Log-concave Sequences Research Articles

The extremals of the Kahn-Saks inequality

A classical result of Kahn and Saks states that given any partially ordered set with two distinguished elements, the number of linear extensions in which the ranks of the distinguished elements differ by k is log-concave as a function of k. The log-concave sequences that can arise in this manner prove to exhibit a much richer structure, however, than is evident from log-concavity alone. The main result of this paper is a complete characterization of the extremals of the Kahn-Saks inequality: we obtain a detailed combinatorial understanding of where and what kind of geometric progressions can appear in these log-concave sequences. This settles a partial conjecture of Chan-Pak-Panova, while the analysis uncovers new extremals that were not previously conjectured. The proof relies on a much more general geometric mechanism—a hard Lefschetz theorem for nef classes that was obtained in the setting of convex polytopes by Shenfeld and Van Handel—which forms a model for the investigation of such structures in other combinatorial problems.

Lorentzian polynomials, Segre classes, and adjoint polynomials of convex polyhedral cones

We consider polynomials expressing the cohomology classes of subvarieties of products of projective spaces, and limits of positive real multiples of such polynomials. We study the relation between these covolume polynomials and Lorentzian polynomials. While these are distinct notions, we prove that, like Lorentzian polynomials, covolume polynomials have M-convex support and generalize the notion of log-concave sequences. In fact, we prove that covolume polynomials are ‘sectional log-concave’, that is, the coefficients of suitable restrictions of these polynomials form log-concave sequences.We observe that Chern classes of globally generated bundles give rise to covolume polynomials, and use this fact to prove that certain polynomials associated with Segre classes of subschemes of products of projective spaces are covolume polynomials. We conjecture that the same polynomials may be Lorentzian after a standard normalization operation.Finally, we obtain a combinatorial application of a particular case of our Segre class result. We prove that the adjoint polynomial of a convex polyhedral cone contained in the nonnegative orthant, and sharing a face with it, is a covolume polynomials. This implies that these adjoint polynomials are M-convex and sectional log-concave, and in fact dually Lorentzian, that is, Lorentzian after a certain change of variables.

Lagrangian geometry of matroids

We introduce the conormal fan of a matroid M \operatorname {M} , which is a Lagrangian analog of the Bergman fan of M \operatorname {M} . We use the conormal fan to give a Lagrangian interpretation of the Chern–Schwartz–MacPherson cycle of M \operatorname {M} . This allows us to express the h h -vector of the broken circuit complex of M \operatorname {M} in terms of the intersection theory of the conormal fan of M \operatorname {M} . We also develop general tools for tropical Hodge theory to prove that the conormal fan satisfies Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann relations. The Lagrangian interpretation of the Chern–Schwartz–MacPherson cycle of M \operatorname {M} , when combined with the Hodge–Riemann relations for the conormal fan of M \operatorname {M} , implies Brylawski’s and Dawson’s conjectures that the h h -vectors of the broken circuit complex and the independence complex of M \operatorname {M} are log-concave sequences.

Poisson processes and a log-concave Bernstein theorem

We discuss interplays between log-concave functions and log-concave sequences. We prove a Bernstein-type theorem, which characterizes the Laplace transform of log-concave measures on the half-line in terms of log-concavity of the alternating Taylor coefficients. We establish concavity inequalities for sequences inspired by the Pr\'ekopa-Leindler and the Walkup theorems. One of our main tools is a stochastic variational formula for the Poisson average.

Log-concavity of some independence polynomials via a partial ordering

Schwenk proved in 1981 that the edge independence polynomial of a graph is unimodal. It has been known since 1987 that the (vertex) independence polynomial of a graph need not be unimodal. Alavi et al. have asked whether the independence polynomial of a tree is unimodal. We apply some results on the log-concavity of combinations of log-concave sequences toward establishing the log-concavity (and, thus, the unimodality) of the independence numbers of some families of trees and related graphs.

Convolution preserves partial synchronicity of log-concave sequences

In a recent proof of the log-concavity of genus polynomials of some families of graphs, Gross et al. defined the weakly synchronicity relation between log-concave sequences, and conjectured that the convolution operation by any log-concave sequence preserves weakly synchronicity. We disprove it by providing a counterexample. Furthermore, we find the so-called partial synchronicity relation between log-concave sequences, which is (i) weaker than the synchronicity, (ii) stronger than the weakly synchronicity, and (iii) preserved by the convolution operation.

On Multiple and Infinite Log-Concavity

Following Boros-Moll, a sequence (an) is m-log-concave if \({\mathcal{L}^{j}(a_{n})\geqslant0}\) for all j = 0, 1, . . . , m. Here, \({\mathcal{L}}\) is the operator defined by \({\mathcal{L}(a_{n}) = a^{2}_{n}-a_{n-1}a_{n+1}}\). By a criterion of Craven-Csordas and McNamara-Sagan it is known that a sequence is ∞-log-concave if it satisfies the stronger inequality \({a^{2}_{k}\geqslant ra_{k-1}a_{k+1}}\) for large enough r. On the other hand, a recent result of Branden shows that ∞-log-concave sequences include sequences whose generating polynomial has only negative real roots. In this paper, we investigate sequences which are fixed by a power of the operator \({\mathcal{L}}\) and are therefore ∞-log-concave for a very different reason. Surprisingly, we find that sequences fixed by the non-linear operators \({\mathcal{L}}\) and \({\mathcal{L}^{2}}\) are, in fact, characterized by a linear 4-term recurrence. In a final conjectural part, we observe that positive sequences appear to become ∞-log-concave if convoluted with themselves a finite number of times.

Log-Concavity of Combinations of Sequences and Applications to Genus Distributions

We formulate conditions on a set of log-concave sequences, under which any linear combination of those sequences is log-concave, and further, of conditions under which linear combinations of log-concave sequences that have been transformed by convolution are log-concave. These conditions involve relations on sequences called synchronicity and ratio-dominance, and a characterization of some bivariate sequences as lexicographic. We are motivated by the 25-year-old conjecture that the genus distribution of every graph is log-concave. Although calculating genus distributions is NP-hard, they have been calculated explicitly for many graphs of tractable size, and the three conditions have been observed to occur in the partitioned genus distributions of all such graphs. They are used here to prove the log-concavity of the genus distributions of graphs constructed by iterative amalgamation of double-rooted graph fragments whose genus distributions adhere to these conditions, even though it is known that the genus polynomials of some such graphs have imaginary roots. A blend of topological and combinatorial arguments demonstrates that log-concavity is preserved through the iterations.

Mixed volumes, log-concave sequences and B-splines

In this paper, a series of problems emerged in discrete geometry and combinatorics related to spline functions are systematically studied. For example, in discrete geometry, the spline representations of cube slicing and mixed volumes of polytopes are considered. With the geometric interpretations of B-splines, the volume of cube slicing can be considered as an equivalent problems in spline theory. Based on the connection, a simple proof for Laplace and P olya's formulas in cub slicing is given by spline theory. A class of mixed volumes are given by the relations between splines and the mixed volumes. A class of log-concave sequence are derived by the connection. Therefore, the open problem proposed by Schmidt and Simion is partially solved. In combinatorics, the combinatorial polynomials and log-concavity for some combinatorial sequences are investigated by spline theory. With the well developed spline theory, not only the existing results in discrete geometry and combinatorics have been veri ed, but also a novel method for solving related discrete mathematics problems has been studied. Splines as functions of a continuous nature provide an analysis method in combinatorial enumerations which are usually considered as counting discrete objects. This method provides a novel analysis method for studying discrete objects.

Positivity of Toeplitz determinants formed by rising factorial series and properties of related polynomials

In this note, we prove the positivity of Maclaurin coefficients of polynomials written in terms of rising factorials and arbitrary log-concave sequences. These polynomials arise naturally in studying log-concavity of rising factorial series. Several conjectures concerning zeros and coefficients of a generalized form of those polynomials are advanced. Also polynomials whose generating functions are higher-order Toeplitz determinants formed by rising factorial series are considered. Three conjectures about these polynomials are made. All conjectures proposed are supported by numerical evidence. Bibliography: 16 titles.

Khinchine type inequalities with optimal constants via ultra log-concavity

We derive Khinchine type inequalities for even moments with optimal constants from the result of Walkup (J Appl Probab 13:76–85, 1976) which states that the class of log-concave sequences is closed under the binomial convolution.

Combinatorial proofs of inverse relations and log-concavity for Bessel numbers

Let the Bessel number of the second kind B ( n , k ) be the number of set partitions of [ n ] into k blocks of size one or two, and let the Bessel number of the first kind b ( n , k ) be the coefficient of x n − k in − y n − 1 ( − x ) , where y n ( x ) is the n th Bessel polynomial. In this paper, we show that Bessel numbers satisfy two properties of Stirling numbers: The two kinds of Bessel numbers are related by inverse formulas, and both Bessel numbers of the first kind and those of the second kind form log-concave sequences. By constructing sign-reversing involutions, we prove the inverse formulas. We review Krattenthaler’s injection for the log-concavity of Bessel numbers of the second kind, and give a new explicit injection for the log-concavity of signless Bessel numbers of the first kind.

Unbounded Regions of Infinitely Logconcave Sequences

We study the properties of a logconcavity operator on a symmetric, unimodal subset of finite sequences. In doing so we are able to prove that there is a large unbounded region in this subset that is $\infty$-logconcave. This problem was motivated by the conjecture of Boros and Moll that the binomial coefficients are $\infty$-logconcave.

Log Concave Sequences of Symmetric Functions and Analogs of the Jacobi-Trudi Determinants

We prove that various sequences of elementary and complete homogeneous symmetric functions are log concave or PF. As corollaries we show that certain sequences of $q$-binomial coefficients and $q$-Stirling numbers have these properties. The principal technique used is a combinatorial interpretation of determinants using lattice paths due to Gessel and Viennot [G-V 85].

Log concave sequences of symmetric functions and analogs of the Jacobi-Trudi determinants

We prove that various sequences of elementary and complete homogeneous symmetric functions are log concave or PF. As corollaries we show that certain sequences of q q -binomial coefficients and q q -Stirling numbers have these properties. The principal technique used is a combinatorial interpretation of determinants using lattice paths due to Gessel and Viennot [G-V 85].

On log concave sequences

It is shown how a log concave sequence generates a log super-modular function on the lattice being the direct product of two finite chains. The FKG inequality is applied to this function, and results in an inequality that is a 2-dimensional version of an inequality shown by Seymour and Welsh (1975).