Difference between revisions of "Bird's array notation"

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(Created page with "Bird's array notation is a parallel notation to BEAF. == Linear arrays == *'''Rule 1'''. With one or two entries, we have \(\{a\} = a\), \(\{a,b\} = a^b\). *'''Rule 2'''...")
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Revision as of 23:55, 13 April 2017

Bird's array notation is a parallel notation to BEAF.

Linear arrays

  • Rule 1. With one or two entries, we have \(\{a\} = a\), \(\{a,b\} = a^b\).
  • Rule 2. If the last entry is 1, it can be removed: \(\{\#,1\} = \{\#\}\).
  • Rule 3. If the second entry is 1, the value is just the first entry: \(\{a,1 \#\} = a\).
  • Rule 4. If the third entry is 1:
    \(\{a,b,1,1,\cdots,1,1,c \#\} = \{a,a,a,a,\cdots,a,\{a,b-1,1,1,\cdots,1,1,c \#\},c-1 \#\}\)
  • Rule 5. Otherwise:
    \(\{a,b,c \#\} = \{a,\{a,b-1,c \#\},c-1 \#\}\)

Example

\begin{eqnarray*} \{3,3,1,2\} &=& \{3,3,\{3,2,1,2\},1\} \\ &=& \{3,3,\{3,3,\{3,1,1,2\},1\}\} \\ &=& \{3,3,\{3,3,3\}\} \\ &=& \{3,3,\{3,\{3,2,3\},2\}\} \\ &=& \{3,3,\{3,\{3,\{3,1,3\},2\},2\}\} \\ &=& \{3,3,\{3,\{3,3,2\},2\}\} \\ &=& \{3,3,\{3,\{3,\{3,2,2\},1\},2\}\} \\ &=& \{3,3,\{3,\{3,\{3,\{3,1,2\},1\}\},2\}\} \\ &=& \{3,3,\{3,\{3,\{3,3\}\},2\}\} \\ &=& \{3,3,\{3,7625597484987,2\}\} \\ &=& 3\uparrow^{3\uparrow\uparrow7625597484987}3