I have a set of vertices $V$ such each vertex $v\in V$ is labelled with a color $col(v)$. I want to consider the class of partitions of $V$ that have the following property in common: all vertices having the same color must belong to the same part.
Partition the vertices of $V$ according to their colors. $V(col)$ is a subset of vertices of $V$ whose color is $col$.
Partition the vertices of $V$ according to the first letter of their color. $V(r)$ is the set of vertices of $V$ whose color starts with letter (r), for example (red).
Partition the vertices of $V$ according to the number of the letters of their color. $V(4)$ is the set of vertices of $V$ whose colors have 4 letters (blue, ....).
I need a (concise) notation to describe the set of all these partitions. The notation have to give me access to the parameter of the partition, for example I need to be able to speak about $V(x)$ where $x$ may be a color, a first letter in a color, .... Thank you.