Using path counting arguments, we prove
\(min\{\binom{x_1+x_2+y_1+y_2}{x_1,x_2,(y_1+y_2)},\binom{(x_1+x_2+y_1+y_2)}{(x_1+x_2),y_1,y_2}\}\leq\binom{x_1+y_1}{x_1}\binom{x_1+y_2}{x_1}\binom{x_2+y_1}{x_2}\binom{x_2+y_2}{x_2}\)
This inequality, motivated by graph coloring considerations, has an interesting geometric interpretation.
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