We introduce and study a new restricted family of overpartitions, called block-separated overpartitions, in which no two consecutive distinct part-size blocks may both be overlined. Using a two-state transfer-matrix automaton, we derive a closed matrix-product expression for the ordinary generating function, establish an Euler-type factorization, and obtain an explicit normalized recurrence suitable for computation of arbitrary coefficients. We further prove that the possible overlining patterns on the distinct blocks are counted by Fibonacci numbers, giving natural bijections with independent sets on paths, pattern-avoiding binary words, and Fibonacci tilings.

翻译：本文引入并研究了一类新的受限超分割族，称为块分离超分割，其中任意两个连续的不同尺寸块不能同时被上划线标记。通过使用双状态转移矩阵自动机，我们推导了普通生成函数的闭式矩阵乘积表达式，建立了欧拉型因式分解，并得到了适用于任意系数计算的显式归一化递推关系。进一步证明，不同块上可能的划线模式数量由斐波那契数列计数，从而自然地建立了与路径上独立集、模式规避二进制词以及斐波那契铺砌之间的双射关系。