The cycle construction $\mathrm{CYC}(\mathcal{A})$

Let $\mathcal{A}$ be a combinatorial class. For each $m\ge 1$, let $C_m=\mathbb{Z}\mathrm{/}m$ act on the set of $m$-tuples

${\mathcal{A}}^m=\{(a_0,\dots ,a_{m-1}):a_i\in \mathcal{A}\}$

by cyclic rotation:

$[r{]}_m\cdot (a_0,\dots ,a_{m-1}):=(a_r,a_{r+1},\dots ,a_{m-1},a_0,\dots ,a_{r-1})\mathrm{.}$

The cycle construction $\mathrm{CYC}(\mathcal{A})$ is the disjoint union, over all $m\ge 1$, of the orbit sets of these actions. An object of $\mathrm{CYC}(\mathcal{A})$ is therefore a cyclic arrangement of finitely many $\mathcal{A}$-objects, and its size is

$\mathrm{\mid }(a_0,\dots ,a_{m-1})\mathrm{\mid }:=\mathrm{\mid }{a}_0\mathrm{\mid }+\cdots +\mathrm{\mid }{a}_{m-1}\mathrm{\mid },$

which is constant on each orbit.

If $\mathcal{A}$ has a size-zero object, then arbitrarily long cycles may have the same total size. The generating-function theorem for $\mathrm{CYC}$ therefore assumes that $\mathcal{A}$ has no size-zero objects.