D-Dyck Paths:Enumeration And Q-Log-Concavity

Lattice paths are important research objects in enumerative combinatorics.In this paper,we study d-Dyck paths,which go from(0,0)to(n,m),with each steps in Z~2,consisting of(1,0)and(0,1),and going below x=dy(d∈Z*).In the first chapter,we review the concepts and research background of d-Dyck paths and q-log-concavity.In the second chapter,we give the enumeration results of d-Dyck paths by both com-binatorial and generating function methods.Moreover,we obtain a recurrence of the q-polynomial f_q(n,m,d)of d-Dyck paths ending at(n,m)and a closed formula,here q records the inversion number of respected permutations.In the third chapter,we prove the q-log-concavity of_{f_q(n,m,d)}with respect to both n and m by establishing in-jections between sets of d-Dyck paths.