| Schr?der numbers is one of the important combination sequences in combinato-rial mathematics.rn is the number of lattice paths in the plane from(0,0)to(2n,0)with steps(1,1),(2,0),and(1,-1),that never go below the horizontal axis.In this paper,we consider some counting problems on the weighted generalized paths.The paths that use the steps up,level,the first kind of down and the second kind of down with assigned weighted 1,a,b and c.We prove the number of lat-tice paths in the plane from(0,0)to(2n,0)with assigned weighted 1,2,1 and 1 is Schr?der numbers by symbolization method.We give a new combinatorial interpre-tation of Schr?der numbers.We study some of enumerations of peaks,horizontal steps and up steps in weighted generalized Schr?der paths by means of the Lagrange inversion formula.By changing the value of weight in weighted generalized Schr?der paths,we obtain the combinatorial interpretation of other combinatorial sequences.The numbers of planted trees with nodes of degree at most two and having all branches of odd length is Schr?der numbers,which called Schr?der trees.We give the generating functions of the Schr?der trees and some of enumerations of nodes,leaves and branches by symbolization method.The ordered trees with no nodes of degree one and with n+1 leaves are small Schr?der trees.We give new combina-torial explanation of the Schr?der numbers according to the characteristics of small Schr?der trees and the relationship between Schr?der numbers and small Schr?der numbers. |
