including Sampford sampling and Pareto sampling, are CNA+. As a consequence, statistics based on such samples automatically satisfy a version of the Central Limit Theorem for triangular arrays.

  • 32.
    Brändén, Petter
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Krasikov, Ilia
    Shapiro, Boris
    ELEMENTS OF POLYA-SCHUR THEORY IN THE FINITE DIFFERENCE SETTING2016In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 144, no 11, p. 4831-4843Article in journal (Refereed)
    Abstract [en]

    The Polya-Schur theory describes the class of hyperbolicity preservers, i.e., the class of linear operators acting on univariate polynomials and preserving real-rootedness. We attempt to develop an analog of Polya-Schur theory in the setting of linear finite difference operators. We study the class of linear finite difference operators preserving the set of real-rooted polynomials whose mesh (i.e., the minimal distance between the roots) is at least one. In particular, we prove a finite difference version of the classical Hermite-Poulain theorem and several results about discrete multiplier sequences.

  • 33.
    Brändén, Petter
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Leander, M.
    Visontai, Mirkó
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Multivariate Eulerian Polynomials and Exclusion Processes2016In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 25, no 4, p. 486-499Article in journal (Refereed)
    Abstract [en]

    We give a new combinatorial interpretation of the stationary distribution of the (partially) asymmetric exclusion process on a finite number of sites in terms of decorated alternative trees and coloured permutations. The corresponding expressions of the multivariate partition functions are then related to multivariate generalisations of Eulerian polynomials for coloured permutations considered recently by N. Williams and the third author, and others. We also discuss stability and negative dependence properties satisfied by the partition functions.

  • 34. Brändén, Petter
    et al.
    Mansour, Toufik
    Finite automata and pattern avoidance in words2005In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 110, no 1, p. 127-145Article in journal (Refereed)
    Abstract [en]

    We say that a word w on a totally ordered alphabet avoids the word v if there are no subsequences in w order-equivalent to v. In this paper we suggest a new approach to the enumeration of words on at most k letters avoiding a given pattern. By studying an automaton which for fixed k generates the words avoiding a given pattern we derive several previously known results for these kind of problems, as well as many new. In particular, we give a simple proof of the formula (Electron. J. Combin. 5(1998) #R15) for exact asymptotics for the number of words on k letters of length n that avoids the pattern 12...(l + 1). Moreover, we give the first combinatorial proof of the exact formula (Enumeration of words with forbidden patterns, Ph.D. Thesis, University of Pennsylvania, 1998) for the number of words on k letters of length n avoiding a three letter permutation pattern.

  • 35.
    Brändén, Petter
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Moci, L.
    The multivariate arithmetic Tutte polynomial2012In: Discrete Mathematics & Theoretical Computer Science, ISSN 1462-7264, E-ISSN 1365-8050, p. 661-672Article in journal (Refereed)
    Abstract [en]

    We introduce an arithmetic version of the multivariate Tutte polynomial recently studied by Sokal, and a quasi-polynomial that interpolates between the two. We provide a generalized Fortuin-Kasteleyn representation for representable arithmetic matroids, with applications to arithmetic colorings and flows. We give a new proof of the positivity of the coefficients of the arithmetic Tutte polynomial in the more general framework of pseudo-arithmetic matroids. In the case of a representable arithmetic matroid, we provide a geometric interpretation of the coefficients of the arithmetic Tutte polynomial.

  • 36.
    Brändén, Petter
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Ottergren, Elin
    A Characterization of Multiplier Sequences for Generalized Laguerre Bases2014In: Constructive approximation, ISSN 0176-4276, E-ISSN 1432-0940, Vol. 39, no 3, p. 585-596Article in journal (Refereed)
    Abstract [en]

    We give a complete characterization of multiplier sequences for generalized Laguerre bases. We also apply our methods to give a short proof of the characterization of Hermite multiplier sequences achieved by Piotrowski.

1 - 36 of 36
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