Point Concurrences

Each entry in the point_concurrences is a function on the $t$-element sets of points, for some positive integer $t$, giving the number of blocks containing each $t$-set. We use the general mechanism for function_on_ksubsets_of_indices with $k=t$, to do this. Note that a block design is $t$-wise balanced (see 7.3.5) if and only if the point concurrence function for $k=t$ 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 $t$-wise point concurrences for $t = 1,2$:

<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>