Polynomials With Symmetric And Unimodal Coeiffcients

The unimodality of sequences is one of the primary branches of combinatorics. Even though the unimodality has "one-line" definition, as noticed by Stanley, it may be quite difficult to show that a sequence is unimodal, and there are an abundance of sequences conjectured to be unimodal but for which no idea of a proof is known. However, the unimodality of symmetric sequences is much better behaved and is expected to be systematically studied. Many sequences of combinatorial interest are known to be unimodal and symmetric. For example, binomial coef-ficients, Eulerian numbers, Narayana numbers as well as Gaussian binomial coefficients. In the monograph "generatingfunctionology", Wilf pointed out that generating functions are a bridge between discrete mathematics and continuous analysis. We may study properties of discrete ob-jects through the relatively mature theory and research methods of continuous mathematics. On the other hand, polynomials with symmetric and unimodal coefficients often appear in algebra, analysis and combinatorics. The main object of this work is to systematically study symmet-ric unimodal polynomials from three different viewpoints. The organization of the paper is as follows.(1) The first part is devoted to study the unimodality of symmetric polynomials from a viewpoint of linear algebra. We verify that all symmetric polynomials with symmetric center n/2constitute a linear space of dimension [n/2]+1, and give transition matrices between basis (?)={qj(1+q+…+qn-2i)},(?)=[qj(1+q)n-2j} and the standard basis S={qj(1+qn-2j)}, j=0,1,...,[n/2]. We present some characterizations and sufficient conditions for symmetric polynomials that can be expressed in term of the first two bases with nonnegative coefficients. Furthermore some known results obtained in a unified manner.(2) The second part establishes the link between symmetric unimodal polynomials (spe-cially,(?)-positive polynomials, symmetric polynomials are expressed under basis (?) with non-negative coefficients) and rank-generating functions of posets. Take classic Eulerian polynomi-als, derangement polynomials and Narayana polynomials as examples, we construct appropriate ranked posets whose rank-generating functions is the same as or related to these polynomials. Further we decompose these posets into symmetric Boolean sublattices and explain that these classic polynomials are (?)-positive polynomials. (3) The third part researches (?)-positive polynomials by the method of group action on sets. By defining group actions on the set of derangements and the set of noncrossing partitions re-spectively, new proofs that derangement polynomials and Narayana polynomials are (?)-positive polynomials are presented. Among other things, we give natural and intuitive combinatorial interpretations of the coefficients under basis (?) of these two types of polynomials.(4) The Catalan-like numbers unify many well-known counting coefficients, including the Catalan numbers, the Motzkin numbers, the Bell numbers, the middle binomial coefficients, the (restricted) hexagonal numbers, the Schroder numbers, and so on. The last part gives a combinatorial proof of the log-convexity of Catalan-like numbers from the view of weighted Motzkin paths.