  Normally ordered forms of functions of boson operators are important in many
contexts in particular concerning Quantum Field Theory and Quantum Optics.
Beginning with the seminal work of Katriel [Lett. Nuovo Cimento,
10(13):565--567, 1974], in the last few years, normally ordered forms have been
shown to have a rich combinatorial structure, mainly in virtue of a link with
the theory of partitions. In this paper, we attempt to enrich this link. By
considering linear representations of noncrossing partitions, we define the
notion of noncrossing normal ordering. Given the growing interest in
noncrossing partitions, because of their many unexpected connections (like, for
example, with free probability), noncrossing normal ordering appears to be an
intriguing notion. We explicitly give the noncrossing normally ordered form of
the functions (a^{r}(a^{\dag})^{s})^{n}) and (a^{r}+(a^{\dag})^{s})^{n}, plus
various special cases. We are able to establish for the first time bijections
between noncrossing contractions of these functions, k-ary trees and sets of
lattice paths.
