This is an extension to factorial i have been working on. It can grow to be as powerful, if not more so than the original array notation of Jonathan Bowers. It starts by looking at standard factorial. This takes a number and multiplies it by everything below it, for example:

  • 6! = 6x5x4x3x2x1 = 720

This is actually a suprisingly powerful tool for making huge numbers, because when taking the factorial of a large number, there are a lot of smaller numbers, so a lot of multiplications. But we want better than this, as just factorial really is easily beaten by even the weakest of arrow notations. To achieve this, first realise that factorial uses multiplication to combine numbers, but this is not the only way of combining numbers, as you can add or power or tetrate or pentate them etc. And therefore, it is possible to have 'hyperfactorials', which use operations other than multiplying. To represent this, square brackets after the exclaimation mark indicate which operation is being used. [1] is adding, [2] is multiplying, [3] is powering, and generally [n] means \uarr^{n-2}\,\!. For example,

  • 4![4] = 4\uarr\uarr3\uarr\uarr2\uarr\uarr1 = 4\uarr\uarr3^3 = 4\uarr\uarr27 = 4^{4^{...^{4^{4}}}}\,\! with 27 4's
  • EDIT: this has been changed to [n] meaning \uarr^n, so the example should now start 4![2]

Now that's more like it! This tool can be used to generate very large numbers, and it is easy to start beating some of the weaker competition. g(1), the first term in the Graham's number sequence, can be beaten (easily) by 4![6]. In fact, in the List of googologisms, we can beat everything up to the up-arrow notation level (and some of the stuff in it) with this. The only number on the wiki which i can see on the list as being definable by this is the Zootzootplex, which equals googolplex![3]. There will be more soon, involving a much more powerful version of the notation and some named numbers defined this way. (including the challenger to array notation as promised at the start).