Background:Consider the usual two-party model of communication complexity where Alice and Bob are given $n$-bit strings $x$ and $y$ and have to compute some Boolean function $f(x,y)$, where $f:\{0,1\}^n \times \{0,1\}^n \to \{0,1\}$.We define the following quantities:$D(f)$ (the deterministic communication complexity of $f$): The minimum number of bits that Alice and Bob need to communicate to compute $f(x,y)$ deterministically.$Pn(f)$ (the partition number of $f$): The logarithm (base 2) of the smallest number of monochromatic rectangles in a partition (or a disjoint cover) of $\{0,1\}^n \times \{0,1\}^n$.A monochromatic rectangle in $\{0,1\}^n \times \{0,1\}^n$ is a subset $R \times C$ such that $f$ takes the same value (i.e., is monochromatic) on all elements of $R \times C$.Also note that the partition number is different from the protocol partition number, which was the subject of this question.See the text by Kushilevitz and Nisan for more information. In their notation, what I have defined to be $Pn(f)$ is $\log_2 C^D(f)$.Note: These definitions are easily generalized to non-Boolean functions $f$, where the output of $f$ is some larger set.Known results:It is known that $Pn(f)$ is a lower bound on $D(f)$, i.e., for all (Boolean or non-Boolean) $f$, $Pn(f) \leq D(f)$.Indeed, most lower bound techniques (or perhaps all?) for $D(f)$ actually lower bound $Pn(f)$. (Can anyone confirm that this is true of all lower bound techniques?)It is also known that this bound is at most quadratically loose (for Boolean or non-Boolean functions), i.e., $D(f) \leq (Pn(f))^2$. To summarize, we know the following:$Pn(f) \leq D(f) \leq (Pn(f))^2$It is conjectured that $Pn(f) = \Theta(D(f))$. (This is open problem 2.10 in the text by text by Kushilevitz and Nisan.)  However, to the best of my knowledge, the best known separation between these two for Boolean functions is only by a multiplicative factor of 2, as shown in The Linear-Array Conjecture in Communication Complexity Is False by Eyal Kushilevitz,    Nathan Linial, and Rafail Ostrovsky.More precisely, they exhibit an infinite family of Boolean functions $f$, such that $D(f) \geq (2 - o(1)) Pn(f)$.Question:What is the best known separation between $Pn(f)$ and $D(f)$ for non-Boolean functions? Is it still the factor-2 separation referenced above?Added in v2: Since I haven't received an answer in a week, I'm also happy to hear partial answers, conjectures, hearsay, anecdotal evidence, etc. 
Deterministic communication complexity vs partition number
lower bounds;communication complexity
This question has just been resolved! As I mentioned, it was known that $Pn(f) \leq D(f) \leq (Pn(f))^2$,but it was a major open problem to show that either $Pn(f) = \Theta(D(f))$ or that there exists a function for which $Pn(f) = o(D(f))$. A few days ago this was resolved by Mika Gs, Toniann Pitassi, Thomas Watson (http://eccc.hpi-web.de/report/2015/050/). They show that there exists a function $f$ which satisfies$Pn(f) = \tilde{O}((D(f))^{2/3})$.They also show an optimal result for the one-sided version of $Pn(f)$, which I'll denote by $Pn_1(f)$, where you only need to cover the 1-inputs with rectangles. $Pn_1(f)$ also satisfies  $Pn_1(f) \leq D(f) \leq (Pn_1(f))^2$,and they show that this is the best possible relation between the two measures, since they exhibit a function $f$ which satisfies$Pn_1(f) = \tilde{O}((D(f))^{1/2})$.