is $t\leq k$.

In Category 1, since in the $Q$-graph, each down edge must coincide with one and only one up edge, the total number of non-coincident edges is $2k$. Besides, due to the restriction that the base $B(Q)$ is a tree, we have $s+t-1=2k$ (according to the definition of a tree that $\#\{\text{edges}\}=\#\{\text{vertices}\}-1$), and this means that the summation over the elements in the corresponding characteristic sequence is zero (it is because we have the number of “1” equals $t-1$ and the number of “-1” equals $2k-s$).

The characteristic sequence of a $M$-pillar whose corresponding $Q$-graph lies in Category 1 has the following features (see Remark \[re1\] and \[re2\]):

1.  The total length of the characteristic sequence is $4k$;

2.  The sequence starts with a zero and ends with a zero (the first down edge is a down innovation and the last up edge is not an innovation);

3.  The number “1” appears only in the even position in the sequence and the number “-1” only in the odd position (down edges are in the odd position while up edges are in the even);

4.  The total number of “1” in the characteristic sequence is $t-1$ ($t-1$ up innovation), 