  Incomplete rankings on a set of items $\{1,\; \ldots,\; n\}$ are orderings of
the form $a_{1}\prec\dots\prec a_{k}$, with $\{a_{1},\dots
a_{k}\}\subset\{1,\dots,n\}$ and $k < n$. Though they arise in many modern
applications, only a few methods have been introduced to manipulate them, most
of them consisting in representing any incomplete ranking by the set of all its
possible linear extensions on $\{1,\; \ldots,\; n\}$. It is the major purpose
of this paper to introduce a completely novel approach, which allows to treat
incomplete rankings directly, representing them as injective words over $\{1,\;
\ldots,\; n\}$. Unexpectedly, operations on incomplete rankings have very
simple equivalents in this setting and the topological structure of the complex
of injective words can be interpretated in a simple fashion from the
perspective of ranking. We exploit this connection here and use recent results
from algebraic topology to construct a multiresolution analysis and develop a
wavelet framework for incomplete rankings. Though purely combinatorial, this
construction relies on the same ideas underlying multiresolution analysis on a
Euclidean space, and permits to localize the information related to rankings on
each subset of items. It can be viewed as a crucial step toward nonlinear
approximation of distributions of incomplete rankings and paves the way for
many statistical applications, including preference data analysis and the
design of recommender systems.
