  For Schauder bases, Dilworth et al. introduced and characterized the
partially greedy property, which is strictly weaker than the (almost) greedy
property. Later, Berasategui et al. defined and studied the strong partially
greedy property for general bases. Let $\mathbf n$ be any strictly increasing
sequence of positive integers. In this paper, we define the strong partially
greedy property with respect to $\mathbf n$, called the ($\mathbf n$, strong
partially greedy) property. We give characterizations of this new property,
study relations among ($\mathbf n$, strong partially greedy) properties for
different sequences $\mathbf n$, establish Lebesgue-type inequalities for the
($\mathbf n$, strong partially greedy) parameter, investigate ($\mathbf n$,
strong partially greedy) bases with gaps, and weighted ($\mathbf n$, strong
partially greedy) bases, to name a few. Furthermore, we introduce the ($\mathbf
n$, almost greedy) property and equate the property to a strengthening of the
($\mathbf n$, strong partially greedy) property. This paper can be viewed both
as a survey of recent results regarding strong partially greedy bases and as an
extension of these results to an arbitrary sequence instead of $\mathbb{N}$.
