Log-Concave Properties Of Polynomials In Combinatorics

The main results of this thesis are some progress in log-concave properties of polynomials in combinatorics and related problems, including the general crite-rion for the limiting distribution of the coefficients of a class of the polynomials; the balanced property of the polynomials; the log-concavity and log-convexity of the polynomials; the interlacing log-concavity of a sequence of polynomials and the q-log-concavity and the q-log-convexity of a sequence of polynomials.In Chapter 1, we give a background on combinatorial properties of the poly-nomials, and introduce some definitions and notations that are used throughout the thesis. Meanwhile, we display some important classical theorems on the com-binatorial properties of the polynomials.In Chapter 2, we start with an estimate for an infinite series sum of Bernoulli numbers. Employing this estimate we show the limiting distribution of the q-Catalan numbers by calculating the exponential moment generating function. Moreover, using a general criterion concerning the limiting distribution of the coefficients of a quotient of products the result is extended to a different q-analog of the Catalan numbers, and also to generalized q-Catalan numbers. Despite the fact that these coefficients are not unimodal for small n, we conjecture that for sufficiently large n, the coefficients are unimodal and even log-concave except for a few terms of the head and tail. Furthermore, based on Bona's study for the bal-anced property of the number of of cycles of permutations, we show the q-Catalan numbers satisfy the balanced property. At last, using Janson's criterion condition we show the limiting distribution of the number of the alternating subsequences of length k contained in a random permutation of [n] is asymptotically normal.In Chapter 3, We first give refinement of the balancing 2-colorings of the n-cube, and obtain a relation between the number of balanced 2-colorings of the n-cube with 2k vertices and the number of balanced 2-colorings of the n-cube with 2k+2 vertices by a bijection. Applying this relation we prove a conjecture of Palmer, Read and Robinson [90] on the unimodality of the number of balancing 2-coloring of n-cube in the n-dimensional Euclidean space. Furthermore, we propose a conjecture the numbers of balancing 2-colorings of n-cube with 2k black vertices for fixedκpossess the log-concavity. And by probabilistic method we show that it holds when n is sufficiently large.In Chapter 4, we define the interlacing log-concavity which implies the log-concavity. Then employing the four recurrence formulae for the Boros-Moll poly-nomials we deduce some inequalities for the coefficients of the Boros-Moll polyno-mials. Using these inequalities, we show the Boros-Moll polynomials possess the interlacing log-concavity by induction. Furthermore we give a sufficient condition to guarantee that an array will satisfy the interlacing log-concavity. These results can be applied to several sequences which arise from combinatorics.In Chapter 5, we creatively get help from the symmetric function to study the log-behavior of the polynomials. Using Schur positivity and the principal specialization of Schur functions, we prove two recent conjectures of Liu and Wang on the q-log-convexity of the Narayana. polynomials. We begin with a formula, of Branden expressing the q-Narayana numbers as a specialization of Schur functions. By establishing several symmetric function identities, we obtain the strong q-log-convexity of the Narayana polynomials as well as the strong q-log-concavity of the q-Narayana numbers. It should be noted that the q-log-concavity of the q-Narayana numbers Nq(n,k) for fixedκis a special case of a conjecture of McNamara and Sagan on the infinite q-log-concavity of the Gaussian polynomials. Furthmore, We present a class of strongly q-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials are strongly q-log-convex. We also prove that the Bessel transformation preserves log-convexity.