Section \[graph\], the indices in ${\bf i}=(i_1, \cdots, i_{2k})$ run over $1,2,\cdots,T$ and the indices in ${\bf j}=(j_1, \cdots, j_{2k})$ run over $1,2,\cdots,p$. See the following Figure \[exp\] for an illustration.

![The $Q(i,j)$-graph that corresponds to .[]{data-label="exp"}](expection.eps){width="13cm"}

Now suppose the pillar of the $Q$-graph in Figure \[exp\] has $t$ noncoincident $I$-vertices and $s$ noncoincident $J$-vertices, which results in $s$ down innovation and $t-1$ up innovation (we make the convention that the first down edge is always a down innovation and the last up edge is not an innovation). Then in the corresponding characteristic sequence $(d_1~u_1~\cdots~ d_{2k}~u_{2k})$, the number of “1” (“1” only appears in the even position as it corresponds to the up edge) is $t-1$, the number of “-1” (“-1” only appears in the odd position) is $2k-s$ and the sequence starts and ends with “0”.

We classify the $Q$-graphs in Figure \[exp\] into three categories:

Category 1 (denoted by $Q_1$) contains all the $Q$-graphs that in its pillar $M(Q)$, each down edge must coincide with one and only one up edge and its base $B(Q)$ is a tree of $2k$ edges. In this category, $t+s-1=2k$ and thus $s$ is suppressed for simplicity.

Category 2 (denoted by $Q_2$) contains 