In the second chapter, we give some combinatorial proof of log-convexity and log-concavity of some combinatorial numbers by lattice paths, Dyck paths and the Reflection Principle. In Section 2.1, we provide the combinatorial proofs of some combinatorial numbers, such as the large Schroder numbers, the central Delannoy numbers and the Fine numbers. In Section 2.2, we prove the log-concavity of the Delannoy numbers and present the generalization of the conjecture of Simion.
Weighted lattice paths are enumerated by counting function which is a natural extension of Gaussian multinomial coefficient in the case of unrestricted paths.
Recently, Mikhalkin has found a way to reinterpret the numbers N(d, g) in terms of tropical geometry and to compute them by counting certain lattice paths in integral polytopes.
We relate these two results by defining an analogue of the relative Gromov-Witten invariants and rederiving the Caporaso-Harris formula in terms of both tropical geometry and lattice paths.
Inhomogeneous Lattice Paths, Generalized Kostka Polynomials and An-1 Supernomials
Inhomogeneous lattice paths are introduced as ordered sequences of rectangular Young tableaux thereby generalizing recent work on the Kostka polynomials by Nakayashiki and Yamada and by Lascoux, Leclerc and Thibon.
An independent method for paper  is presented. Weighted lattice paths are enumerated by counting function which is a natural extension of Gaussian multno-mial coefficient in the case of unrestricted paths. Convolutions for path counts are investigated, which yields some Vandermonde-type identities for multinomial and q-multinomial coefficients.
The paper is concerned with the partially ordered set consisting of lattice-points onthe plane N￣2.The formula of incidence function is presented.Based on the symmetry of theorder relation,some Vandermonde-type identities are established.