Color Function not correctly mapping to range

Question

Observe the following code:

nd[y_, a_, b_, n_] := (Binomial[b - a + 1, y]*Sum[((-1)^i)*Binomial[y, i]*((y - i)/(b - a + 1))^n, {i, 0, y}]);
mycolorfun = Function[Blend[{Black, Purple, Blue, Green, Yellow, Orange, Red, White}, Rescale[#, {0, 1}]]]
Manipulate[
 ArrayPlot[Table[nd[y, 1, K, n], {K, 1, 25}, {n, 1, 25}], 
  ColorFunction -> mycolorfun, DataReversed -> True, Frame -> True, 
  FrameTicks -> Automatic, 
  FrameLabel -> {Rotate["K", -90 Degree], "N"}], {y, 1, 15, 1}]

Notice how when the slider is moved up to y=15, there is a white square that appears at N = 25 and K = 21:

My goal is for the color function to map the individual value of the output of my function to a color as dependent on its closeness to 0 or 1 (since my function is a probability density function, this makes sense).

However, look what happens when you check the actual value of the function for N = 25 and b=K=21:

nd[15, 1, 21, 25.0]
(out) 0.268495

I get 26%! 26% should be mapping to a color somewhere around Purple/Blue, but it's getting mapped all the way up to the color white! White is what's supposed to be mapped to when the output of my function is really close to 1! Clearly something is amiss here, and I cannot figure out what it is. Can somebody please explain what is going on and what I need to do to get the desired color output?

Answer 1

Short answer: Add ColorFunctionScaling -> False.

From my understanding, the ColorFunction option takes all the arguments supplied to it and scales them so that they go from 0 to 1.

So, if you look at the max value in the table at y = 15,

N@Max[Table[nd[15, 1, K, n], {K, 1, 25}, {n, 1, 25}]]

you can see that it's 0.268495. ColorFunction will take this as its "1", which, using mycolorfun, corresponds to a reddish color.

I think most of the other non-black colors in your plot are, in that sense, "wrong" as well, but correct me if that seems off to you:

Anyway, hopefully that helps!