Some Positivity Problems In Combinatorics

The work presented in this thesis is concerned with positivity problems arising in com-binatorics. Many combinatorial problems can be expressed in terms of the notion of positivity, i.e., proving that certain real numbers are nonnegative. The combinatorial significance of positivity is that a nonnegative real number have both a combinatorial interpretation and an algebraic interpretation. Besides,"Notions of Positivity and the Geometry of Polynomials" is a topic of series Trends in Mathematics, which devoted to publish volumes arising from conferences and lecture series focusing on a particu-lar topic from any area of mathematics. We mainly studied some positivity problems related to the theory of symmetric functions and the theory of unimodality.The first part of the thesis is to answer a positivity question on Schur functions. Schur functions form a very important basis of the ring of symmetric functions. In the study of two integer sequences related to Catalan numbers, Michel Lassalle introduced a specialization of symmetric functions, and conjectured that any Schur function is pos-itive with respect to this specialization. Based on the relationship between symmetric functions and totally positivity sequences, Lassalle’s conjecture is transformed into a problem to determining the total positivity of certain sequence. Then by a connection between totally positive sequences and multiplier sequences, we find that the desired total positivity can be derived from a classical result due to Laguerre, and thereby con-firm Lassalle’s positivity conjecture on Schur functions.In the second part of this thesis, we present a criterion for determining a sequence of self-reciprocal polynomials to be q-log-convex. This is motivated by the q-log-convexity conjecture for a class of polynomials, which were introduced by Zhi-Wei Sun in the study of the Ramanujan-Sato type series for powers of n. Li Liu and Y-i Wang gave a sufficient condition to determine the q-log-convexity of polynomials. Our q-log-convexity criterion is based on their result, and is easier to use by consider-ing the symmetry of the coefficients of self-reciprocal polynomials. With this criterion, Sun’s conjecture is transformed to a problem of determining the sign changes of certain polynomial of degree6over some intervals. By a careful analysis of the properties of the successive derivatives of this polynomial, we obtain their sign changes, and hereby confirm Sun’s q-log-convexity conjecture.The third part is a continuation of the second part. In this part, we study the q-log-convexity of Domb’s polynomials and the log-convexity of Domb’s sequence. It is clear that the former implies the latter. The q-log-convexity of Domb’s polynomials was proposed as another conjecture by Sun in his study of the series expansion for powers of π. By using our q-log-convexity criterion for self-reciprocal polynomials, this conjecture is transformed to a problem of determining the sign changes of certain polynomial of degree8over some intervals. This is achieved in a similar manner as in the second part, and, together with the q-log-convexity of Narayana polynomials of type B, leads to the q-log-convexity of Domb’s polynomials.