
186 Polytope Projects
Figure 10.6 illustrates the commutation condition for permutation
trees.
The permutation graphs may be considered as trees of elementary
designs.
Figure 10.7 shows dual permutation trees of designs.
The generating function for permutation species is (Bergeron et al. 1998):
∑
∞
=
=
−
=
0n
n
!n
x
!n
x1
1
)x(S
(10.1)
The number of permutations at the level n is n!
Fig. 10.6 Commutation condition for permutation trees.
Fig. 10.7 Dual permutation trees of designs.
U ( )
D ( )
DU (
UD ( )
DU ( ) UD ( )
-
21
1
321
2121 12
21
21
21
)
++
21 +
=
=
=
=
= 213 312++
2121
12
21
0
ˆ
0 0 1 0 1 00 01 1 0 0 001
01
10
1
0 0 1 0 01 0 1 0 1 1 0 00 0
1 0
10
ab
0 0 1 0 0 1
1
0
ˆ
10
10 0
001
010