| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
Here is a problem I have been working on for some time, but have
not solved:
Find a closed form solution for the number of transitive relations
on a set of N elements. This form is obviously some function of
N.
I have a closed form solution to the number of equivalence relations
on a set of N elements, which is a count of the partions of a set
of N elements.
Any help will be gladly appreciated.
R.J.Kolker
KIRK::KOLKER
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 712.1 | CHOVAX::YOUNG | Back from the Shadows Again, | Wed Jun 03 1987 19:35 | 7 | |
Sounds interesting, but some of us dinosaurs (me) have a lot of
cobwebs in the attic. Which is to say I don't follow your terminology
(correct though it may be).
So how about more detail and maybe a short example or two?
-- Barry
| |||||
| 712.2 | clarification for dinosaurs | KIRK::KOLKER | Thu Jun 04 1987 11:41 | 34 | |
re .1
Greetings Saurian.
Let S be a set. A relation on S is a subset of S X S (X is cartesian
product). A a relation r is transitive iff:
r(s1,s2) and r(s2,s3) imply r(s1,s3) for all s1,s2,s3 in S.
another way of putting is r*r is contained r where * is the relation
composition operation. If r,s are relations on S the r*s is defined
to be the set of pairs (i.e. a relation) (s1,s3) for which there
exists s2 in S such that r(s1,s2) and r(s2,s3).
An equivalence relation on S x S is a relation e such that:
1. e(s,s) for all s in S (e is reflexive)
2. e(s1,s2) implies e(s2,s1) for s1,s2 in S (e is symmetric)
3. e is transitive
It turns out each equivalence relation on S X S corresponds in a
1 - 1 fashion to a partitioning of S into disjoint non empty subsets.
Thus e(x,y) iff x,y in the same subset of the partition.
A binary relation can be represented by a matrix whose elements
are 0,1 in a two element boolean algebra. The index set of the
matrix is S itself so if S has N elements a binary relation can
be represented by an N x N matrix. The multipication of such matrices
is defined in a manner analogous to matrices defined over rings.
Conjunction takes the place of multiplication and disjunction takes
the place of addition.
A binary matrix M represents a transitive relation iff the (s1,s2)
element of M x M implies the corresponding element of M.
Thus the problem is equivalent to counting the transitive matrices.
| |||||
| 712.3 | small correction to .2 | KIRK::KOLKER | Thu Jun 04 1987 12:39 | 8 | |
correction to .2
Let r,s be relations on S then the composition r*s is the relation
v such that
v(s1,s3) iff there exist s2 in S such that r(s1,s2) and s(s2,s3).
If you are a database theorist and like relational databases you
will recognize r*s as the natural join of r and s projected on the
first and third domains of the join.
| |||||
| 712.4 | Appreciation from extinct species | CHOVAX::YOUNG | Back from the Shadows Again, | Thu Jun 04 1987 13:21 | 5 |
Re .2,.3:
Thanks. Good explanation, I think it has even sunk into my hindbrain.
-- Barry
| |||||
| 712.5 | CLT::GILBERT | eager like a child | Thu Jun 04 1987 14:11 | 6 | |
This problem sounds similar to the problem of minimal-comparison
sorting. See Knuth's "Art of Computer Programming", Vol.3 for
more information.
I suspect this problem is very difficult. Simply determining the
numbers for N <= 15 seems a more realistic goal.
| |||||
| 712.6 | Now you know why Dinosaurs are extinct | KIRK::KOLKER | Thu Jun 04 1987 15:36 | 6 | |
re .5
I have computed the results for n <= 4. For n = 5 I estimated about
20 days on a VAX 750. For n = 6, I should live so long.
| |||||
| 712.7 | See 598 | CHOVAX::YOUNG | Back from the Shadows Again, | Thu Jun 04 1987 17:27 | 9 |
I believe that this problem is isomorphic with the Push-Button
Lock problem that I presented in 598. We never did fully solve
it, but Peter (how could you forget this Peter?) did give a good
recursive formula in 598.2, and a list of the values up to n=50
in 598.5.
"Proof" available upon demand.
-- Barry
| |||||
| 712.8 | BEING::POSTPISCHIL | Always mount a scratch monkey. | Thu Jun 04 1987 18:10 | 26 | |
Re .7:
Solutions for two buttons:
1, 2
2, 1
1-2
Transitive relations for two items:
{ }
{ (1,1) }
{ (2,2) }
{ (1,1), (2,2) }
{ (1,2) }
{ (1,1), (1,2) }
{ (2,2), (1,2) }
{ (1,1), (2,2), (1,2) }
{ (2,1) }
{ (1,1), (2,1) }
{ (2,2), (2,1) }
{ (1,1), (2,2), (2,1) }
{ (1,1), (2,2), (1,2), (2,1) }
-- edp
| |||||
| 712.9 | KIRK::KOLKER | Thu Jun 04 1987 19:50 | 13 | ||
re .7
the item in note 598 is the ennumeration of partitions of a 5 element
set.
ennumerating the partitions is not the problem I have posed. It
is the ennumeration of transitive relations. There are more transitive
relation on a set of N elements than there are partitions on a set
of N elements because a transitive relation need not be either
symmetric or reflexive.
Thank you for the xref.
R.J.Kolker
| |||||
| 712.10 | Sorry...cobwebs in the attic. | CHOVAX::YOUNG | Back from the Shadows Again, | Thu Jun 04 1987 23:41 | 31 |
Re .8:
Yes after I had thought about this for a while it became clear to
me that I had missed something in my line of reasoning somewhere
(ie. the "proof" I was going to offer was a crock).
Re .9:
I agree the 598 is not an enumeration of the transitive relations
on a set, however neither is 598 an enumeration of the partitions
of a set. A partition is I believe, an unordered set of sets, that
is the partition {{1,2,3},{4},{5,6}} is identical to
{{5,6},{4},{1,2,3}} whereas the PBL problem deals with
'ordered'-partitions. In this case we would have combination
<{1,2,3},{4},{5,6}> which is clearly different from <{5,6},{4},{1,2,3}>
because they are in a different order (hope I'm getting my symbology
right).
Thus we might say that the PBL problem is equivilant to the enumeration
of the 'ordered' partitons of a set. To translate this into
set-relations terms, the PBL problem is equivilant to the enumeration
of the transitive relations on a set that satisfy the following
additional conditions:
1) r is reflexive.
2) For any s1,s2 in S, either r(s1,s2) or r(s2,s1).
I had missed the need for these extra conditions to establish the
isomorphism when I entered .7.
-- Barry
| |||||
| 712.11 | equivalences .=. partitions | KIRK::KOLKER | Fri Jun 05 1987 09:55 | 28 | |
re .10
Let S be a set and let S = union over some collection of subsets
of S where no two distinct sets of the collection have any elements
in common. Define an equivalence relation based on the partition
by saying two elements of S are equivalent if both are in the same
subset of the partition. This relation is reflexive since an element
is in the same set as itself, it is symmetric since if s1 and s2
are in the same member of the partition then s2 and s1 are in the
same member of the partition. Like wise transitivity can be shown.
This proves that to a partition there is a corresponding equivalence
reltion. Now to the converse: Let e be an equivalence relation on
S. For s in S define P(s) = set of s1 in S such that e(s,s1).
P(s) is not empty since it contains s. ( oops I forgot to mention
that the subsets forming a partition are non empty, sorry) To continue,
consider P(s) and P(s1) for s,s1 in S. If P(s) intersection P(s1)
has an element s2 in common, then by definition of P e(s,s2), e(s2,s1),
hence by transitivity of e, e(s,s2) which implies s2 in P(s). Likewise
s in P(s2). Use this facts to show P(s) contained in P(s1) and
P(s1) contained in P(s). This if P(s) intersects P(s2) they are
the same set. This shows the collection P(s) for s in S form a
partition .
To sum up to each partition an equivalence relation. To each
equivalence relation a partition. QED.
R.J.Kolker
| |||||
| 712.12 | CLT::GILBERT | Builder | Tue Sep 08 1987 14:43 | 25 | |
Transitive relations for three items:
(re)labellings e(n,n)
{} 1 2**3
{(1,2)} 6 2**3
{(1,2),(1,3)} 3 2**3
{(1,2),(3,2)} 3 2**3
{(1,2),(2,3),(1,3)} 6 2**3
{(1,2),(2,1),(1,1),(2,2)} 3 2**1
{(1,2),(2,1),(1,1),(2,2),(1,3),(2,3)}
3 2**1
{(1,2),(2,1),(1,1),(2,2),(3,1),(3,2)}
3 2**1
{(1,2),(2,1),(1,1),(2,2),(1,3),(2,3),(3,1),(3,2),(3,3)}
1 2**0
The columns marked "(re)labellings" and "e(n,n)" provide multipliers.
The first is straight-forward; the second considers whether relations
of the form (n,n) are included.
Thus, for N=3, the number of transitive relations is:
(1+6+3+3+6)*2**3 + (3+3+3)*2**1 + 1*2**0 = 171
| |||||
| 712.13 | CLT::GILBERT | Builder | Tue Sep 08 1987 15:34 | 37 | |
There are 3994 transitive relations for 4 items:
{ } 1 4
{ (3,0) } 12 4
{ (3,0) (3,1) } 12 4
{ (3,0) (3,1) (3,2) } 4 4
{ (2,0) (3,0) } 12 4
{ (2,0) (3,1) } 12 4
{ (2,0) (3,0) (3,1) } 24 4
{ (2,0) (3,0) (3,2) } 24 4
{ (2,0) (3,0) (3,1) (3,2) } 24 4
{ (2,0) (2,1) (3,0) (3,1) } 6 4
{ (2,0) (2,1) (3,0) (3,1) (3,2) } 12 4
{ (2,2) (2,3) (3,2) (3,3) } 6 2
{ (2,0) (2,2) (2,3) (3,0) (3,2) (3,3) } 12 2
{ (2,0) (2,1) (2,2) (2,3) (3,0) (3,1) (3,2) (3,3) } 6 2
{ (1,0) (2,0) (3,0) } 4 4
{ (1,0) (2,0) (3,0) (3,1) } 24 4
{ (1,0) (2,0) (3,0) (3,1) (3,2) } 12 4
{ (1,0) (2,0) (2,1) (3,0) (3,1) } 12 4
{ (1,0) (2,0) (2,1) (3,0) (3,1) (3,2) } 24 4
{ (1,0) (2,2) (2,3) (3,2) (3,3) } 12 2
{ (1,0) (2,0) (2,2) (2,3) (3,0) (3,2) (3,3) } 12 2
{ (1,0) (2,0) (2,1) (2,2) (2,3) (3,0) (3,1) (3,2) (3,3) } 12 2
{ (1,1) (1,2) (2,1) (2,2) (3,1) (3,2) } 12 2
{ (1,1) (1,2) (2,1) (2,2) (3,0) (3,1) (3,2) } 12 2
{ (1,0) (1,1) (1,2) (2,0) (2,1) (2,2) (3,0) (3,1) (3,2) } 12 2
{ (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) (3,1) (3,2) (3,3) } 4 1
{ (1,0) (1,1) (1,2) (1,3) (2,0) (2,1) (2,2) (2,3) (3,0) (3,1) (3,2) (3,3) } 4 1
{ (0,0) (0,1) (1,0) (1,1) (2,0) (2,1) (3,0) (3,1) } 6 2
{ (0,0) (0,1) (1,0) (1,1) (2,0) (2,1) (3,0) (3,1) (3,2) } 12 2
{ (0,0) (0,1) (1,0) (1,1) (2,2) (2,3) (3,2) (3,3) } 3 0
{ (0,0) (0,1) (1,0) (1,1) (2,0) (2,1) (2,2) (2,3) (3,0) (3,1) (3,2) (3,3) } 6 0
{ (0,0) (0,1) (0,2) (1,0) (1,1) (1,2) (2,0) (2,1) (2,2) (3,0) (3,1) (3,2) } 4 1
{ (0,0) (0,1) (0,2) (0,3) (1,0) (1,1) (1,2) (1,3) (2,0) (2,1) (2,2) (2,3) (3,0) (3,1) (3,2) (3,3) } 1 0
Total = 3994
| |||||
| 712.14 | CLT::GILBERT | Builder | Tue Sep 08 1987 15:40 | 143 | |
There are 154303 transitive relations for 5 items:
{ } 1 5
{ 40 } 20 5
{ 40 41 } 30 5
{ 40 41 42 } 20 5
{ 40 41 42 43 } 5 5
{ 30 40 } 30 5
{ 30 41 } 60 5
{ 30 40 41 } 120 5
{ 30 41 42 } 60 5
{ 30 40 41 42 } 60 5
{ 30 40 43 } 60 5
{ 30 40 41 43 } 120 5
{ 30 40 41 42 43 } 60 5
{ 30 31 40 41 } 30 5
{ 30 31 40 42 } 60 5
{ 30 31 40 41 42 } 60 5
{ 30 31 40 41 43 } 60 5
{ 30 31 40 41 42 43 } 60 5
{ 30 31 32 40 41 42 } 10 5
{ 30 31 32 40 41 42 43 } 20 5
{ 33 34 43 44 } 10 3
{ 30 33 34 40 43 44 } 30 3
{ 30 31 33 34 40 41 43 44 } 30 3
{ 30 31 32 33 34 40 41 42 43 44 } 10 3
{ 20 30 40 } 20 5
{ 20 30 41 } 60 5
{ 20 30 40 41 } 60 5
{ 20 30 40 42 } 120 5
{ 20 30 40 41 42 } 120 5
{ 20 30 40 42 43 } 60 5
{ 20 30 40 41 42 43 } 60 5
{ 20 31 40 41 } 60 5
{ 20 31 40 42 } 120 5
{ 20 31 40 41 42 } 120 5
{ 20 31 40 41 42 43 } 60 5
{ 20 30 31 40 41 } 60 5
{ 20 30 31 40 42 } 120 5
{ 20 30 31 40 41 42 } 120 5
{ 20 30 31 40 41 43 } 120 5
{ 20 30 31 40 41 42 43 } 120 5
{ 20 30 32 40 42 } 60 5
{ 20 30 32 40 41 42 } 120 5
{ 20 30 32 40 42 43 } 120 5
{ 20 30 32 40 41 42 43 } 120 5
{ 20 30 31 32 40 41 42 } 60 5
{ 20 30 31 32 40 41 42 43 } 120 5
{ 20 33 34 43 44 } 60 3
{ 20 30 33 34 40 43 44 } 60 3
{ 20 31 33 34 41 43 44 } 60 3
{ 20 30 31 33 34 40 41 43 44 } 60 3
{ 20 30 32 33 34 40 42 43 44 } 60 3
{ 20 30 31 32 33 34 40 41 42 43 44 } 60 3
{ 20 21 30 31 40 41 } 10 5
{ 20 21 30 31 40 41 42 } 60 5
{ 20 21 30 31 40 41 42 43 } 30 5
{ 20 21 30 31 32 40 41 42 } 30 5
{ 20 21 30 31 32 40 41 42 43 } 60 5
{ 20 21 33 34 43 44 } 30 3
{ 20 21 30 33 34 40 43 44 } 60 3
{ 20 21 30 31 33 34 40 41 43 44 } 30 3
{ 20 21 30 31 32 33 34 40 41 42 43 44 } 30 3
{ 22 23 32 33 42 43 } 30 3
{ 22 23 32 33 40 42 43 } 60 3
{ 22 23 32 33 40 41 42 43 } 30 3
{ 20 22 23 30 32 33 40 42 43 } 60 3
{ 20 22 23 30 32 33 40 41 42 43 } 60 3
{ 20 21 22 23 30 31 32 33 40 41 42 43 } 30 3
{ 22 23 24 32 33 34 42 43 44 } 10 2
{ 20 22 23 24 30 32 33 34 40 42 43 44 } 20 2
{ 20 21 22 23 24 30 31 32 33 34 40 41 42 43 44 } 10 2
{ 10 20 30 40 } 5 5
{ 10 20 30 40 41 } 60 5
{ 10 20 30 40 41 42 } 60 5
{ 10 20 30 40 41 42 43 } 20 5
{ 10 20 30 31 40 41 } 60 5
{ 10 20 30 31 40 42 } 60 5
{ 10 20 30 31 40 41 42 } 120 5
{ 10 20 30 31 40 41 43 } 120 5
{ 10 20 30 31 40 41 42 43 } 120 5
{ 10 20 30 31 32 40 41 42 } 30 5
{ 10 20 30 31 32 40 41 42 43 } 60 5
{ 10 20 33 34 43 44 } 30 3
{ 10 20 30 33 34 40 43 44 } 30 3
{ 10 20 30 31 33 34 40 41 43 44 } 60 3
{ 10 20 30 31 32 33 34 40 41 42 43 44 } 30 3
{ 10 20 21 30 31 40 41 } 20 5
{ 10 20 21 30 31 40 41 42 } 120 5
{ 10 20 21 30 31 40 41 42 43 } 60 5
{ 10 20 21 30 31 32 40 41 42 } 60 5
{ 10 20 21 30 31 32 40 41 42 43 } 120 5
{ 10 20 21 33 34 43 44 } 60 3
{ 10 20 21 30 33 34 40 43 44 } 60 3
{ 10 20 21 30 31 33 34 40 41 43 44 } 60 3
{ 10 20 21 30 31 32 33 34 40 41 42 43 44 } 60 3
{ 10 22 23 32 33 42 43 } 60 3
{ 10 22 23 32 33 40 42 43 } 60 3
{ 10 22 23 32 33 40 41 42 43 } 60 3
{ 10 20 22 23 30 32 33 40 42 43 } 60 3
{ 10 20 22 23 30 32 33 40 41 42 43 } 60 3
{ 10 20 21 22 23 30 31 32 33 40 41 42 43 } 60 3
{ 10 22 23 24 32 33 34 42 43 44 } 20 2
{ 10 20 22 23 24 30 32 33 34 40 42 43 44 } 20 2
{ 10 20 21 22 23 24 30 31 32 33 34 40 41 42 43 44 } 20 2
{ 11 12 21 22 31 32 41 42 } 30 3
{ 11 12 21 22 31 32 40 41 42 } 60 3
{ 11 12 21 22 31 32 41 42 43 } 60 3
{ 11 12 21 22 31 32 40 41 42 43 } 60 3
{ 11 12 21 22 30 31 32 40 41 42 } 30 3
{ 11 12 21 22 30 31 32 40 41 42 43 } 60 3
{ 11 12 21 22 33 34 43 44 } 15 1
{ 11 12 21 22 30 33 34 40 43 44 } 30 1
{ 11 12 21 22 31 32 33 34 41 42 43 44 } 30 1
{ 11 12 21 22 30 31 32 33 34 40 41 42 43 44 } 30 1
{ 10 11 12 20 21 22 30 31 32 40 41 42 } 30 3
{ 10 11 12 20 21 22 30 31 32 40 41 42 43 } 60 3
{ 10 11 12 20 21 22 30 33 34 40 43 44 } 15 1
{ 10 11 12 20 21 22 30 31 32 33 34 40 41 42 43 44 } 30 1
{ 11 12 13 21 22 23 31 32 33 41 42 43 } 20 2
{ 11 12 13 21 22 23 31 32 33 40 41 42 43 } 20 2
{ 10 11 12 13 20 21 22 23 30 31 32 33 40 41 42 43 } 20 2
{ 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 } 5 1
{ 10 11 12 13 14 20 21 22 23 24 30 31 32 33 34 40 41 42 43 44 } 5 1
{ 00 01 10 11 20 21 30 31 40 41 } 10 3
{ 00 01 10 11 20 21 30 31 40 41 42 } 60 3
{ 00 01 10 11 20 21 30 31 40 41 42 43 } 30 3
{ 00 01 10 11 20 21 30 31 32 40 41 42 } 30 3
{ 00 01 10 11 20 21 30 31 32 40 41 42 43 } 60 3
{ 00 01 10 11 20 21 33 34 43 44 } 30 1
{ 00 01 10 11 20 21 30 31 33 34 40 41 43 44 } 30 1
{ 00 01 10 11 20 21 30 31 32 33 34 40 41 42 43 44 } 30 1
{ 00 01 10 11 22 23 32 33 40 41 42 43 } 15 1
{ 00 01 10 11 20 21 22 23 30 31 32 33 40 41 42 43 } 30 1
{ 00 01 10 11 22 23 24 32 33 34 42 43 44 } 10 0
{ 00 01 10 11 20 21 22 23 24 30 31 32 33 34 40 41 42 43 44 } 10 0
{ 00 01 02 10 11 12 20 21 22 30 31 32 40 41 42 } 10 2
{ 00 01 02 10 11 12 20 21 22 30 31 32 40 41 42 43 } 20 2
{ 00 01 02 10 11 12 20 21 22 30 31 32 33 34 40 41 42 43 44 } 10 0
{ 00 01 02 03 10 11 12 13 20 21 22 23 30 31 32 33 40 41 42 43 } 5 1
{ 00 01 02 03 04 10 11 12 13 14 20 21 22 23 24 30 31 32 33 34 40 41 42 43 44 } 1 0
Total = 154303
| |||||
| 712.15 | CLT::GILBERT | Builder | Tue Sep 08 1987 18:18 | 1 | |
There are 9415189 transitive relations for 6 items. | |||||
| 712.16 | BEING::POSTPISCHIL | Always mount a scratch monkey. | Tue Sep 08 1987 20:19 | 4 | |
There is no unused CPU time on the CLT cluster.
-- edp
| |||||
| 712.17 | Re.15: Wheres the list? :-) | CHOVAX::YOUNG | Back from the Shadows Again, | Wed Sep 09 1987 00:17 | 1 |
| 712.18 | CLT::GILBERT | Builder | Wed Sep 09 1987 10:52 | 724 | |
Here's the list for N=6. The notation [xy..z] means that x, y, ... z
form an equivalence class -- for example, [345] means that the pairs
33, 34, 35, 43, 44, 45, 53, 54, and 55 are all in the relation.
There are 9415189 transitive relations for 6 items.
{ } 1 6
{ 50 } 30 6
{ 50 51 } 60 6
{ 50 51 52 } 60 6
{ 50 51 52 53 } 30 6
{ 50 51 52 53 54 } 6 6
{ 40 50 } 60 6
{ 40 51 } 180 6
{ 40 50 51 } 360 6
{ 40 51 52 } 360 6
{ 40 50 51 52 } 360 6
{ 40 51 52 53 } 120 6
{ 40 50 51 52 53 } 120 6
{ 40 50 54 } 120 6
{ 40 50 51 54 } 360 6
{ 40 50 51 52 54 } 360 6
{ 40 50 51 52 53 54 } 120 6
{ 40 41 50 51 } 90 6
{ 40 41 50 52 } 360 6
{ 40 41 50 51 52 } 360 6
{ 40 41 52 53 } 90 6
{ 40 41 50 52 53 } 360 6
{ 40 41 50 51 52 53 } 180 6
{ 40 41 50 51 54 } 180 6
{ 40 41 50 51 52 54 } 360 6
{ 40 41 50 51 52 53 54 } 180 6
{ 40 41 42 50 51 52 } 60 6
{ 40 41 42 50 51 53 } 180 6
{ 40 41 42 50 51 52 53 } 120 6
{ 40 41 42 50 51 52 54 } 120 6
{ 40 41 42 50 51 52 53 54 } 120 6
{ 40 41 42 43 50 51 52 53 } 15 6
{ 40 41 42 43 50 51 52 53 54 } 30 6
{ [45] } 15 4
{ [45] 40 } 60 4
{ [45] 40 41 } 90 4
{ [45] 40 41 42 } 60 4
{ [45] 40 41 42 43 } 15 4
{ 30 40 50 } 60 6
{ 30 40 51 } 360 6
{ 30 40 50 51 } 360 6
{ 30 40 51 52 } 180 6
{ 30 40 50 51 52 } 180 6
{ 30 40 50 53 } 360 6
{ 30 40 50 51 53 } 720 6
{ 30 40 50 51 52 53 } 360 6
{ 30 40 50 53 54 } 180 6
{ 30 40 50 51 53 54 } 360 6
{ 30 40 50 51 52 53 54 } 180 6
{ 30 41 50 51 } 360 6
{ 30 41 52 } 120 6
{ 30 41 50 52 } 720 6
{ 30 41 50 51 52 } 360 6
{ 30 41 50 53 } 720 6
{ 30 41 50 51 53 } 720 6
{ 30 41 50 52 53 } 720 6
{ 30 41 50 51 52 53 } 720 6
{ 30 41 50 51 53 54 } 360 6
{ 30 41 50 51 52 53 54 } 360 6
{ 30 40 41 50 51 } 360 6
{ 30 40 41 50 52 } 360 6
{ 30 40 41 51 52 } 720 6
{ 30 40 41 50 51 52 } 720 6
{ 30 40 41 50 53 } 720 6
{ 30 40 41 50 51 53 } 720 6
{ 30 40 41 50 52 53 } 720 6
{ 30 40 41 50 51 52 53 } 720 6
{ 30 40 41 50 51 54 } 720 6
{ 30 40 41 50 51 52 54 } 720 6
{ 30 40 41 50 51 53 54 } 720 6
{ 30 40 41 50 51 52 53 54 } 720 6
{ 30 41 42 51 52 } 180 6
{ 30 41 42 50 51 52 } 360 6
{ 30 41 42 50 53 } 360 6
{ 30 41 42 50 51 53 } 720 6
{ 30 41 42 50 51 52 53 } 360 6
{ 30 41 42 51 52 54 } 360 6
{ 30 41 42 50 51 52 54 } 360 6
{ 30 41 42 50 51 52 53 54 } 360 6
{ 30 40 41 42 50 51 52 } 180 6
{ 30 40 41 42 50 53 } 360 6
{ 30 40 41 42 50 51 53 } 720 6
{ 30 40 41 42 50 51 52 53 } 360 6
{ 30 40 41 42 50 51 52 54 } 360 6
{ 30 40 41 42 50 51 52 53 54 } 360 6
{ 30 40 43 50 53 } 180 6
{ 30 40 43 50 51 53 } 720 6
{ 30 40 43 50 51 52 53 } 360 6
{ 30 40 43 50 53 54 } 360 6
{ 30 40 43 50 51 53 54 } 720 6
{ 30 40 43 50 51 52 53 54 } 360 6
{ 30 40 41 43 50 51 53 } 360 6
{ 30 40 41 43 50 52 53 } 360 6
{ 30 40 41 43 50 51 52 53 } 720 6
{ 30 40 41 43 50 51 53 54 } 720 6
{ 30 40 41 43 50 51 52 53 54 } 720 6
{ 30 40 41 42 43 50 51 52 53 } 180 6
{ 30 40 41 42 43 50 51 52 53 54 } 360 6
{ [45] 30 } 180 4
{ [45] 30 40 } 180 4
{ [45] 30 41 } 360 4
{ [45] 30 40 41 } 360 4
{ [45] 30 41 42 } 180 4
{ [45] 30 40 41 42 } 180 4
{ [45] 30 40 43 } 180 4
{ [45] 30 40 41 43 } 360 4
{ [45] 30 40 41 42 43 } 180 4
{ 30 31 40 41 50 51 } 60 6
{ 30 31 40 41 50 52 } 360 6
{ 30 31 40 41 50 51 52 } 180 6
{ 30 31 40 41 50 51 53 } 360 6
{ 30 31 40 41 50 51 52 53 } 360 6
{ 30 31 40 41 50 51 53 54 } 180 6
{ 30 31 40 41 50 51 52 53 54 } 180 6
{ 30 31 40 42 51 52 } 120 6
{ 30 31 40 42 50 51 52 } 360 6
{ 30 31 40 42 50 51 53 } 720 6
{ 30 31 40 42 50 51 52 53 } 720 6
{ 30 31 40 42 50 51 52 53 54 } 360 6
{ 30 31 40 41 42 50 51 52 } 180 6
{ 30 31 40 41 42 50 51 53 } 360 6
{ 30 31 40 41 42 50 51 52 53 } 360 6
{ 30 31 40 41 42 50 51 52 54 } 360 6
{ 30 31 40 41 42 50 51 52 53 54 } 360 6
{ 30 31 40 41 43 50 51 53 } 180 6
{ 30 31 40 41 43 50 51 52 53 } 360 6
{ 30 31 40 41 43 50 51 53 54 } 360 6
{ 30 31 40 41 43 50 51 52 53 54 } 360 6
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{ [01] [45] 20 30 32 40 42 43 } 180 2
{ [01] [34] 20 53 } 90 2
{ [01] [34] 20 50 53 } 180 2
{ [01] [34] 20 50 52 53 } 180 2
{ [01] [34] 20 30 50 53 } 180 2
{ [01] [34] 20 30 50 52 53 } 180 2
{ [01] [34] 20 30 32 50 52 53 } 180 2
{ [01] [345] 20 } 60 1
{ [01] [345] 20 30 } 60 1
{ [01] [345] 20 30 32 } 60 1
{ [01] [23] 40 42 50 52 } 45 2
{ [01] [23] 40 42 50 52 54 } 90 2
{ [01] [23] [45] } 15 0
{ [01] [23] [45] 40 } 90 0
{ [01] [23] [45] 40 42 } 45 0
{ [01] [23] 20 40 42 50 52 } 90 2
{ [01] [23] 20 40 42 50 52 54 } 180 2
{ [01] [23] [45] 20 40 } 45 0
{ [01] [23] [45] 20 40 42 } 90 0
{ [01] [234] 52 } 60 1
{ [01] [234] 50 52 } 60 1
{ [01] [234] 20 50 52 } 60 1
{ [01] [2345] } 15 0
{ [01] [2345] 20 } 15 0
{ [012] 30 40 50 } 20 3
{ [012] 30 40 50 53 } 120 3
{ [012] 30 40 50 53 54 } 60 3
{ [012] 30 40 43 50 53 } 60 3
{ [012] 30 40 43 50 53 54 } 120 3
{ [012] [45] 30 40 } 60 1
{ [012] [45] 30 40 43 } 60 1
{ [012] [34] 30 50 53 } 60 1
{ [012] [345] } 10 0
{ [012] [345] 30 } 20 0
{ [0123] 40 50 } 15 2
{ [0123] 40 50 54 } 30 2
{ [0123] [45] 40 } 15 0
{ [01234] 50 } 6 1
{ [012345] } 1 0
| |||||