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Originally motivated by a conjecture of Chung et al. on the distribution ofcycles in certain restricted permutations, we use the theory of analyticcombinatorics in several variables (ACSV) to establish local central limittheorems for parameters in a family of combinatorial classes. Our results applyto multivariate generating functions whose denominators have the form$H(\mathbf{z},t) = 1-q(t)-\sum_{k=1}^dq_k(t)z_k$, where each of the $q_k$ is apolynomial and $q$ is a power series. Key to establishing these results inarbitrary dimension is an interesting symbolic determinant, which we compute byconjecturing and then proving an appropriate $LU$-factorization. In addition tothe motivating family of permutations, the method is applied to prove localcentral limit theorems for (integer) compositions and $n$-colour compositionswith varying restrictions and values tracked. This work also provides ablueprint to aid the reader in using the powerful tools of analyticcombinatorics in several variables to prove their own limit theorems.

A Family of Multivariate Limit Theorems for Combinatorial Parameters via ACSV