objects of size and , there are ways to do the relabelling. Therefore, the total number of objects of size is This binomial convolution relation for the terms is equivalent to multiplying the EGFs, Sequence The sequence construction is defined similarly to the unlabelled case: and again, as above, Set In labelled structures, a set of elements corresponds to exactly sequences. This is different from the unlabelled case, where some of the permutations may coincide. Thus for , we have Cycle Cycles are also easier than in the unlabelled case. A cycle of length corresponds to distinct sequences. Thus for , we have Boxed product In labelled structures, the min-boxed product is a variation of the original product which requires the element of in the product with the minimal label. Similarly, we can also define a max-boxed product, denoted by , by the same manner. Then we have, or equivalently, Example An increasing Cayley tree is a labelled non-plane and rooted tree whose labels along any branch stemming from the root form an increasing sequence. Then, let be the class of such trees. The recursive specification is now Other elementary constructions The operators and represent cycles of even and odd length, and sets of even and odd cardinality. Example Stirling numbers of the second kind may be derived and analyzed using the structural decomposition The decomposition is used to study unsigned Stirling numbers of the first kind, and in the derivation of the statistics of random permutations. A detailed examination of the exponential generating functions associated to Stirling numbers within symbolic combinatorics may be found on the page on Stirling numbers and exponential generating functions in symbolic combinatorics. See also Combinatorial species References François Bergeron, Gilbert Labelle, Pierre Leroux, Théorie des espèces et combinatoire des structures arborescentes, LaCIM, Montréal (1994). English version: Combinatorial Species and Tree-like Structures, Cambridge University Press (1998). Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge University Press (2009). (available online: http://algo.inria.fr/flajolet/Publications/book.pdf) Micha Hofri, Analysis of Algorithms: Computational Methods and Mathematical Tools, Oxford University Press (1995). Combinatorics