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G-structures and Cartan geometries are two major approaches to thedescription of geometric structures (in the sense of differential geometry) onmanifolds of some fixed dimension $n$. We show that both descriptions naturallyextend to the setting of manifolds of dimension $\geq n$ which are endowed witha distinguished involutive distribution $F$ of rank $n$. The resulting``partial'' structures are most naturally interpreted as smooth families ofstandard G-structures or Cartan geometries on the leaves of the foliationdefined by $F$.  We prove that for the special class of AHS-structures (also known as$|1|$-graded parabolic geometries) the construction of a canonical Cartangeometry associated to a G-structure extends to this general setting. As anapplication, we prove that for partial AHS-structures there is an analog of themachinery of BGG sequences. This constructs sequences of differential operatorsof arbitrarily high order intrinsic to the structures. Under appropriateflatness conditions, these sequence are fine resolutions of sheaves whichlocally can be realized as pullbacks of sheaves on local leaf spaces for thefoliation defined by $F$.

Partial AHS-Structures, their Cartan description and partial BGG sequences