Higher Order Cascading Hyper Notation (HCHN) is the 11th part of HN.

Rules

An expression is of the form H_m<n,A>, where n is the base, m is the secondary base, and A is an array.

1. If there is no array, H_m<n> = m^n
2. If there is no secondary base, H<n> = H_n<n>
3. If the array's first entry is 1, delete it.
4. If the array is a sequence array, decrease it by one, then nest the whole expression into the base n times.
5. If the array is a limit array, change the array into the nth member of it's FS.

Arrays

As always, the definitions from NCHN apply.

X_1 = X.

Define [A] is first order block, and X is cascader for first order block.

Define H_X<X,A> is second order block, and X_2 is cascader for second order block. That means X_2 copies H_X<X,A> and not [A].

Define H_X_(B-1)<X_(B-1),A> (where B is sequence array) is (B-1)th order block, and X_B is cascader for (B-1)th order block. That means X_B copies H_X_(B-1)<X_(B-1),A> and not [A] or H_X_C<X_C,A> (where C can be any array that smaller than B)

Finally, the nth member of X_A's FS (where A is limit array) is X_FS(A,n).

To compare arrays, use NCHN rules, and if you need to compare X_A and X_B, compare A and B, and that is the result.