Block-Separated Overpartitions and Their Fibonacci-Type Structure
El-Mehdi Mehiri
TL;DR
Block-separated overpartitions introduce a mild restriction on overpartitions by forbidding two consecutive overlined distinct blocks. A two-state transfer-matrix automaton over part-sizes yields a matrix-product expression for the generating function and an Euler-type factorization with a stable, computable recurrence. The internal decoration patterns follow Fibonacci-type combinatorics, connecting to 11-avoiding binary words, independent sets on paths, and Fibonacci tilings, and leading to a convolution form F(q) = (q)^{-1}_∞ times a Fibonacci-weighted sum of elementary symmetric polynomials. These results illuminate a new bridge between partition-type generating functions and Fibonacci-type combinatorics, with promising directions for higher separation, q-difference equations, and congruence studies.
Abstract
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.