A maximization problem related to Huffman codes

Let L = (l₀, l₁, ..., l_n-1) be a finite sequence of positive integers with Kraft sum (Σ_i 2^-li) equal to 1. We call such a sequence a complete length sequence. Let P_n(L) be the set of all probability mass functions (PMFs) that have a Huffman code with codeword lengths L. For a PMF P, let L(P) denote the minimum expected length of a binary prefix-free code for P. In this paper, we show that the set {L(P) : P E P_n(L)} has a maximum value. A P E P_n(L) at which this maximum is attained is explicitly determined for a class of complete length sequences.