YY sequence or Yabai Y sequence is a difference sequence system introduced by a Japanese googologist Yukito on 17/07/2020. It is intended to be a notation stronger than Y sequence. Although it has not been formalised yet, the expression \((1,2,5)\) in YY sequence is expected to correspond to the limit of Bashicu matrix system version 2.3 and the expression \((1,3)\) in Y sequence, and the expression \((1,3)\) in YY sequence corresponds to the limit of Y sequence.
In order to distinguish a sequence in YY sequence and other sequence notations, people frequently put a shorthand of the notation in front of the expression. For example, (1,3) in Y sequence is denoted by Y(1,3), (1,3) in YY sequence is denoted by YY(1,3), and so on.
As Y sequence allows ω as an entry only when we consider the limit expression Y(1,ω), the limit expression of YY sequence is denoted by YY(1,ω). It does not mean that Y sequence and YY sequence are extended to be systems of sequences of transfinite entries.
Yukito also indicated the existence of higher extensions on 19/07/2020: YYY sequence, YYYY sequence, and Yω sequence. According to Yukito, there is a sequence "Y1 sequence, Y2 sequence, Y3 sequence, …" such that Y1 sequence is Y sequence, Y2 sequence is YY sequence, Y3 sequence is YYY sequence, Y4 sequence is YYYY sequence, and the limit of Yn(1,ω) with n<ω "coincides" with Yω(1,3).
The last equality probably means that there exists a correspondence from standard expressions in Yn sequence to ordinals for any ordinals n≦ω such for which expansions give fundamental sequences and the limit of ordinals corresponding to Yn(1,ω) coincides with the ordinal corresponding to Yω(1,3). Although none of the systems has been formalised, it means that the whose system is expected to be much stronger than Y sequence, which is expected to be much stronger than BMS version 2.3.
Although there was a table of analysis among BMS, Y sequence, YY sequence, YYY sequence, and YYYY sequence which tells us how higher extension outgrows YY sequence, it has been deleted since 19/07/2020. According to Yukito, he found a bug in Yn sequence for all n≧3.
Nayuta mentioned that Yn sequence employs a similar idea to an unborn version of N primitive called NΩ.0, and some weird property of YYY(1,3,3) implies that NΩ.0 might not work as intended. Therefore NΩ.0 is canceled.
Relation to Bell numbers
A Japanese googologist koteitan discovered that the sequence YY(1,2,5,15,52,203) appearing in the expansion of YY(1,3) is an initial segment of the sequence of Bell numbers. Yukito agreed that the expansion of YY(1,3) coincides with the sequence of Bell numbers. According to Yukito, he actually knew Bell numbers but the equality is a coincidence.
- The user page of Yukito in the Japanese Googology Wiki.
- Yukito, YYvsY, Google Spreadsheet.
- The user page of koteitan in Googology Wiki.
- koteitan, Relation between YY sequence and Bell numbers.
- Bell numbers, A0000110 - OEIS.
- E. T. Bell, Exponential polynomials], Annals of Mathematics, Second Series, Vol. 35, No. 2 (1934), pp. 258--277.