Here we consider the problem of all the possible orders of a linguistic
structure formed by $n$ elements, for instance, subject, direct object and verb
($n=3$) or subject, direct object, indirect object and verb ($n=4$). We
investigate if the frequency of the $n!$ possible orders is constrained by two
principles. First, entropy minimization, a principle that has been suggested to
shape natural communication systems at distinct levels of organization. Second,
swap distance minimization, namely a preference for word orders that require
fewer swaps of adjacent elements to be produced from a source order. Here we
present average swap distance, a novel score for research on swap distance
minimization, and investigate the theoretical distribution of that score for
any $n$: its minimum and maximum values and its expected value in die rolling
experiments or when the word order frequencies are shuffled. We investigate
whether entropy and average swap distance are significantly small in distinct
linguistic structures with $n=3$ or $n=4$ in agreement with the corresponding
minimization principles. We find strong evidence of entropy minimization and
swap distance minimization with respect to a die rolling experiment. The
evidence of these two forces with respect to a Polya urn process is strong for
$n=4$ but weaker for $n=3$. We still find evidence of swap distance
minimization when word order frequencies are shuffled, indicating that swap
distance minimization effects are beyond pressure to minimize word order
entropy.

考虑语言结构的所有可能排列顺序的问题，研究了熵最小化和交换距离最小化原则对于 $n=3$ 或 $n=4$ 的语言结构中的影响，并通过掷骰实验和词序频率随机排列验证了熵最小化和交换距离最小化的强有力证据。