Cluster algebraic interpretation of generalized Markov numbers and their matrixizations

Markov numbers, i.e. positive integers appearing in solutions to $x^2 + y^2 + z^2 = 3xyz$, can be viewed as specializations of cluster variables. The second author and Matsushita gave a generalization of the Markov equation, $x^2 + y^2 + z^2 + k_1yz + k_2xz + k_3xy = (3+k_1+k_2+k_3)xyz$, whose solutions can be viewed as specializations of cluster variables in generalized cluster algebras. We give two families of matrices in $SL(2,\mathbb{Z}[x_1^\pm,x_2^\pm,x_3^\pm])$ associated to these cluster structures. These matrix formulas relate to previous matrices appearing in the context of Markov numbers, including Cohn matrices and generalized Cohn matrices given by the second author, Maruyama, and Sato, as well as matrices appearing in the context of cluster algebras, including matrix formulas given by Kanatarc{\i} O\u{g}uz and Y{\i}ld{\i}r{\i}m. We provide a classification of the two families of matrices and exhibit an explicit family of each. The latter is done by realizing cluster variables in generalized Markov cluster algebras as weight-generating functions of order ideals in certain fence posets which are related to Christoffel words. An interesting observation is that these functions resemble Caldero-Chapoton functions for string modules, and a byproduct of our proofs is a new skein-like formula for such functions.