Rules

Rule 1 (1 / 2 entries)

{a} = a
{a, b} = ab

Rule 2 (last entry is 1)

{a, b, c, ..., z, 1} = {a, b, c, ..., z}

Rule 3 (second entry is 1)

{a, 1, c, d, ..., z} = a

Rule 4 (third entry is 1)

{a, b, 1, ..., 1, d, e, ..., z} = {a, a, a, ..., {a, b-1, 1, ..., 1, d, e, ..., z}, d-1, e, ..., z}

The ... between the 1s represents an unbroken string of 1s - there can be any number of 1s, from one 1 (third entry alone) to a string of 1s up to the penultimate entry - it is the last 1 of this unbroken string (not necessarily the last 1 in the array) that is replaced by a copy of the entire array with its second entry reduced by 1, and all entries prior to this become an unbroken string of as. This is the only way that a fourth or subsequent entry in the array can be reduced in number (albeit by 1); if there are n 1s in the unbroken string from the third entry onwards then the (n+3)th entry (represented by d) is reduced by 1.

Rule 5 (rules 1-4 do not apply)

{a, b, c, d, ... , z} = {a, {a, b-1, c, d, ... , z}, c-1, d, ... , z}

The second entry is replaced by a copy of the entire array with its second entry reduced by 1, in order to reduce the third entry by 1.