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ESMN (Extended Steinhaus-Moser Notation)

Rules

1) a•1 = a^a

2) a•(n+1) = ((...((a•n)•n)...)•n)•n with a brackets

3) [a, b] = a•b

4) [a, b, c] = a••...••b with c dots

5) [a, b, c] = [a, [a, [ ... [a, [a, b, c-1], c-1] ... ], c-1], c-1]

6) [a, b, c, d] = [a, b, [a, b, c, d-1], d-1]

7) [#, a, b] = [#, [#, a, b-1], b-1] where # denotes a vaild array

8) [a, b, #, 1, d] with c 1's = [a, b, #, [a, b, ß, d-1], d-1] where # denotes an chain of 1's of length c and ß denotes an array of a of length c

9) [a, 1, #] = [a] = a•1 = a^a 

10) [#, 1] = [#] where # denotes a valid array

Examples

3•3 = ((3•2)•2)•2 (Rule 2) = ((((3•1)•1)•1)•2)•2  (Ditto) ≈ ((4.4*10^38)•2)•2 ≈ (((...((4.4*10^38)•1)•1)...)•1)•1)•2

[3, 3, 1, 2] = [3, 3, [3, 3, 1, 1], 1] (Rule 5) = [3, 3, [3, 3]] = [3, 3, 3•3] = quite large

Comparisons

Since the first rule is a rehash of Steinhaus-Moser notation n•n ≈ \(f_\omega[n]\)

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