\(\omega\)-Y sequence is a difference sequence system introduced by a Japanese googologist Yukito in July 2021. It is intended to be much stronger than the creator's past work Y sequence, as we will explain later.
\(\omega\)-Y sequence has new difference sequences called galaxy, galaxy group(or super cluster) and above, which are the extension of Mt. Fuji for the multidimensional structure.
Y sequence has "diagonal difference sequence" which consists of the numbers picked up from the top of the Mt. Fuji and it make a new Mt.Fuji (1,4,24 on the top of the Mt.Fuji based on (1,5,33) in the figure). On the other hand, \(\omega\)-Y sequence has not only diagonal difference sequences but also galactic difference sequence(or spiral difference sequence), which consists of the numbers picked up at the bottom-left corners of Mt.Fujis (1,3,14 on the (1,5,33)-based galaxy).
The galactic difference sequence (1,3,14) has an own difference sequence (2,11), and it makes a new galaxy based on the difference sequence (2,11) (in the right side of (1,5,33)). The galaxy difference sequence of the (2,1)-based galaxy is (2,7) and it makes another (5)-based galaxy. Now we got the 3 galaxies and it makes a new galaxy group based on (1,3) (in the bottom side of the (1,5,33)) by the difference sequence of the galaxy group difference sequence (1,2,5).
In this way, \(\omega\)-Y sequence can continue it infinitely, sequences, Mt.Fujis, galaxies, galaxy groups, 6th structures, 7th structures, 8th structures, ... and so on. The n-th structures consist of n-th dimensional structure. Especially, \(\omega\)-Y(1,n) consist of n-th dimensional simplex.
The difference of the structure of \(\omega\)-Y sequence to the one of Y sequence is that Y sequence makes a next Mt.Fuji by putting the diagonal sequence itself into the base of the new Mt.Fuji and \(\omega\)-Y sequence makes the next Mt.Fuji (or the next structure) by putting the "difference sequence" of the diagonal sequence. For example, the basic Mt.Fuji of Y(1,4) is ((1,4),(3)) and the diagonal sequence is (1,3), and the next Mt.Fuji becomes (1,3) itself in Y-sequence. On the other hand in \(\omega\)-Y sequence, the diagonal sequence of ((1,4),(3)) is (1,3) and the next Mt.Fuji becomes (2) which is the difference sequence of (1,3).
Yukito said he had tried to make Y^n sequences whose limit Y^n(1,\(\omega\)) is equal to Y^n+1(1,3) at first time, however, it had critical bug and he changed the way. While Y^ns unlock the new dimensions, \(\omega\)-Y sequence unlocks the all dimensions at once.
The expansion rule of \(\omega\)-Y sequence to solve an expression like \(\omega\)-Y(s)[n] for a valid sequence \(s\) and a natural number \(n\) is defined by the function expand(s,n,stringify) on the code by Naruyoko.
Yukito said that he named f^2000(1) using f(n)=\(\omega\)-Y(1,\(\omega\))[n] as \(\omega\)-Y sequence number.
User:Naruyoko made a program MEGA whY mountain to draw the structure of \(\omega\)-sequence. It draws a multi-dimensional Mt. Fuji with a structure that looks like a pile of Mt.Fuji on top of Mt.Fuji, one after another. Yukito calls it Mega-Mt.Fuji.