| dc.description.abstract | The thesis is devoted to the study of certain combinatorial
objects called \emph{non-crossing partitions}. The enumeration
properties of the lattice ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ of
\emph{non-crossing partitions} were studied since
the work of G. Kreweras in 1972. An important
feature of ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$, observed by P.
Biane in 1997, is that it embeds into the symmetric group
$\mathfrak{S}_n$; via this embedding,
${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ is canonically identified to
the interval $[\varepsilon, \gamma_o] \subseteq \mathfrak{S}_n$
(considered with respect to a natural partial order on
$\mathfrak{S}_n$), where $\varepsilon$ is the unit
of $\mathfrak{S}_n$ and $\gamma_o$ is the forward cycle.\\
There are two extensions of the concept of non-crossing partitions
that were considered in the recent research literature. On the one
hand, V. Reiner introduced in 1997 the analogue of \emph{type B} for
${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$. This poset is denoted
\textsf{NC$^{\textsf{\,B}}$(n)} and it is isomorphic to the interval
$[\varepsilon, \gamma_o]$ of the hyperoctahedral group $B_n$, where
now $\gamma_o$ stands for the natural forward cycle of $B_n$. On the
other hand, J. Mingo and A. Nica studied in 2004 a set of
\emph{annular} non-crossing partitions (diagrams drawn inside an
annulus -- unlike the partitions from
${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ or from
${\textsf{NC$^{\textsf{\,B}}$(n)}}\,$, which are drawn inside a
disc).\\
In this thesis the type B and annular objects are considered in a
unified framework. The forward cycle of $B_n$ is replaced by a
permutation which has two cycles, $\gamma=
[1,2,\ldots,p][p+1,\ldots,p+q]$, where $p+q = n$. Two equivalent
characterizations of the interval $[ \varepsilon , \gamma ]
\subseteq B_n$ are found -- one of them is in terms of a \emph{genus
inequality}, while the other is in terms of \emph{annular crossing
patterns}. A corresponding poset
\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}} of \emph{annular
non-crossing partitions of type B} is introduced, and it is proved
that $[\varepsilon, \gamma] \simeq
\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$, where the partial
order on $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$ is the
usual reversed refinement order for partitions.\\
The posets $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$ are not
lattices in general, but a remarkable exception is found to occur in
the case when $q=1$. Moreover, it is shown that the meet operation
in the lattice $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(n-1, 1)}\,}}$
is the usual
``intersection meet'' for partitions. Some results concerning
the enumeration properties of this lattice are obtained, specifically concerning its rank
generating function and its M\"{o}bius function.\\
The results described above in type B are found to also hold in
connection to the Weyl groups of
\emph{type D}. The poset
\mbox{{\textsf{NC$^{\textsf{\,D}}$\,(n-1, 1)}\,\,}} turns out to be
equal to the poset {\textsf{NC$^{\textsf{\,D}}$(n)}} constructed by
C. Athanasiadis and V. Reiner in a paper in 2004. The non-crossing
partitions of type D of Athanasiadis and Reiner are
thus identified as annular objects.\\
Non-crossing partitions of type A are central objects in the
combinatorics of free probability. A parallel concept of \emph{free
independence of type B}, based on non-crossing partitions of type B,
was proposed by P. Biane, F. Goodman and A. Nica in a paper in
2003. This thesis introduces a concept of \emph{scarce
$\mathbb{G}$-valued probability spaces}, where $\mathbb{G}$ is the
algebra of Gra{\ss}man numbers, and recognizes free independence of
type B as free independence in the ``scarce $\mathbb{G}$-valued''
sense. | en |
