User blog:VoidSansXD/Factorial Notation

Factorial notation

Part 1:  Up to 4 factorial entries.
Before we get into the rules, let's adress some things:  1.)  If someone could get me the growth rate for this, that'd be helpful.  2.)  If the entry arrays look like BEAF arrays, that's because the arrays are inspired by BEAF arrays.

Rules:
1.)  n! = n*(n-1)*(n-2)..........*4*3*2*1

2.)  n!! = (n*(n-1)*(n-2)..........*4*3*2*1-1)*(n*(n-1)*(n-2)..........*4*3*2*1-2)*(n*(n-1)*(n-2)..........*4*3*2*1)*(n*(n-1)*(n-3)..........*4*3*2*1-3)*(n*(n-1)*(n-2)..........*4*3*2*1-4)..........*3*2*1, or (n*(n-1)*(n-2)..........*4*3*2*1)!

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

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

5.)  n[3, x] = n(n(n(x)))

6.)  n[y, x] = n(n(n(n......n(n(n(x)))))....)))))) with y n's.

7.)  n[y, 1, 2] = n[y, y[y, y[y, y[y, y...y, y[y, y]]]]]]...]]]]]]]] with y nests.

8.)  n[y, 2, 2] = n[y, y[y, y[y, y[y, y...y, y[y, y]]]]]]...]]]]]]]] with (n[y, y[y, y[y, y[y, y...y, y[y, y]]]]]]...]]]]]]]] with y nests) nests.

9.)  n[y, 1, 3] = n[y, y[y, y[y, y[y, y...y, y[y, y]]]]]]...]]]]]]]] with (n[y, y[y, y[y, y[y, y...y, y[y, y]]]]]]...]]]]]]]] with (n[y, y[y, y[y, y[y, y...y, y[y, y]]]]]]...]]]]]]]] with (...............n[y, y[y, y[y, y[y, y...y, y[y, y]]]]]]...]]]]]]]] with (n[y, y[y, y[y, y[y, y...y, y[y, y]]]]]]...]]]]]]]] with y nests.) nests.) nests.)............nests)nests) nests, with the amount of "with (x) nests" being y.

10.)  n[y, 2, 3] = n[y, y[y, y[y, y[y, y...y, y[y, y]]]]]]...]]]]]]]] with (n[y, y[y, y[y, y[y, y...y, y[y, y]]]]]]...]]]]]]]] with (n[y, y[y, y[y, y[y, y...y, y[y, y]]]]]]...]]]]]]]] with (...............n[y, y[y, y[y, y[y, y...y, y[y, y]]]]]]...]]]]]]]] with (n[y, y[y, y[y, y[y, y...y, y[y, y]]]]]]...]]]]]]]] with y nests.) nests.) nests.)............nests)nests) nests, with the amount of "with (x) nests" being n[y, y[y, y[y, y[y, y...y, y[y, y]]]]]]...]]]]]]]] with (n[y, y[y, y[y, y[y, y...y, y[y, y]]]]]]...]]]]]]]] with (n[y, y[y, y[y, y[y, y...y, y[y, y]]]]]]...]]]]]]]] with (...............n[y, y[y, y[y, y[y, y...y, y[y, y]]]]]]...]]]]]]]] with (n[y, y[y, y[y, y[y, y...y, y[y, y]]]]]]...]]]]]]]] with y nests.) nests.) nests.)............nests)nests) nests, with the amount of "with (x) nests" being y.  (Two sets)

11.)  n[y, 1, 4] = n[y, y, 3]

12.)  Repeats

13.)  n[y, 1, 1, 2] = n[y, 1, y[y, 1, y[y, 1, y[y, 1, y[........y, 1, y[y, 1, y[y, 1, y]]]....]]]] with y nests.

Thanks for reading this thing lol