CHAPTER I
P R E L I M I N A R I E S O N NON-CROSSIN G PARTITION S
In this first chapter, we want to collect all relevant combinatorial facts about
the lattice of non-crossing partitions. In particular, we will examine the notion of
(scalar-valued) multiplicative functions on this lattice, a notion which will be gen-
eralized in the next chapter to the crucial concept of operator-valued multiplicative
functions. The material presented here is essentially taken from [Spe4] and deeply
inspired by Rota's [Rotl,Rot2] combinatorial point of view on classical probability
theory.
1.1. Non-crossing partitions
We start by recalling the basic definitions and facts about non-crossing partitions.
1.1.1. DEFINITION. Let 5 be a linearly ordered set. Then ir = {Vi,... , Vp} is a
partition of 5, if the Vi ^ 0 are disjoint sets, whose union is S. The partition TT is
called non-crossing if for all z, j = 1,.. . ,p with Vi = {^i,..., vn} (vi vn)
and Vj = {wi,..., iu
m
} (wi wm) we have
wk vi Wk+i OWkvn Wk+i (k = 1 , . . . , m - 1).
We can reformulate the definition of 'non-crossing' in a recursive way: The partition
7r = {Vi,... , Vp} is non-crossing if at least one Vi is an interval in S (i.e. it contains
exactly all points of S lying between two points) and 7r\{Vj} is a non-crossing
partition of S\Vi.
We will denote the set of all non-crossing partitions of 5 by NC(S).
1.1.2. EXAMPLE. Non-crossing partitions of {1,2,3,4,5} are
{{1,3, 5}, {2}, {4}} and {{1,5}, {2,4}, {3}},
crossing ones are
{{1,3}, {2,4,5}} and {{1,4}, {2},{3,5}}.
1.1.3. NOTATIONS. 1) The Vi are called the blocks of a given partition TT =
{Vi,...,V^}. If we write a block in the form Vi (vi, •.. ,vm) then this will
always imply that vm. A ir G NC(S) determines an equivalence relation
on S in a canonical way and we will write v ^ w, if v, w E S belong to the same
block of 7r. By |7r| we will denote the number of blocks of 7r, and in the same way
by |V| the number of elements of the set V. For si, S2 S with si 52, we call
[si, s2] := {s G S I si s s2} and ]si, s2[:= {s G S \ Si s s2}
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