Down-arrow notation

Down-arrow notation is an left-associative extension for addition, multiplication and exponentiation. Formally it is defined as follows:

\[a \downarrow^1 b = a^b \text{ (if n=1)}\]

\[a \downarrow^n 1 = a \text{ (if b=1)}\]

\[a \uparrow^{n + 1} (b + 1) = (a \uparrow^{n +1} b) \uparrow^n a \text{ (otherwise)}\]