import matplotlib.pyplot as plt
import numpy as np
from matplotlib.colors import ListedColormap

#discrete color scheme
cMap = ListedColormap(['white', 'green', 'blue','red'])

#data
np.random.seed(42)
data = np.random.rand(4, 4)
fig, ax = plt.subplots()
heatmap = ax.pcolor(data, cmap=cMap)

#legend
cbar = plt.colorbar(heatmap)

cbar.ax.get_yaxis().set_ticks([])
for j, lab in enumerate(['$0$','$1$','$2$','$>3$']):
    cbar.ax.text(.5, (2 * j + 1) / 8.0, lab, ha='center', va='center')
cbar.ax.get_yaxis().labelpad = 15
cbar.ax.set_ylabel('# of contacts', rotation=270)


# put the major ticks at the middle of each cell
ax.set_xticks(np.arange(data.shape[1]) + 0.5, minor=False)
ax.set_yticks(np.arange(data.shape[0]) + 0.5, minor=False)
ax.invert_yaxis()

#labels
column_labels = list('ABCD')
row_labels = list('WXYZ')
ax.set_xticklabels(column_labels, minor=False)
ax.set_yticklabels(row_labels, minor=False)

plt.show()

You were very close. Once you have a reference to the color bar axis, you can do what ever you want to it, including putting text labels in the middle. You might want to play with the formatting to make it more visible.

Using the handle for the colorbar (in your case, the variable hcb), you can locate the colorbar handle title using the |get| function. Once you've found the handle for the colorbar title, you can directly change the title string via the |set| function. When working with figures in MATLAB, you'll often find yourself referencing graphic handles, so I recommend brushing up on them! * * In your case, you can change the colormap title with just a few lines of code! Here is an example, which you can add after your example code above, to get you started: colorTitleHandle = get(hcb,'Title'); titleString = 'A title'; set(colorTitleHandle ,'String',titleString); Hope this clarifies things a bit!

To your second question: you can use a negative labelpad value to move the label back towards the ticklabels, like this:

import numpy as np
import matplotlib.pyplot as plt

data = np.linspace(0, 10, num=256).reshape(16,16) 

cf = plt.contourf(data, levels=(0, 2.5, 5, 7.5, 10))
cb = plt.colorbar(cf)

cb.set_ticklabels([r'', 1, 2, r'', r'$10^{14}+12345678$'])
cb.set_label(r' in ', labelpad=-40, y=0.45)

plt.show()

Using the parameter y, you can additionally move the label up or down for better symmetry.

The argument of labelpad is given in points (1/72 inch). y accepts values in [0, 1], 0.0 is the lower border and 1.0 the upper.

The result: