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n! = n*(n-1)*(n-2)*...*3*2*1

n!! = (((...(((n!)!)!)...)!)!)!)! with n levels

n!!! = (((...(((n!!)!!)!!)...)!!)!!)!!)!! with n levels

n!!!...!!! = (((...(((n!!!..!!)!!!...!!)!!!...!!)...)!!!...!!)!!!...!!)!!!...!!)!!!...!! with n levels

n!2 = n!!!...!!! with n !'s

n!2! = (((...(((n!2)!2)!2)...)!2)!2)!with n levels

n!2!2 = n!2!!!...!!! with n !'s

n!= n!2!2!2 ...!2!2!with n !2's

n!! = n!n

n!!+1 = n!!!!!! ...!!!!!with n !!'s

n!(!)! = n!!*(!-1)*...*3*2*1

n!!! = n!(((...((!)!)!...)!)!)! with n !'s

n!!!! = n!(((...((!!)!!)!!...)!!)!!)!! with n !!'s

n!!2 = n!!!!...!!! with n !'s

n!!! = n!!n with n !'s

Replace n!_! into n!^!, and n!_!_! into n!^(!^!)

Then:

n!^1^2 = n!^(!^(!^(...(^(!^(!^!)))...))) exponentiated n times

n!^1^3 = n!^1^(!^(!^(...(!^(!^(!^()^2)^2)^2)...)^2)^2) with n levels

n!^1^1^2 = n!^1^(!^1^(!^1^(...(!^1^(!^1^(!^1^())))...))) with n levels

n!^^2 = n!^1^1^1^...^1^1^2 with n ^1^'s

etc.

(sorry, too lazy)

-- UNDER CONSTRUCTION --

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