Suppose an even number of red dots and an even number of blue dots are placed arbitrarily onto the circumference of a circle. The placement may be described by the placement vector
$P = (i_1, j_1, i_2, j_2, \ldots, i_k, j_k)$
which reflects $i_1$ red dots, followed by $j_1$ blue dots, followed by $i_2$ red dots... so on and so forth, where $\exists m, n \in \mathbb{Z}_{\geq 0}$ such that $\sum i_k = 2m$ and $\sum j_k = 2n$.
The blue dots all need to be paired with blue chords; the red dots all need to be paired with red chords. Blue chords cannot intersect any chords, but red chords are allowed to intersect other red chords.
For instance, here is a legal pairing configuration for $P = (2,2,2,2)$.
How many ways are there to accomplish this for a arbitrary placement vector P? A closed form would be ideal, but failing that, can this quantity be sharply bounded from above?