Bashicu matrix system: Difference between revisions

 

# The parent of an entry \( x \) (an entry is a natural number in the array) is the last entry \( y \) before it in the same row, such that the entry directly above \( y \) (if it exists) is an ancestor of the entry above \( x \), and \( y<x \). The ancestors of an entry \( x \) are defined recursively as the parent of \( x \) and the ancestors of the parent of \( x \).

# If \( A \) is empty, then \( A[n]=A \) for all natural numbers \( n \). Otherwise let \( C \) be the last column of \( A \), and let \( m_0 \) be maximal such that the \( m_0 \)-th element of \( C \) has a parent if such an \( m_0 \) exists, otherwise \( m_0 \) is undefined. Let \( G \) and \( B_0 \) be arrays such that \( A=G+B_0+(C) \), where \( + \) is concatenation, and the first column in \( B_0 \) contains the parent of the \( m_0 \)-th element of \( C \) if \( m_0 \) is defined, otherwise \( B_0 \) is empty.

# Say that an entry in \( B_0 \) "ascends" if it is in the first column of \( B_0 \) or has an ancestor in the first column of \( B_0 \). Define \( B_1,B_2,...,B_n \) as copies of \( B_0 \), but in each \( B_i \), each ascending entry \( x \) is increased by \( i \) times the difference between the entry in \( C \) in the same row as \( x \) and the entry in the first column of \( B_0 \) in the same row as \( x \).

# \( A[n]=G+B_0+B_1+...+B_n \), where \( + \) is again concatenation.