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Motivated by the Novikov equation and its peakon problem, we propose a newmixed type Hermite--Pad\'{e} approximation whose unique solution is a sequenceof polynomials constructed with the help of Pfaffians. These polynomials belongto the family of recently proposed partial-skew-orthogonal polynomials. Therelevance of partial-skew-orthogonal polynomials is especially visible in theapproximation problem germane to the Novikov peakon problem so that we obtainexplicit inverse formulae in terms of Pfaffians by reformulating the inversespectral problem for the Novikov multipeakons. Furthermore, we investigate twoHermite--Pad\'{e} approximations for the related spectral problem of thediscrete dual cubic string, and show that these approximation problems can alsobe solved in terms of partial-skew-orthogonal polynomials and nonsymmetricCauchy biorthogonal polynomials. This formulation results in a newcorrespondence among several integrable lattices.

Hermite--Pad\'{e} approximations with Pfaffian structures: Novikov peakon equation and integrable lattices