  Given integers $1 \le k_1 < \cdots < k_l \le n-1$, let
$\text{Fl}_{k_1,\dots,k_l;n}$ denote the type $A$ partial flag variety
consisting of all chains of subspaces $(V_{k_1}\subset\cdots\subset V_{k_l})$
inside $\mathbb{R}^n$, where each $V_k$ has dimension $k$. Lusztig (1994, 1998)
introduced the totally positive part $\text{Fl}_{k_1,\dots,k_l;n}^{>0}$ as the
subset of partial flags which can be represented by a totally positive $n\times
n$ matrix, and defined the totally nonnegative part
$\text{Fl}_{k_1,\dots,k_l;n}^{\ge 0}$ as the closure of
$\text{Fl}_{k_1,\dots,k_l;n}^{>0}$. On the other hand, following Postnikov
(2007), we define $\text{Fl}_{k_1,\dots,k_l;n}^{\Delta>0}$ and
$\text{Fl}_{k_1,\dots,k_l;n}^{\Delta\ge 0}$ as the subsets of
$\text{Fl}_{k_1,\dots,k_l;n}$ where all Pl\"{u}cker coordinates are positive
and nonnegative, respectively. It follows from the definitions that Lusztig's
total positivity implies Pl\"{u}cker positivity, and it is natural to ask when
these two notions of positivity agree. Rietsch (2009) proved that they agree in
the case of the Grassmannian $\text{Fl}_{k;n}$, and Chevalier (2011) showed
that the two notions are distinct for $\text{Fl}_{1,3;4}$. We show that in
general, the two notions agree if and only if $k_1, \dots, k_l$ are consecutive
integers. We give an elementary proof of this result (including for the case of
Grassmannians) based on classical results in linear algebra and the theory of
total positivity. We also show that the cell decomposition of
$\text{Fl}_{k_1,\dots,k_l;n}^{\ge 0}$ coincides with its matroid decomposition
if and only if $k_1,\dots,k_l$ are consecutive integers, which was previously
only known for complete flag varieties, Grassmannians, and $\text{Fl}_{1,3;4}$.
Finally, we determine which notions of positivity are compatible with a natural
action of the cyclic group of order $n$ that rotates the index set.
