On the signs of the principal minors of Hermitian matrices

The signed enhanced principal rank characteristic sequence (sepr-sequence) of a given $n\times n$ Hermitian matrix $B$ is the sequence $t_1{t}_2\cdots t_n$, where $t_k$ is ${\mathtt{A}}^{\ast }$, ${\mathtt{A}}^{+}$, ${\mathtt{A}}^{-}$, $\mathtt{N}$, ${\mathtt{S}}^{\ast }$, ${\mathtt{S}}^{+}$, or ${\mathtt{S}}^{-}$, based on the following criteria: $t_k={\mathtt{A}}^{\ast }$ if all the order-$k$ principal minors of $B$ are nonzero, and two of those minors are of opposite sign; $t_k={\mathtt{A}}^{+}$ (respectively, $t_k={\mathtt{A}}^{-}$) if all the order-$k$ principal minors of $B$ are positive (respectively, negative); $t_k=\mathtt{N}$ if all the order-$k$ principal minors of $B$ are zero; $t_k={\mathtt{S}}^{\ast }$ if $B$ has a positive, a negative, and a zero order-$k$ principal minor; $t_k={\mathtt{S}}^{+}$ (respectively, $t_k={\mathtt{S}}^{-}$) if $B$ has both a zero and a nonzero order-$k$ principal minor, and all the nonzero order-$k$ principal minors of $B$ are positive (respectively, negative). A complete characterization of the sequences of order $2$ and order $3$ that do not occur as a subsequence of the sepr-sequence of any Hermitian matrix is presented (a sequence has order $k$ if it has $k$ terms). An analogous characterization for real symmetric matrices is presented as well.