A general method for proving the non-trivial linear homogeneous partition inequalities

摘要

An asymptotic classification for the linear homogeneous partition inequalities of the form n-ary sumation i=1rp(n+xi)<= n-ary sumation i=1sp(n+yi) has recently been introduced. In this paper, we investigate partition inequalities of this form when r=s. From an asymptotic point of view, such partition inequalities are considered to be non-trivial because they have the same number of terms on both sides. In this context, we provide a very general method for proving the non-trivial partition inequalities. This is a numerical method that does not involve q series and connects the non-trivial linear homogeneous partition inequalities with the Prouhet-Tarry-Escott problem: if {x1,x2, horizontal ellipsis ,xr}=k{y1,y2, horizontal ellipsis ,yr}(k > 0), then for n large enough the expression n-ary sumation i=1r(p(n+xi)-p(n+yi)) has the same sign as n-ary sumation i=1r(xik+1-yik+1). The method can be adopted to other inequalities of a similar nature which involve sequences that are asymptotically completely monotone.