Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

If $p (- TTn$ consists of $ci$ blocks of size $i$ for $i (- n$ , then $p$ is said to be of block-type $c = (c1,...,cn)$ . From the definition it is obvious that $sum ni=1 ici = n$ . Furthermore, it is clear that $p$ is a partition of size $sum ni=1ci$ . The set of mosaics of block-type $c$ will be indicated as $TTc$ . Since the action of $G$ on $TTn$ can be restricted to an action of $G$ on $TTc$ , we want to determine the number of $G$ -isomorphism classes of mosaics of type $c$ . For doing that, let $c$ be a particular partition of type $c$ . (For instance, $c$ can be defined such that the blocks of $c$ of size 1 are given by ${1}$ , ${2}$ , $...$ , ${c1}$ , the blocks of $c$ of size 2 are given by ${c1 + 1,c1 + 2}$ , ${c1 + 3,c1 + 4}$ , $...$ , ${c1 + 2c2- 1,c1 + 2c2}$ , and so on.) According to Kerber (1991, 1999), the stabilizer $Hc$ of $c$ in the symmetric group $Sn-$ is isomorphic to the direct product

n × Si-|Sci i=1

of wreath products of symmetric groups. In other words, $Hc$ is the set of all permutations $s (- Sn$ , which map each block of the partition $c$ again onto a block (of the same size) of the partition.

Hence, the $G$ -isomorphism classes of mosaics of type $c$ can be described as $Hc × G$ -orbits of bijections from $Zn$ to $n-$ under the following group action:

Zn Zn - 1 (Hc × G)× nbij-- > nbij, (s,g)*f := s o f o g .

When interpreting the bijections from $Zn$ to $n-$ as permutations of the $n$ -set $n-$ , then $G$ -mosaics of type $c$ correspond to double cosets (cf. Kerber, 1991, 1999) of the form

Hc\Sn/G.

Theorem 6. The number $Mc$ of $G$ -isomorphism classes of mosaics of type $c$ is given by

1 sum prod n Mc = ------- ai(g)!iai(s), |Hc ||G |(s,zg)(g (- )=Hzc(×s)Gi=1

where $z(g)$ and $z(s)$ are the cycle types of $g$ and of $s$ respectively, given in the form $(ai(g))i (- n$ or $(ai(s))i (- n$ . In other words, we are summing over pairs $(s,g)$ such that $g$ and $s$ determine permutations of the same cycle type.

In conclusion, in this section we demonstrated how to classify the isomorphism classes of mosaics. We applied methods from step 1 or step 2 of the general scheme of classification of discrete structures.

3 Enumeration of Non-Isomorphic Canons

The present concept of a canon is described in Mazzola (2002) and was presented by Guerino Mazzola to the author in the following way: A canon is a subset $K (_ Zn$ together with a covering of $K$ by pairwise different subsets $Vi /= Ø$ for $1 < i < t$ , the voices, where $t > 1$ is the number of voices of $K$ , in other words,

t K = U Vi, i=1