adjacencies, it is equivalent to Definition \[defn:normal\] when the number of descents is fixed. To see this note that if $\sigma$ and $\pi$ have the same number of descents then any letters in $\pi$ that form a decreasing adjacency, and thus a descent, must be nonzero in all embeddings otherwise that descent would not be in $\sigma$, contradicting the assumption that $\sigma$ and $\pi$ have the same number of descents.

One route to simplifying Equation is answering the following question:

\[que:sum\] Given an interval $[\sigma,\pi]$, for which $\lambda\in[\sigma,\pi)$ is the following sum nonzero?: $$\label{eq:sum}\displaystyle\sum_{S\in EZ^{\lambda,\pi}}(-1)^{|S|}.$$

One case where the sum in equals zero is when $EZ^{\lambda,\pi}$ is always empty, which leads us to the following definition and proposition:

We say an interval $[\sigma,\pi]$ has a *single block* if there exists some $i$ such that $\eta_i=0$ for any $\eta\in \widehat{E}^{\sigma,\pi}$. That is, there is a letter in $\pi$ that is not contained in any of the occurrences in $\widehat{E}^{\sigma,\pi}$.

We say an interval is *single* if for all $\lambda\in[\sigma,\pi)$ the interval $[\lambda,\pi]$ has a single block.

Our notation here follows from the idea of interval blocks in [@SteTen10]. Computer tests show that $78.6\%$ of intervals $[\sigma,\pi]$, where $|\pi|<9$, are single and 