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()

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

plt.show()