Bowers Exploding Array Function

BEAF (Bowers' Exploding Array Function) is a notation invented by Jonathan Bowers, similiar to Chained Arrow Notation. It was created by rolling Array Notation and Extending Array Notation (both invented by Bowers) into one notation with a master set of rules.

Definitions

 * The "base" (b) is the first entry in the array.
 * The "prime" (p) is the second entry in the array.
 * The "pilot" is the first non-1 entry after the prime. It can be as early as the third entry.
 * The "copilot" is the entry immediately before the pilot. The copilot does not exist if the pilot is the first entry in its row.
 * A "structure" is a part of the array that consists of a lower-dimensional group. This could be an entry (written $$X^0$$,) a row (written $$X^1$$,) a plane ($$X^2$$), a realm ($$X^3$$), or a flune $$X^4$$, not to mention higher-dimensional structures ($$X^5$$, $$X^6$$, etc.)
 * A "previous entry" is an entry that occurs before the pilot, but is on the same row as all other previous entries. A "previous row" is a row that occurs before the pilot's row, but is on the same plane as all other previous rows. A "previous plane" is a plane that occurs before the pilot's plane, but is on the same realm as all other previous planes, etc. These are called "previous structures."
 * A "prime block" of a structure $$X^n$$ is the first $$p^n$$ block of entries in that structure.
 * The "airplane" includes the pilot, all previous entries, and the prime block of all previous structures.
 * The "passengers" are the entries in the airplane that are not the pilot or copilot.
 * The value of the array is notated $$v(A)$$, where A is the array.

Rules

 * 1) If $$p = 1$$, $$v(A) = b$$.
 * 2) If there is no pilot, $$v(A) = b^p$$.
 * 3) If neither 1 nor 2 apply, then:
 * 4) pilot decreases by 1,
 * 5) copilot takes on the value of the original array with the prime decreased by 1,
 * 6) all passengers become b,
 * 7) and the rest of the array remains unchanged.