Fibonacci, Motzkin, Schroder, Fuss-Catalan and other Combinatorial Structures: Universal and Embedded Bijections

A combinatorial structure, $\mathcal{F}$, with counting sequence $\{a_n\}_{n\ge 0}$ and ordinary generating function $G_\mathcal{F}=\sum_{n\ge0} a_n x^n$, is positive algebraic if $G_\mathcal{F}$ satisfies a polynomial equation $G_\mathcal{F}=\sum_{k=0}^N p_k(x)\,G_\mathcal{F}^k$ and $p_k(x)$ is a polynomial in $x$ with non-negative integer coefficients. We show that every such family is associated with a normed $\mathbf{n}$-magma. An $\mathbf{n}$-magma with $\mathbf{n}=(n_1,\dots, n_k)$ is a pair $\mathcal{M}$ and $\mathcal{F}$ where $\mathcal{M}$ is a set of combinatorial structures and $\mathcal{F}$ is a tuple of $n_i$-ary maps $f_i\,:\,\mathcal{M}^{n_i}\to \mathcal{M}$. A norm is a super-additive size map $||\cdot||\,:\, \mathcal{M}\to \mathbb{N}$. If the normed $\mathbf{n}$-magma is free then we show there exists a recursive, norm preserving, universal bijection between all positive algebraic families $\mathcal{F}_i$ with the same counting sequence. A free $\mathbf{n}$-magma is defined using a universal mapping principle. We state a theorem which provides a combinatorial method of proving if a particular $\mathbf{n}$-magma is free. We illustrate this by defining several $\mathbf{n}$-magmas: eleven $(1,1)$-magmas (the Fibonacci families), seventeen $(1,2)$-magmas (nine Motzkin and eight Schr\"oder families) and seven $(3)$-magmas (the Fuss-Catalan families). We prove they are all free and hence obtain a universal bijection for each $\mathbf{n}$. We also show how the $\mathbf{n}$-magma structure manifests as an embedded bijection.