Distribution On Descents Of Tableaux Of Two Rows

Young tableaux are important research structure in enumerative combinatorics.They are closely related to representation theory,algebraic geometry and other branches of mathematics.Enumeration on tableaux is also a hot research topic of combinatorics during recent years.In this paper we mainly study the distribution and enumeration on the descents of tableaux of shape(n-a,a)with k repeating numbers.Here tableaux of shape(n-a,a)include increasing tableaux,row-increasing tableaux and standard Young tableaux.At the beginning,we prove that the major index of rowincreasing tableaux have the same distribution by constructing the bijection between row-increasing tableaux.Then we enumerate the distribution and give another combinatorial proof of two row increasing tableaux.Finally,we obtain major index polynomials of Motzkin path and Riordan path.