User blog comment:Deedlit11/Is BEAF well-defined?/@comment-25418284-20121023211454

Intriguing stuff. I've read the Exploding Array Function page many times, and Jabe always loses me at the pentationals. Last year, I used to spend all my free time at high school doodling on a whiteboard and trying to figure this out...


 * X = {b, p (1) 2}
 * X^2 = {b, p (2) 2}
 * X^3 = {b, p (3) 2}
 * X^4 = {b, p (4) 2}
 * X^X = {b, p (0, 1) 2} (Since X solves to p, this is X^p.)
 * X^(1 + X) = {b, p (1, 1) 2}
 * X^(2 + X) = {b, p (2, 1) 2}
 * X^2X = {b, p (0, 2) 2}
 * X^(1 + 2X) = {b, p (1, 2) 2}
 * X^3X = {b, p (0, 3) 2}
 * X^(X^2) = {b, p (0, 0, 1) 2}
 * X^(1 + X^2) = {b, p (1, 0, 1) 2}
 * X^(1 + 2X + X^2) = {b, p (1, 2, 1) 2} (The first row of the dimensional separator describes a polynomial in X.)
 * X^(2X^2) = {b, p (0, 0, 2) 2}
 * X^(X^3) = {b, p (0, 0, 0, 1) 2}
 * X^X^X = {b, p ((1) 1) 2} = {b, p (0, 0, ..., 0, 1) 2} with p zeroes
 * X^(1 + X^X) = {b, p (1 (1) 1) 2}
 * X^X^(X + 1) = {b, p ((1) 0, 1) 2} (The second row describes a polynomial in X multiplied by X^X.)
 * X^X^(X + 2) = {b, p ((1) 0, 0, 1) 2}
 * X^X^2X = {b, p ((1) (1) 1) 2}
 * X^X^3X = {b, p ((1) (1) (1) 1) 2}
 * X^X^4X = {b, p ((1) (1) (1) (1) 1) 2}
 * X^X^X^2 = {b, p ((2) 1) 2} = {b, p ((1) (1) ... (1) (1) 1) 2} with p (1)'s
 * X^X^(X^2 + 1) = {b, p ((2) 0, 1) 2} (Going forward entry by entry still multiplies the exponent only by X.)
 * X^X^X^3 = {b, p ((2) (2) 1) 2}
 * X^X^X^X = {b, p ((3) 1) 2}
 * X^X^X^X^X = {b, p ((4) 1) 2}
 * X^X^X^X^X^X = {b, p ((5) 1) 2}
 * X^^n = {b, p ((n - 1) 1) 2}
 * X^^X = {b, p ((0, 1) 1) 2}
 * Now what?