Monotonicity of the rank functions for concave compositions

Abstract

A (strongly) concave composition of an integer $n$ is a sequence of positive integers that is (strictly) decreasing to a point and then (strictly) increasing thereafter, such that the sum of the entries equals $n$. The value at the low point is called the center part. The difference between the number of entries before and after the low point of the sequence is referred to as the rank of the (strongly) concave composition. The rank functions $V_d(m,n)$ and $V(m,n)$ are defined as the number of concave compositions and strongly concave compositions, respectively, of $n$ with rank $m$. By constructing the difference systems that characterize the rank generating functions, we establish monotonicity properties for the rank functions of both strongly concave compositions and concave compositions for all positive integers $n$. Moreover, we also study the monotonicity properties for the rank functions of (strongly) concave compositions with fixed center parts.