On the proportion of prefix codes in the set of three-element codes

Abstract: Let $L$ be a finite sequence of natural numbers. In Woryna (2017,2018), we derived some interesting properties for the ratio $\rho_{n,L}=|PR_n(L)|/|UD_n(L)|$, where $UD_n(L)$ denotes the set of all codes over an $n$-letter alphabet and with length distribution $L$, and $PR_n(L)\subseteq UD_n(L)$ is the corresponding subset of prefix codes. In the present paper, we study the case when the length distributions are three-element sequences. We show in this case that the ratio $\rho_{n,L}$ is always greater than $\alpha_n$, where $\alpha_n=(n-2)/n$ for $n>2$ and $\alpha_2=1/6$. Moreover, the number $\alpha_n$ is the best possible lower bound for this ratio, as the length distributions of the form $L=(1,1,c)$ and $L=(1,2,c)$ assure that the ratios asymptotically approach $\alpha_n$. Namely, if $L=(1,1,c)$, then $\rho_{n,L}$ tends to $(n-2)/n$ with $c\to\infty$, and, if $L=(1,2,c)$, then $\rho_{2,L}$ tends to $1/6$ with $c\to\infty$.