# How many combinations are there of a commutative operation acting over a collection of objects?

We have a collection of objects $$P$$ and an operator $$(\star)$$ such that for all of our objects: $$\rho_i \star \rho_j = \rho_j \star \rho_i = \rho_k$$

Where $$\{\rho_i, \rho_j\, \rho_k \} \in P$$, and $$\{i, j, k\}$$ may or may not be equal to each other.

Suppose we have 4 objects from $$P$$. How many different ways are there to operate on all of them with $$(\star)$$? I.E.

$$(\rho_1 \star \rho_2) \star (\rho_3 \star \rho_4)$$

$$\rho_1 \star (\rho_2 \star (\rho_3 \star \rho_4))$$

... Etc.

What is the generalisation of this problem to $$n$$ such objects?

These products are in bijection with the unordered full binary tree with $$n$$ labeled leaves, of which there are $$(2n-3)!!$$ (where $$(-1)!!=1$$). See double factorial and OEIS sequence A001147.
You can see this by noting that removing the $$n$$-th element from a product generates each of the products of the remaining $$n-1$$ elements $$2n-3$$ times, once for each node of the corresponding tree at which the $$n$$-th element could be inserted.