A statistical model for points expanding in higher dimensions while being tied to bijective involutions

Let $\mathcal{M}$ be a set with $M$ elements, let $\psi :\mathcal{M}\to\mathcal{M}$ be a bijective involution, and let~$\boldsymbol{\mathcal{X}}_{\psi}$ be the set of sequences $(x_1,\dots,x_M)\in\mathcal{M}^M$ with the property that $x_{M+1-j} = \psi(x_j)$ for $1\le j\le M$. This framework can be used to infer the possible distribution of sequences, such as the modular ones, that pose challenges for conventional methods. We prove that when $M$ is even, there exists a limit probability density function that weighs the parameter $k$ that counts the appearances of the elements of $\mathcal{M}$ among the terms of sequences $\textbf{x}\in\boldsymbol{\mathcal{X}}_{\psi}$. It turns out that the number of fixed points of $\psi$ influences the probability density function, which decomposes into two pieces, each multiplied by complementary factors, and the smaller of the two pieces appears only when $k$ is even. Applying the model, we find a threshold from which almost all sequences contain related terms with prescribed frequencies.