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Point Concurrences

Each entry in the point_concurrences is a function on the -element sets of points, for some positive integer , giving the number of blocks containing each -set. We use the general mechanism for function_on_ksubsets_of_indices with , to do this. Note that a block design is -wise balanced (see 7.3.5) if and only if the point concurrence function for takes only a single value.

For example, here is a small block design:

<block_design b="5" id="v3-b5-r3-1" v="3">
<blocks ordered="true">
<block><z>0</z></block>
<block><z>2</z></block>
<block><z>0</z><z>1</z></block>
<block><z>1</z><z>2</z></block>
<block><z>0</z><z>1</z><z>2</z></block>
</blocks>
</block_design>

and here are its -wise point concurrences for :

<point_concurrences>
<function_on_ksubsets_of_indices domain_base="points" k="1" n="3"
ordered="true" title="replication_numbers">
<map>
<preimage>
<entire_domain>
</entire_domain>
</preimage>
<image><z>3</z></image>
</map>
</function_on_ksubsets_of_indices>
<function_on_ksubsets_of_indices domain_base="points" k="2" n="3"
ordered="true" title="pairwise_point_concurrences">
<map>
<preimage>
<ksubset><z>0</z><z>2</z></ksubset>
</preimage>
<image><z>1</z></image>
</map>
<map>
<preimage>
<ksubset><z>0</z><z>1</z></ksubset>
<ksubset><z>1</z><z>2</z></ksubset>
</preimage>
<image><z>2</z></image>
</map>
</function_on_ksubsets_of_indices>
</point_concurrences>

Peter Dobcsanyi 2003-12-15