Download The bijection between partitions and integral simplices

A Bijection between the
d-dimensional simplices with
all distances in {1,2} and the
partitions of d+1
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Sascha Kurz
An integral point set is a set of n points in the
euclidean Ed with integral distances between the points.
We use the term simplex for a set of d+1 points in the
euclidean Ed such that not all the points are contained
in a hyperplane .
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Sascha Kurz
Similar to integral point sets we define integral simplices
as simplices with integral distances between its points.
The largest distance of a point set is called its diameter.
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Number of integral simplices by
diameter and dimension.
Dimension
Diameter
1
2
3
4
4
3
4
5
6
1 1
1
1
4 6
10
14
16 56 197
656
45 336 3133 21771
Sascha Kurz
7
8
9
1
21
2127
329859
1
29
6548
3336597
1
41
19130
32815796
There is clearly an unique integral simplex with
diameter 1 in any dimension.
By testing the sequence of the numbers of simplices
with diameter 2 with N.J.A. Sloane's marvellous
„Online-Encyclopedia of Integer Sequences“ we
learned that it is one less than the sequence of partions.
A partition of an integer n is an r-tuple of integers
i1 i 2... i r0
i1 i 2 ...i rn.
(i1,...,ir) with
and
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Theorem: The number of integral d-dimensional
simplices with diameter at most 2 is the number
of partitions of d+1.
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3
065
605
550
5
5
6
1
2
An integer triangle with its distance matrix.
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The bijection between partitions and integral simplices
Distance matrix D:
Partition: i=(4,3,3,2,1)
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0222222222222
2022222222221
2202222222112
222 0222211222
2222 011122222
2222101122222
2222110122222
2222111 022222
22212222 01222
222122221 0222
2212222222012
2212222222102
212222222222 0
2222222222222
2
2
2
2
2
2
2
2
2
2
2
2
2
0
We consider only the part
above the main diagonal.
The matrix contains strings of
bold printed 1's. The length of
such a block of 1's is one less
than the corresponding
summand of the partition.
i=( 4
9
3
3
2
1 )
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2222222222222
2222222222 12
2222222 1 122
22221 12222
11 1222222
11222222
1222222
222222
12222
2222
122
22
2
Each such block of 1's is
completed to an upper
triangular matrix at the
bottom of the corresponding
columns.
i=( 4
10
3
3
2
1 )
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2222222222222
2222222222 12
2222222 1 122
22221 12222
11 1222222
11222222
1222222
222222
12222
2222
122
22
2
Proof of the bijection
Every partition yields an integral simplex.
Two different partitions yield two
nonisomorphic integral simplices.
For every integral simplex there is a
corresponding partition.
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Sascha Kurz
Every partition yields an integral simplex
Not every symmetric matrix can be realized as
a distance matrix in the euclidean space.
There is, for example, no triangle with side
lengths 4, 2, and 1.
Definition:
For a matrix A we define
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1
A 
A :
1
110
Theorem (Menger): If M is a set of d+1 points with
distance maxtrix D=(di,j) and A=(di,j2), then M is
realizeable in the euclidean d-dimensional space,
iff
1
d 1
det A  0
and each subset of M is realizeable in the (d-1)dimensional space.
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Sascha Kurz
Two different partitions yield two
nonisomorphic integral simplices.
In general there are different distance matrices
which describe the same integral simplex. So we
define an unique representant for the set of these
matrices.
Now we only have to show that two different
partitions yield two different representants
of the corresponding matrices.
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For every integral simplex there is a
corresponding partition
For a proof we only need the triangle inequallity
and the properties of the unique representant
of a distance matrix.
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Generalizations
Integral simplices with all distances in {1,2,3}.
Simplices with at most two different side lengths a and b.
Maximum number of such nonisomorphic
simplices  number of graphs
Minimum number of nonisomorphic simplices.
Description of graph classes corresponding to
the simplices for special side lenghts a and b
 threshold functions
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