  In this paper, we study the generalized gapped k-mer filters and derive a
closed form solution for their coefficients. We consider nonnegative integers
$\ell$ and $k$, with $k\leq \ell$, and an $\ell$-tuple
$B=(b_1,\ldots,b_{\ell})$ of integers $b_i\geq 2$, $i=1,\ldots,\ell$. We
introduce and study an incidence matrix $A=A_{\ell,k;B}$. We develop a
M\"obius-like function $\nu_B$ which helps us to obtain closed forms for a
complete set of mutually orthogonal eigenvectors of $A^{\top} A$ as well as a
complete set of mutually orthogonal eigenvectors of $AA^{\top}$ corresponding
to nonzero eigenvalues.
  The reduced singular value decomposition of $A$ and combinatorial
interpretations for the nullity and rank of $A$, are among the consequences of
this approach.
  We then combine the obtained formulas, some results from linear algebra, and
combinatorial identities of elementary symmetric functions and $\nu_B$, to
provide the entries of the Moore-Penrose pseudo-inverse matrix $A^{+}$ and the
Gapped k-mer filter matrix $A^{+} A$.
