Block concurrences

Similarly, here we record the functions giving the numbers of points in the intersection of $t$-sets of blocks. The blocks are indexed from $0$ to $b-1$, and we again use the general mechanism for function_on_ksubsets_of_indices.

In practice, we almost always use the compressed representation of this function where we give only the preimage cardinalities (as described in section 4.2 on Functions and index flags).

For example, in the Fano plane, any block contains three points, and any two blocks meet in one point. This is recorded as follows:

<block_concurrences>
    <function_on_ksubsets_of_indices domain_base="blocks" k="1" n="7"
     ordered="unknown" title="block_sizes">
        <map>
            <preimage_cardinality><z>7</z></preimage_cardinality>
            <image><z>3</z></image>
        </map>
    </function_on_ksubsets_of_indices>
    <function_on_ksubsets_of_indices domain_base="blocks" k="2" n="7"
     ordered="unknown" title="pairwise_block_intersection_sizes">
        <map>
            <preimage_cardinality><z>21</z></preimage_cardinality>
            <image><z>1</z></image>
        </map>
    </function_on_ksubsets_of_indices>
</block_concurrences>