 Compressed coefficient vector

A compressed coefficient vector is a finite coefficient sequence with no zero entries and no equal consecutive entries. Its compression is obtained by first deleting zeros and then replacing each consecutive run of equal values by one value. For positive vectors, the zero-deletion step is vacuous.

For example, $(0,1,1,0,2,2,1)$ compresses to $(1,2,1)$, and $(1,0,1)$ compresses to $(1)$. Repeated values separated by a genuinely different value remain: $(1,2,1)$ is already compressed. Thus compression is a run operation, not replacement by the set of distinct entries.

The notion records coefficient patterns in Milliken-Taylor systems and their separation under finite colorings. It does not assert that replacing a vector by its compression leaves the exact generated set unchanged for a fixed sequence: $(1,1)$ requires two nonempty ordered index blocks, whereas $(1)$ permits a single block with one index. The canonical pattern and the precise block constraints are different data.^[ref]