Asymptotically trivial linear homogeneous partition inequalities

Abstract

We consider linear homogeneous partition inequalities of the form(⁎)∑k=1rakp(n+μk)≤∑ℓ=1sbℓp(n+νℓ), where p(n) is the number of integer partitions of n, {a1,a2,⋯,ar}, {b1,b2,⋯,bs} are positive integers, and 0≤μ1<μ2<⋯<μr, 0≤ν1<ν2<⋯<νs are integers. From the fact that limn→∞⁡p(n+μ)p(n)=1 (μ an integer) it follows that the inequality (⁎) can only hold if ∑k=1rak≤∑ℓ=1sbℓ. If the last relation is a strict inequality than (⁎) holds for all n>N, for an appropriately specified N, and can be established for all n≥1 by verifying that it holds for the finite set of cases specified by 1≤n≤N. Such inequalities will be referred to as asymptotically trivial. Several examples of such inequalities are presented. The inequality (⁎) is trivial if the stronger condition ∑k=1rakp(μk−min⁡(μ1,ν1)+1)≤∑ℓ=1sbℓ holds, i.e., the supremum of the left-hand side of (⁎) is smaller than or equal to the infimum of its right-hand side. If ∑k=1rak=∑ℓ=1sbℓ then we say that (⁎) is non-trivial. In this case (⁎) can be an identity. A “conventional” proof, establishing the nature of (⁎) for all n, is required.