  In filtering, each output is produced by a certain number of different
inputs. We explore the statistics of this degeneracy in an explicitly treatable
filtering problem in which filtering performs the maximal compression of
relevant information contained in inputs (arrays of zeroes and ones). This
problem serves as a reference model for the statistics of filtering and related
sampling problems. The filter patterns in this problem conveniently allow a
microscopic, combinatorial consideration. This allows us to find the statistics
of outputs, namely the exact distribution of output degeneracies, for arbitrary
input sizes. We observe that the resulting degeneracy distribution of outputs
decays as $e^{-c\log^\alpha \!d}$ with degeneracy $d$, where $c$ is a constant
and exponent $\alpha>1$, i.e. faster than a power law. Importantly, its form
essentially depends on the size of the input data set, appearing to be closer
to a power-law dependence for small data set sizes than for large ones. We
demonstrate that for sufficiently small input data set sizes typical for
empirical studies, this distribution could be easily perceived as a power law.
We extend our results to filter patterns of various sizes and demonstrate that
the shortest filter pattern provides the maximum informative representations of
the inputs.
