Unimodality Problems In Combinatorial Triangles

Unimodality problem is one of the primary topics in combinatorics, including the study of unimodality, log-concavity, log-convexity, strong q-log-concavity and Polya frequency property. Many triangular arrays arise in combinatorics, such as Pascal triangle. The thesis is devoted to investigate unimodality problems in some conventional combinatorial triangles. The main frame of the thesis is as follows.The first part investigates the unimodality problems in Pascal triangle. Belbachir et al, Tanny et al, and Benoumhani independently studied the unimodality problems of the sequences of binomial coefficients locating on some transversals of Pascal triangle. We examine various unimodality properties of the binomial sequences locating on all kinds of transversals. For instance, it is shown in this part that a finite sequence on a transversal is log-concave, an infinite sequence on a transversal not parallel to the boundaries of Pascal triangle is asymptotically log-convex, while an infinite sequence on a vertical transversal changes from log-concavity to log-convexity. The results unify many known results related to the unimodality problem of binomial sequences. In particular, we answer a question of Belbachir et al on the unimodality problem in Pascal triangle. Our work has attracted other researchers'attention.The second part studies the unimodality problems in other triangles, such as triangles composed of multinomial coefficients, symmetric functions, and the generalized binomial coefficients. The first subsection shows that the triangle of multinomial coefficients not only shares common unimodality properties with Pascal triangle, but also owns peculiar unimodality properties. For instance, a finite sequence locating on a transversal of the triangle of multinomial coefficients is not log-concave in general. The second subsection gives an answer to a question of Sagan on symmetric functions, and applies the result to the unimodality problems of q-binomial coefficients and q-Stirling numbers of two kinds. Finally, we prove a unimodality result on the generalized binomial coefficients.The third part presents a necessary condition of a finite PF sequence. Newton's in-equality illustrates that log-concavity is necessary for a nonnegative finite sequence to be PF. We prove that the concavity and convexity of a finite PF sequence change at most twice. Then the sequence can be divided into at most three segments, where the middle segment is concave and the other segments are convex. This result can be viewed as a new necessary condition of a finite PF sequence. This part also proves that the triangle of the generalized binomial coefficients preserves strong q-log-concavity.