Necklaces are the equivalence classes of words under the action of the cyclic group. Let a transition in a word be any change between two adjacent letters modulo the word’s length. We present a closed-form solution for the enumeration of necklaces in n beads, k colours and t transitions. We show that our result provides a more general solution to the problem of counting alternating (proper) colourings of the vertices of a regular n-gon.

中文翻译：

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枚举带有过渡的项链

项链是词在循环群作用下的等价类。让单词中的转换为两个相邻字母之间的任何变化，以单词的长度为模。我们提出了一个封闭形式的项链枚举解决方案n珠子，ķ颜色和吨过渡。我们表明，我们的结果为计算规则顶点的交替（正确）着色问题提供了更通用的解决方案n-贡。