Bird's array notation is a parallel notation to [BEAF](BEAF.md "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: \(\{a,b,\underbrace{1,1,\cdots,1,1}_n,c \\#\} = \{\underbrace{a,a,a,a,\cdots,a}_{n+1},\{a,b-1,\underbrace{1,1,\cdots,1,1}_n,c \\#\},c-1 \\#\}\) - **Rule 5**. Otherwise: \(\{a,b,c \\#\} = \{a,\{a,b-1,c \\#\},c-1 \\#\}\) Limit: \(\{n,n\[2\]2\}\) has growth rate \(\omega^\omega\) ## 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 \end{eqnarray*} # Multidimentional arrays - **Rule M1**. If there are only two entries, \(\{a, b\} = a^b\). - **Rule M2**. If \(m < n\), we have \(\{\\# \[m\] 1 \[n\] \\#^*\} = \{\\# \[n\] \\#^*\}\). (This also removes ones from the end of an array.) - **Rule M3**. If the second entry is 1, we have \(\{a,1 \\#\} = a\). - **Rule M4**. If there is a non-zero entry immediately after batch of unfilled separators: \(\{a,b \[m_1\] 1 \[m_2\] \cdots 1 \[m_x\] c \\#\} = \{a \langle m_1-1 \rangle b \[m_1\] a \langle m_2-1 \rangle b \[m_2\] \cdots a \langle m_x-1 \rangle b \[m_x\] (c-1) \\#\}\) - **Rule M5**. If there is a non-zero entry after batch of unfilled separators and a 1. \(\{a,b \[m_1\] 1 \[m_2\] \cdots 1 \[m_x\] 1,c \\#\} = \{a \langle m_1-1 \rangle b \[m_1\] a \langle m_2-1 \rangle b \[m_2\] \cdots a \langle m_x-1 \rangle b \[m_x\] \{a,b-1 \[m_1\] 1 \[m_2\] \cdots 1 \[m_x\] 1,c \\#\},c-1 \\#\}\) - **Rule M6**. Rules M1-M5 don't apply. \(\{a,b,c \\#\} = \{a,\{a,b-1,c \\#\},c-1 \\#\}\) - **Rule A1**. If \(c = 0\), we have \(\textrm\` a \langle 0 \rangle b = a \textrm'\). - **Rule A2**. If \(b = 1\), we have \(\textrm\` a \langle c \rangle 1 = a \textrm'\). - **Rule A3**. Otherwise, \(\textrm\` a \langle c \rangle b \textrm' = \textrm\` \underbrace{a \langle c-1 \rangle b \[c\] \cdots \[c\] a \langle c-1 \rangle b}_b \textrm'\). Limit: \(\{n,n\[1,2\]2\}\) has growth rate \(\omega^{\omega^\omega}\) ## Example \begin{eqnarray*} \{3,2\[3\]2\} &=& \{3 \langle 2 \rangle 2\[3\]1\} \\ &=& \{3 \langle 1 \rangle 2\[2\]3 \langle 1 \rangle 2\} \\ &=& \{3,3\[2\]3,3\} \\ &=& \{3,3,3\[2\]2,3\} \\ &=& \{3,\{3,2,3\[2\]2,3\},2\[2\]2,3\} \\ &=& \{3,\{3,3,2\[2\]2,3\},2\[2\]2,3\} \\ &=& \{3,\{3,\{3,2,2\[2\]2,3\}\[2\]2,3\},2\[2\]2,3\} \\ &=& \{3,\{3,\{3,3\[2\]2,3\}\[2\]2,3\},2\[2\]2,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3\[2\]1,3\}\[2\]1,3\},2\[2\]1,3\}\[2\]2,3\},2\[2\]2,3\} \end{eqnarray*} # Hyperdimentional arrays - **Rule M1**. If there are only two entries, \(\{a, b\} = a^b\). - **Rule M2**. If \(m < n\), we have \(\{\\# \[m\] 1 \[n\] \\#^*\} = \{\\# \[n\] \\#^*\}\). (This also removes ones from the end of an array.) - **Rule M3**. If the second entry is 1, we have \(\{a,1 \\#\} = a\). - **Rule M4**. If there is a non-zero entry immediately after batch of unfilled separators: \(\{a,b \[m_1 \\#_1\] 1 \[m_2 \\#_2\] \cdots 1 \[m_x \\#_x\] c \\#\} = \{a \langle m_1-1 \\#_1 \rangle b \[m_1 \\#_1\] a \langle m_2-1 \\#_2 \rangle b \[m_2 \\#_2\] \cdots a \langle m_x-1 \\#_x \rangle b \[m_x \\#_x\] (c-1) \\#\}\) - **Rule M5**. If there is a non-zero entry after batch of unfilled separators and a 1. \(\{a,b \[m_1 \\#_1\] 1 \[m_2 \\#_2\] \cdots 1 \[m_x \\#_x\] 1,c \\#\}\) \(= \{a \langle m_1-1 \\#_1 \rangle b \[m_1 \\#_1\] a \langle m_2-1 \\#_2 \rangle b \[m_2 \\#_2\] \cdots a \langle m_x-1 \\#_x \rangle b \[m_x \\#_x\] \{a,b-1 \[m_1 \\#_1\] 1 \[m_2 \\#_2\] \cdots 1 \[m_x \\#_x\] 1,c \\#\},c-1 \\#\}\) - **Rule M6**. Rules M1-M5 don't apply. \(\{a,b,c \\#\} = \{a,\{a,b-1,c \\#\},c-1 \\#\}\) - **Rule A1**. If \(c = 0\), we have \(\textrm\` a \langle 0 \rangle b = a \textrm'\). - **Rule A2**. If \(b = 1\), we have \(\textrm\` a \langle A \rangle 1 = a \textrm'\). - **Rule A3**. If the first entry in the angle brackets is zero, and there exists a non-zero entry after it: \(\textrm\` a \langle 0,\underbrace{1,1,\cdots,1,1}_n,c \\# \rangle b \textrm' = \textrm\` a \langle \underbrace{b,b,b,\cdots,b,b}_{n+1},c-1 \\# \rangle b \textrm'\) - **Rule A4**. Otherwise, \(\textrm\` a \langle c \\# \rangle b \textrm' = \textrm\` \underbrace{a \langle c-1 \\# \rangle b \[c \\#\] \cdots \[c \\#\] a \langle c-1 \\# \rangle b}_b \textrm'\). Limit: \(\{n,n\[1\[2\]2\]2\}\) has growth rate \(^4 \omega\) ## Example \begin{eqnarray*} \{3,2\[1,1,2\]2\} &=& \{3 \langle 0,1,2 \rangle 2\} \\ &=& \{3 \langle 2,2 \rangle 2\} \\ &=& \{3 \langle 1,2 \rangle 2\[2,2\]3 \langle 1,2 \rangle 2\} \\ &=& \{3,3\[2\]3,3\[1,2\]3,3\[2\]3,3\[2,2\]3,3\[2\]3,3\[1,2\]3,3\[2\]3,3\} \end{eqnarray*} # Nested arrays Main rules will remain the same forever. - **Rule A1**. If \(c = 0\), we have \(\textrm\` a \langle 0 \rangle b = a \textrm'\). - **Rule A2**. If \(b = 1\), we have \(\textrm\` a \langle A \rangle 1 = a \textrm'\). - **Rule A3**. If \(\[A\] < \[B\]\), \(\textrm\` a \langle \\# \[A\] 1 \[B\] \\#^* \rangle b \textrm' = \textrm\` a \langle \\# \[B\] \\#^* \rangle b \textrm'\). - **Rule A4**. If the first entry in the angle brackets is zero, and there exists a non-zero entry after it: \(\textrm\` a \langle 0 \[x_1 \\#_1\] 1 \[x_2 \\#_2\] \cdots 1 \[x_n \\#_n\] c \\# \rangle b \textrm' = \textrm\` a \langle b \langle x_1-1 \\#_1 \rangle b \[x_1 \\#_1\] b \langle x_2-1 \\#_2 \rangle b \[x_2 \\#_2\] \cdots b \langle x_n-1 \\#_n \rangle b \[x_n \\#_n\] c-1 \\# \rangle b \textrm'\) - **Rule A5**. Otherwise, \(\textrm\` a \langle c \\# \rangle b \textrm' = \textrm\` \underbrace{a \langle c-1 \\# \rangle b \[c \\#\] \cdots \[c \\#\] a \langle c-1 \\# \rangle b}_b \textrm'\). Limit: \(\{n,n\[1\backslash2\]2\}\) has growth rate \(\varepsilon_0\) ## Example \begin{eqnarray*} \{3,2\[1\[2\]2\]2\} &=& \{3 \langle 0\[2\]2 \rangle 2\} \\ &=& \{3 \langle 2 \langle 1 \rangle 2 \rangle 2\} \\ &=& \{3 \langle 2,2 \rangle 2\} \\ &=& \{3,3\[2\]3,3\[1,2\]3,3\[2\]3,3\[2,2\]3,3\[2\]3,3\[1,2\]3,3\[2\]3,3\} \end{eqnarray*} # Hyper-Nested arrays - **Rule A1**. If \(c = 0\), we have \(\textrm\` a \langle 0 \rangle b \textrm' = \textrm\` a \textrm'\) and \(\textrm\` a \langle 0 \rangle \backslash b \textrm' = \textrm\` a \textrm'\). - **Rule A2**. If \(b = 1\), we have \(\textrm\` a \langle A \rangle 1 \textrm' = \textrm\` a \textrm'\) and \(\textrm\` a \langle A \rangle \backslash 1 \textrm' = \textrm\` a \textrm'\). - **Rule A3**. If \(\[A\] < \[B\]\), \(\textrm\` a \langle \\# \[A\] 1 \[B\] \\#^* \rangle b \textrm' = \textrm\` a \langle \\# \[B\] \\#^* \rangle b \textrm'\) and \(\textrm\` a \langle \\# \[A\] 1 \[B\] \\#^* \rangle \backslash b \textrm' = \textrm\` a \langle \\# \[B\] \\#^* \rangle \backslash b \textrm'\). - **Rule A4**. If the first entry in the angle brackets is zero, and there exists a non-zero entry after it: \(\textrm\` a \langle 0 \[x_1 \\#_1\]\backslash 1 \[x_2 \\#_2\]\backslash \cdots 1 \[x_m \\#_m\]\backslash 1 \[y_1 \\#^*_1\] 1 \[y_2 \\#^*_2\] \cdots \[y_n \\#^*_n\] c \\# \rangle b \textrm'\) \(= \textrm\` a \langle b \langle x_1-1 \\#_1 \rangle \backslash b \[x_1 \\#_1\]\backslash b \langle x_2-1 \\#_2 \rangle \backslash b \[x_2 \\#_2\]\backslash \cdots b \langle x_n-1 \\#_n \rangle \backslash b \[x_n \\#_n\]\backslash b \langle y_1-1 \\#^*_1 \rangle b \[y_1 \\#^*_1\] b \langle y_2-1 \\#^*_2 \rangle b \[y_2 \\#^*_2\] \cdots b \langle y_n-1 \\#^*_n \rangle b \[y_n \\#^*_n\] c-1 \\# \rangle b \textrm'\). \(\textrm\` a \langle 0 \[x_1 \\#_1\]\backslash 1 \[x_2 \\#_2\]\backslash \cdots 1 \[x_m \\#_m\]\backslash 1 \[y_1 \\#^*_1\] 1 \[y_2 \\#^*_2\] \cdots \[y_n \\#^*_n\] c \\# \rangle \backslash b \textrm'\) \(= \textrm\` a \langle b \langle x_1-1 \\#_1 \rangle \backslash b \[x_1 \\#_1\]\backslash b \langle x_2-1 \\#_2 \rangle \backslash b \[x_2 \\#_2\]\backslash \cdots b \langle x_n-1 \\#_n \rangle \backslash b \[x_n \\#_n\]\backslash b \langle y_1-1 \\#^*_1 \rangle b \[y_1 \\#^*_1\] b \langle y_2-1 \\#^*_2 \rangle b \[y_2 \\#^*_2\] \cdots b \langle y_n-1 \\#^*_n \rangle b \[y_n \\#^*_n\] c-1 \\# \rangle \backslash b \textrm'\). - **Rule A5**. First non-zero entry is prior to a single backslash: \(\textrm\` a \langle 0 \[x_1 \\#_1\]\backslash 1 \[x_2 \\#_2\]\backslash \cdots 1 \[x_n \\#_n\]\backslash 1 \backslash c \\# \rangle b\textrm'\) \(= \textrm\`a \langle b \langle A_1' \rangle\backslash b \[A_1\]\backslash b \langle A_2' \rangle\backslash b \[A_2\]\backslash \cdots b \langle A_n' \rangle\backslash b \[A_n\]\backslash R_b \backslash c-1 \\# \rangle b\textrm'\); \(\textrm\` a \langle 0 \[x_1 \\#_1\]\backslash 1 \[x_2 \\#_2\]\backslash \cdots 1 \[x_n \\#_n\]\backslash 1 \backslash c \\# \rangle \backslash b\textrm'\) \(= \textrm\`a \langle b \langle A_1' \rangle\backslash b \[A_1\]\backslash b \langle A_2' \rangle\backslash b \[A_2\]\backslash \cdots b \langle A_n' \rangle\backslash b \[A_n\]\backslash R_b \backslash c-1 \\# \rangle \backslash b\textrm'\). - **Rule A6**. Otherwise, \(\textrm\` a \langle c \\# \rangle b \textrm' = \textrm\` \underbrace{a \langle c-1 \\# \rangle b \[c \\#\] \cdots \[c \\#\] a \langle c-1 \\# \rangle b}_b \textrm'\) and \(\textrm\` a \langle c \\# \rangle \backslash b \textrm' = \textrm\` \underbrace{a \langle c-1 \\# \rangle \backslash b \[c \\#\]\backslash \cdots \[c \\#\]\backslash a \langle c-1 \\# \rangle \backslash b}_b \textrm'\). Limit: \(\{n,n\[1/2\]2\}\) has growth rate \(\Gamma_0\). ## Examples \begin{eqnarray*} \{3 \langle 0 \backslash 2 \rangle 2\} &=& \{3 \langle R_2 \rangle 2\} \\ &=& \{3 \langle 2 \rangle 2\} \\ &=& \{3,3\[2\]3,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3\[2\]\{3,\{3,\{3,\{3,\{3,3,3\[2\]\{3,\{\underbrace{3,\cdots,3}_{3\uparrow\uparrow\uparrow3}\},2\[2\]2\}\}\[2\]1,2\},2\[2\]1,2\}\[2\]2,3\}\[2\]2,2\},2\}\[2\]1,3\},2\[2\]1,3\} \end{eqnarray*} ------------------------------------------------------------------------ \begin{eqnarray*} \{3 \langle 0 \[2\]\backslash 2 \rangle 2\} &=& \{3 \langle 2 \backslash 2 \rangle 2\} \\ &=& \{3,3\[2\]3,3\[1 \backslash 2\]3,3\[2\]3,3\[2 \backslash 2\]3,3\[2\]3,3\[1 \backslash 2\]3,3\[2\]3,3\} \end{eqnarray*} # Nested Hyper-Nested arrays This part consists of 2 parts. ## Negations - **Rule A1**. \(\textrm\` a \langle 0 \neg \\# \rangle b \textrm' = \textrm\` a \textrm'\). - **Rule A2**. If \(b = 1\), we have \(\textrm\` a \langle A \neg B \rangle 1 \textrm' = \textrm\` a \textrm'\). - **Rule A3**. If \(\[A\] < \[B\]\), \(\textrm\` a \langle \\# \[A\] 1 \[B\] \\#^* \neg \% \rangle b \textrm' = \textrm\` a \langle \\# \[B\] \\#^* \neg \% \rangle b \textrm'\). - **Rule A4**. If the first entry in the angle brackets is zero: \(\textrm\` a \langle 0 \[A_{1,1}\] 1 \[A_{1,2}\] \cdots \[A_{1,p_1}\] c_1 \\#_1 \neg \\#^* \rangle b \textrm' = \textrm\` a \langle S_1 \neg \\#^* \rangle b \textrm'\). Set i to 1. - *Rule A4a*. \(\[A_{i,p_i}\] = \[1 \neg 1 \[A_{i+1,1}\] 1 \[A_{i+1,2}\] \cdots \[A_{i+1,p_{i+1}}\] c_i \\#_i\]\): \(S_i = \textrm\` b \langle A_{i,1}' \rangle b \[A_{i,1}\] b \langle A_{i,2}' \rangle b \[A_{i,2}\] \cdots \[A_{i,p_i-1}\] b \langle b \neg S_{i+1} \rangle b \[A_{i,p_i}\] c_i \\#_i \textrm'\). Increase i by 1 and follow Rules A4a\~d again. - *Rule A4b*. \(\[A_{i,p_i}\] = \backslash\): \(S_i = \textrm\` R_b \textrm'\), \(R_n = \textrm\` b \langle A_{i,1}' \rangle b \[A_{i,1}\] b \langle A_{i,2}' \rangle b \[A_{i,2}\] \cdots \[A_{i,p_i-1}\] b \langle R_{n-1} \rangle b \[A_{i,p_i}\] c_i-1 \\#_i \textrm'\), \(R_1 = \textrm\` 0 \textrm'\). - *Rule A4c*. Rule A4b don't apply, \(\[A_{i,p_i}\] = \[1 \neg k \\#_{i+1}\] (k>1)\): \(S_i = \textrm\` b \langle A_{i,1}' \rangle b \[A_{i,1}\] b \langle A_{i,2}' \rangle b \[A_{i,2}\] \cdots \[A_{i,p_i-1}\] b \langle R_b \rangle b \[A_{i,p_i}\] c_i-1 \\#_i \textrm'\), \(R_n = \textrm\` b \langle A_{i,1}' \rangle b \[A_{i,1}\] b \langle A_{i,2}' \rangle b \[A_{i,2}\] \cdots \[A_{i,p_i-1}\] b \langle R_{n-1} \rangle b \[A_{i,p_i}\] c_i-1 \\#_i \neg k-1 \\#_{i+1} \textrm'\), \(R_1 = \textrm\` 0 \textrm'\). - *Rule A4d*. Otherwise: \(S_i = \textrm\` b \langle A_{i,1}' \rangle b \[A_{i,1}\] b \langle A_{i,2}' \rangle b \[A_{i,2}\] \cdots \[A_{i,p_i-1}\] b \langle A_{i,p_i}' \rangle b \[A_{i,p_i}\] c_i-1 \\#_i \textrm'\). - **Rule A5**. Otherwise, \(\textrm\` a \langle c \\# \neg \\#^* \rangle b \textrm' = \textrm\` \underbrace{a \langle c-1 \\# \neg \\#^* \rangle b \[c \\#\] \cdots \[c \\#\] a \langle c-1 \\# \neg \\#^* \rangle b}_b \textrm'\). Note: \(\\#, \\#^*, and \%\) does not contain \(\neg\)s. Limit: \(\{n,n\[1\[1\neg1\neg2\]2\]2\}\) has growth rate \(\theta(\Omega^\Omega)\). ### Examples \begin{eqnarray*} \{3 \langle 0 \[1 \neg 1 \backslash 2\] 2 \rangle 3\} &=& \{3 \langle 3 \langle 3 \neg 3,3\[2\]3,3\[3\]3,3\[2\]3,3 \rangle 3 \rangle 3\} \end{eqnarray*} ------------------------------------------------------------------------ \begin{eqnarray*} \{3 \langle 0 \[1 \neg 4\] 2 \rangle 3\} &=& \{3 \langle 3 \langle 3 \langle 3 \neg 3 \rangle 3 \neg 3 \rangle 3 \rangle 3\} \end{eqnarray*} ## Hierarchal backslashes - **Rule A1**. \(\textrm\` a \langle 0 \backslash_n \\# \rangle b \textrm' = \textrm\` a \textrm', n>1\). - **Rule A2**. If \(b = 1\), we have \(\textrm\` a \langle \\# \rangle 1 \textrm' = \textrm\` a \textrm'\). - **Rule A3**. If \(\[A\] < \[B\]\), \(\textrm\` a \langle \\# \[A\] 1 \[B\] \\#^* \rangle b \textrm' = \textrm\` a \langle \\# \[B\] \\#^* \neg \% \rangle b \textrm'\). - **Rule A4**. If the first entry in the angle brackets is zero: \(\textrm\` a \langle 0 \[A_{1,1,1}\] 1 \[A_{1,1,2}\] \cdots \[A_{1,1,p_{1,1}}\] c_{1,1} \\#_{1,1} \backslash_n \\#^* \rangle b \textrm' = \textrm\` a \langle S_{1,1} \backslash_n \\#^* \rangle b \textrm', n>1\). Set i and j to 1. - *Rule A4a*. \(\[A_{i,1,p_{i,1}}\] = \backslash\): \(S_{i,1} = \textrm\` R_b \textrm'\), \(R_n = \textrm\` b \langle A_{i,1,1}' \rangle b \[A_{i,1,1}\] b \langle A_{i,1,2}' \rangle b \[A_{i,1,2}\] \cdots \[A_{i,1,p_{i,1}-1}\] b \langle R_{n-1} \rangle b \[A_{i,1,p_{i,1}}\] c_{i,1}-1 \\#_{i,1} \textrm'\), \(R_1 = \textrm\` 0 \textrm'\). - *Rule A4b*. \(\[A_{i,j,p_{i,1}}\] = \backslash_j, j>1\): \(S_{i,j} = \textrm\` R_{b,j-1} \textrm'\), \(R_{n,j-1}\) \(= \textrm\` b \langle A_{i,j,1}' \rangle b \[A_{i,j,1}\] b \langle A_{i,j,2}' \rangle b \[A_{i,j,2}\] \cdots b \langle A_{i,j,p_{i,j}-1}' \rangle b \[A_{i,j,p_{i,j}-1}\] b \langle A_{i,1,1}' \rangle b \[A_{i,1,1}\] b \langle A_{i,1,2}' \rangle b \[A_{i,1,2}\] \cdots b \langle A_{i,1,p_{i,1}-1}' \rangle b \[A_{i,1,p_{i,1}-1}\] b \langle R_{n-1,1} \rangle b \[A_{i,1,p_{i,1}}\] c_{i,1}-1 \\#_{i,1} \backslash_j c_{i,j}-1 \\#_{i,j} \textrm'\), \(R_{n,k} = \textrm\` b \langle A_{i,k+1,1}' \rangle b \[A_{i,k+1,1}\] b \langle A_{i,k+1,2}' \rangle b \[A_{i,k+1,2}\] \cdots b \langle A_{i,k+1,p_{i,k+1}-1}' \rangle b \[A_{i,k+1,p_{i,k+1}-1}\] b \langle R_{n,k+1} \rangle b \[A_{i,k+1,p_{i,k+1}}\] c_{i,k+1}-1 \\#_{i,k+1} \textrm'\) \(R_{1,1} = \textrm\` 0 \textrm'\). - *Rule A4e*. Otherwise: \(S_i = \textrm\` b \langle A_{i,1,1}' \rangle b \[A_{i,1,1}\] b \langle A_{i,1,2}' \rangle b \[A_{i,1,2}\] \cdots \[A_{i,1,p_{i,1}-1}\] b \langle A_{i,1,p_{i,1}}' \rangle b \[A_{i,1,p_{i,1}}\] c_{i,1}-1 \\#_{i,1} \textrm'\). - **Rule A5**. Otherwise, \(\textrm\` a \langle c \\# \rangle b \textrm' = \textrm\` \underbrace{a \langle c-1 \\# \rangle b \[c \\#\] \cdots \[c \\#\] a \langle c-1 \\# \rangle b}_b \textrm'\). # Hierarchal Hyper-Nested arrays # Nested Hierarchal Hyper-Nested arrays