The allow sequence of distinct eigenvalues for a sign pattern

A sign pattern $\mathcal{A}$ is a matrix with entries in $\{+,-,0\}$. This article introduces the allow sequence of distinct eigenvalues for an $n\times n$ sign pattern $\mathcal{A}$, defined as $q_{\rm seq}(\mathcal{A})=\langle q_1,\ldots,q_n\rangle$, with $q_k=1$ if there exists a real matrix with exactly $k$ distinct eigenvalues having pattern $\mathcal{A}$, and $q_k=0$ otherwise. For example, $q_{\rm seq}(\mathcal{A})=\langle 0,\ldots,0,1\rangle$ is equivalent to $\mathcal{A}$ requiring all distinct eigenvalues, while $q_{\rm seq}(\mathcal{A})=\langle 1,0,\ldots,0\rangle$ is equivalent to the digraph of $\mathcal{A}$ being acyclic. Relationships between the allow sequence for $\mathcal{A}$ and composite cycles of the digraph of $\mathcal{A}$ are explored to identify zeros in the sequence, while methods based on Jacobian matrices are developed to identify ones in the sequence. When $\mathcal{A}$ is an $n\times n$ irreducible sign pattern, the possible sequences for $q_{\rm seq}(\mathcal{A})$ are completely determined when $n\leq 4$ and when the sequence has at least $n-4$ trailing zeros for $n\geq 5$.