Some numerical results
Table 2 gives the numbers of D12-mosaics of type
λ, (in this table the type
(λ1,...,λn) of a
partition is written in the form
(1λ1,2λ2,...),
where all terms with λi=0 are omitted)
which were computed by using SYMMETRICA routines for the
Redfield-cap operator.
Number of D12-mosaics in
twelve tone music of type λ.
| λ |
|
λ |
|
λ |
|
λ |
|
| (12) |
1 |
(1,11) |
1 |
(2,10) |
6 |
(12,10) |
6 |
| (3,9) |
12 |
(1,2,9) |
30 |
(13,9) |
12 |
(4,8) |
29 |
| (1,3,8) |
85 |
(22,8) |
84 |
(12,2,8) |
140 |
(14,8) |
29 |
| (5,7) |
38 |
(1,4,7) |
170 |
(2,3,7) |
340 |
(12,3,7) |
340 |
| (1,22,7) |
510 |
(13,2,7) |
340 |
(15,7) |
38 |
(62) |
35 |
| (1,5,6) |
236 |
(2,4,6) |
610 |
(12,4,6) |
610 |
(32,6) |
424 |
| (1,2,3,6) |
2320 |
(13,3,6) |
781 |
(23,6) |
645 |
(12,22,6) |
1820 |
| (14,2,6) |
610 |
(16,6) |
50 |
(2,52) |
386 |
(12,52) |
386 |
| (3,4,5) |
1170 |
(1,2,4,5) |
3480 |
(13,4,5) |
1170 |
(1,32,5) |
2330 |
| (22,3,5) |
3510 |
(12,2,3,5) |
6960 |
(14,3,5) |
1170 |
(1,23,5) |
3500 |
| (13,22,5) |
3510 |
(15,2,5) |
708 |
(17,5) |
38 |
(43) |
297 |
| (1,3,42) |
2915 |
(22,42) |
2347 |
(12,2,42) |
4470 |
(14,42) |
792 |
| (2,32,4) |
5890 |
(12,32,4) |
5890 |
(1,22,3,4) |
17370 |
(13,2,3,4) |
11580 |
| (15,3,4) |
1170 |
(24,4) |
2325 |
(12,23,4) |
8860 |
(14,22,4) |
4463 |
| (16,2,4) |
610 |
(18,4) |
29 |
(34) |
713 |
(1,2,33) |
7740 |
| (13,33) |
2610 |
(23,32) |
6005 |
(12,22,32) |
17630 |
(14,2,32) |
5890 |
| (16,32) |
424 |
(1,24,3) |
8725 |
(13,23,3) |
11623 |
(15,22,3) |
3510 |
| (17,2,3) |
340 |
(19,3) |
12 |
(26) |
554 |
(12,25) |
2792 |
| (14,24) |
2325 |
(16,23) |
645 |
(18,22) |
84 |
(110,2) |
6 |
| (112) |
1 |
A SYMMETRICA program for computing these numbers could be written
as
main()
{
OP a,b,c,d;
anfang();
a=callocobject(); b=callocobject();
c=callocobject(); d=callocobject();
scan(INTEGER,a);
m_il_v(2L,b);
zykelind_Dn(a,S_V_I(b,0L));
/*
zykelind_Cn(a,b);
zykelind_Dn(a,b);
zykelind_aff1Zn(a,b);
*/
first_part_EXPONENT(a,c);
do
{
zykelind_stabilizer_part(c,S_V_I(b,1L));
/* computes the cycle index of the stabilizer of a partition
of type a */
redf_cap(b,d);
/* computes the Redfield cap of the cycle indices, which occur
in the vector b */
printf(" The number of mosaics of type ");
print(c);
printf(" is ");
println(d);
} while (next(c,c)); /* lists all possible types of partitions */
freeall(a); freeall(b); freeall(c); freeall(d);
ende();
}
If n is even then a mosaic consisting of two blocks of
size n/2 corresponds to a trope introduced by
Hauer. By applying the power group enumeration theorem [8] an explicit formula for the
number of all orbits of tropes under a group action was determined
in [4].
harald.fripertinger "at" uni-graz.at, May 10,
2016
