Equidistribution of mesh patterns of short length on layered permutations

We study the distribution of mesh patterns of length 2 on layered permutations, extending recent work on the distribution of mesh patterns on arbitrary permutations. We classify all mesh patterns of length 2 into exactly 35 equidistribution classes and derive a formula for the distribution of each class. With one exception, the corresponding avoidance sequences appear in the Online Encyclopedia of Integer Sequences, revealing connections with the Fibonacci and Tribonacci numbers, certain integer compositions, and restricted words. We also refine several of our enumerative results by jointly tracking five classical permutation statistics: descents, left-to-right maxima and minima, and right-to-left maxima and minima.