Id Symbol Conducement Reducementals Inverse Uninomial Priming: $〈1〉\circ \left\{1\right\}=\left\{1\right\}$ - - 1 ${\mathbb{U}}_{0}$ $\left\{1\right\}\oplus \left\{\begin{array}{ccc}1& 1& \begin{array}{cc}1& \cdots \end{array}\end{array}\right\}=\left\{\begin{array}{ccc}1& 1& \begin{array}{cc}1& \cdots \end{array}\end{array}\right\}$ $\left\{1\right\}\odot \left\{\left(\begin{array}{c}n\\ n\end{array}\right)\right\}$ 𝕦0 $\left\{1\right\}$ 2 ${\mathbb{U}}_{1}$ $\begin{array}{l}\left\{\begin{array}{ccc}1& 1& \begin{array}{cc}1& \cdots \end{array}\end{array}\right\}\oplus \left\{\begin{array}{ccc}1& 1& \begin{array}{cc}1& \cdots \end{array}\end{array}\right\}\\ =\left\{\begin{array}{ccc}1& 2& \begin{array}{cc}3& \begin{array}{cc}\cdots & \left(\begin{array}{c}n\\ n-1\end{array}\right)\end{array}\end{array}\end{array}\right\}\end{array}$ $\left\{\begin{array}{cc}1& 1\end{array}\right\}\odot \left\{\begin{array}{cc}\left(\begin{array}{c}n\\ n\end{array}\right)& \left(\begin{array}{c}n\\ n-1\end{array}\right)\end{array}\right\}$ 𝕦−1 $\left\{\begin{array}{cc}1& -1\end{array}\right\}$ 3 ${\mathbb{U}}_{2}$ $\begin{array}{l}\left\{\begin{array}{ccc}1& 2& \begin{array}{cc}3& \begin{array}{cc}\cdots & \left(\begin{array}{c}n\\ n-1\end{array}\right)\end{array}\end{array}\end{array}\right\}\oplus \left\{\begin{array}{ccc}1& 1& \begin{array}{cc}1& \cdots \end{array}\end{array}\right\}\\ =\left\{\begin{array}{ccc}1& 3& \begin{array}{cc}6& \begin{array}{cc}\cdots & \left(\begin{array}{c}n+1\\ n-1\end{array}\right)\end{array}\end{array}\end{array}\right\}\end{array}$ $\left\{\begin{array}{cc}1& \begin{array}{cc}2& 1\end{array}\end{array}\right\}\odot \left\{\begin{array}{cc}\left(\begin{array}{c}n\\ n\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n\\ n-1\end{array}\right)& \left(\begin{array}{c}n\\ n-2\end{array}\right)\end{array}\end{array}\right\}$ 𝕦−2 $\left\{\begin{array}{cc}1& \begin{array}{cc}-2& 1\end{array}\end{array}\right\}$ 4 ${\mathbb{U}}_{3}$ $\begin{array}{l}\left\{\begin{array}{ccc}1& 3& \begin{array}{cc}6& \begin{array}{cc}\cdots & \left(\begin{array}{c}n+1\\ n-1\end{array}\right)\end{array}\end{array}\end{array}\right\}\oplus \left\{\begin{array}{ccc}1& 1& \begin{array}{cc}1& \cdots \end{array}\end{array}\right\}\\ =\left\{\begin{array}{ccc}1& 4& \begin{array}{cc}10& \begin{array}{cc}\cdots & \left(\begin{array}{c}n+2\\ n-1\end{array}\right)\end{array}\end{array}\end{array}\right\}\end{array}$ $\left\{\begin{array}{cc}1& \begin{array}{cc}3& \begin{array}{cc}3& 1\end{array}\end{array}\end{array}\right\}\odot \left\{\begin{array}{cc}\left(\begin{array}{c}n\\ n\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n\\ n-1\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n\\ n-2\end{array}\right)& \left(\begin{array}{c}n\\ n-3\end{array}\right)\end{array}\end{array}\end{array}\right\}$ 𝕦−3 $\left\{\begin{array}{cc}1& \begin{array}{cc}-3& \begin{array}{cc}3& -1\end{array}\end{array}\end{array}\right\}$ Binomial 5 Priming: $〈\begin{array}{cc}1& 1\end{array}〉\circ \left\{\begin{array}{cc}1& 1\end{array}\right\}=\left\{\begin{array}{cc}1& 1\end{array}\right\}$ 6* ${\mathbb{B}}_{1}$ $\begin{array}{l}\left\{\begin{array}{cc}1& 1\end{array}\right\}\oplus \left\{\begin{array}{ccc}1& 2& \begin{array}{cc}3& \begin{array}{cc}\cdots & \left(\begin{array}{c}n\\ n-1\end{array}\right)\end{array}\end{array}\end{array}\right\}\\ =\left\{\begin{array}{ccc}1& 3& \begin{array}{ccc}5& 7& \cdots \end{array}\end{array}\right\}\end{array}$ $\left\{\begin{array}{cc}1& 1\end{array}\right\}\odot \left\{\begin{array}{cc}\left(\begin{array}{c}n-1\\ n-1\end{array}\right)& \left(\begin{array}{c}n-1\\ n-2\end{array}\right)\end{array}\right\}$ 𝕓−1 $\left\{\begin{array}{cc}1& \begin{array}{cc}0& -1\end{array}\end{array}\right\}$ 7 ${\mathbb{B}}_{2}$ $\begin{array}{l}\left\{\begin{array}{ccc}1& 3& \begin{array}{ccc}5& 7& \cdots \end{array}\end{array}\right\}\oplus \left\{\begin{array}{cc}1& 1\end{array}\right\}\\ =\left\{\begin{array}{ccc}1& 4& \begin{array}{ccc}8& 12& \cdots \end{array}\end{array}\right\}\end{array}$ $\left\{\begin{array}{cc}1& \begin{array}{cc}2& 1\end{array}\end{array}\right\}\odot \left\{\begin{array}{cc}\left(\begin{array}{c}n-1\\ n-1\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-1\\ n-2\end{array}\right)& \left(\begin{array}{c}n-1\\ n-3\end{array}\right)\end{array}\end{array}\right\}$ 𝕓−2 $\left\{\begin{array}{cc}1& \begin{array}{cc}1& \begin{array}{cc}-1& -1\end{array}\end{array}\end{array}\right\}$ 8 ${\mathbb{B}}_{3}$ $\begin{array}{l}\left\{\begin{array}{ccc}1& 4& \begin{array}{ccc}8& 12& \cdots \end{array}\end{array}\right\}\oplus \left\{\begin{array}{cc}1& 1\end{array}\right\}\\ =\left\{\begin{array}{ccc}1& 5& \begin{array}{ccc}12& 20& \cdots \end{array}\end{array}\right\}\end{array}$ $\left\{\begin{array}{cc}1& \begin{array}{cc}3& \begin{array}{cc}3& 1\end{array}\end{array}\end{array}\right\}\odot \left\{\begin{array}{cc}\left(\begin{array}{c}n-1\\ n-1\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-1\\ n-2\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-1\\ n-3\end{array}\right)& \left(\begin{array}{c}n-1\\ n-4\end{array}\right)\end{array}\end{array}\end{array}\right\}$ 𝕓−3 $\left\{\begin{array}{cc}1& \begin{array}{cc}2& \begin{array}{cc}0& \begin{array}{cc}-2& -1\end{array}\end{array}\end{array}\end{array}\right\}$ Trinomial Sequences 9 ${\mathbb{T}}_{1}$ $\begin{array}{l}\left\{\begin{array}{cc}1& \begin{array}{cc}1& 1\end{array}\end{array}\right\}\oplus \left\{\begin{array}{ccc}1& 2& \begin{array}{ccc}3& 4& \cdots \end{array}\end{array}\right\}\\ =\left\{\begin{array}{ccc}1& 3& \begin{array}{ccc}6& 9& \cdots \end{array}\end{array}\right\}\end{array}$ $\left\{\begin{array}{cc}1& \begin{array}{cc}1& 1\end{array}\end{array}\right\}\odot \left\{\begin{array}{cc}\left(\begin{array}{c}n-2\\ n-2\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-2\\ n-3\end{array}\right)& \left(\begin{array}{c}n-2\\ n-4\end{array}\right)\end{array}\end{array}\right\}$ 𝕥−1 $\left\{\begin{array}{cc}1& \begin{array}{cc}0& \begin{array}{cc}0& -1\end{array}\end{array}\end{array}\right\}$ 10 ${\mathbb{T}}_{2}$ $\begin{array}{l}\left\{\begin{array}{cc}1& \begin{array}{cc}1& 1\end{array}\end{array}\right\}\oplus \left\{\begin{array}{ccc}1& 3& \begin{array}{ccc}6& 9& \cdots \end{array}\end{array}\right\}\\ =\left\{\begin{array}{ccc}1& 4& \begin{array}{ccc}10& 18& \cdots \end{array}\end{array}\right\}\end{array}$ $\begin{array}{l}\left\{\begin{array}{cc}1& \begin{array}{cc}2& \begin{array}{cc}3& \begin{array}{cc}2& 1\end{array}\end{array}\end{array}\end{array}\right\}\\ \odot \left\{\begin{array}{cc}\left(\begin{array}{c}n-2\\ n-2\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-2\\ n-3\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-2\\ n-4\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-2\\ n-5\end{array}\right)& \left(\begin{array}{c}n-2\\ n-6\end{array}\right)\end{array}\end{array}\end{array}\end{array}\right\}\end{array}$ 𝕥−2 $\left\{\begin{array}{cc}1& \begin{array}{cc}1& \begin{array}{cc}1& \begin{array}{cc}-1& \begin{array}{cc}-1& -1\end{array}\end{array}\end{array}\end{array}\end{array}\right\}$ 11 ${\mathbb{T}}_{3}$ $\begin{array}{l}\left\{\begin{array}{cc}1& \begin{array}{cc}1& 1\end{array}\end{array}\right\}\oplus \left\{\begin{array}{ccc}1& 4& \begin{array}{ccc}10& 18& \cdots \end{array}\end{array}\right\}\\ =\left\{\begin{array}{ccc}1& 5& \begin{array}{ccc}15& 32& \cdots \end{array}\end{array}\right\}\end{array}$ $\begin{array}{l}\left\{\begin{array}{cc}1& \begin{array}{cc}3& \begin{array}{cc}6& \begin{array}{cc}7& \begin{array}{cc}6& \begin{array}{cc}3& 1\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right\}\\ \odot \left\{\begin{array}{cc}\left(\begin{array}{c}n-2\\ n-2\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-2\\ n-3\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-2\\ n-4\end{array}\right)& \begin{array}{c}\left(\begin{array}{c}n-2\\ n-5\end{array}\right)\end{array}\end{array}\end{array}\end{array}\\ \begin{array}{cc}\left(\begin{array}{c}n-2\\ n-6\end{array}\right)& \begin{array}{cc}\left(\begin{array}{c}n-2\\ n-7\end{array}\right)& \left(\begin{array}{c}n-2\\ n-8\end{array}\right)\end{array}\end{array}\right\}\end{array}$ 𝕥−3 $\left\{\begin{array}{cc}1& \begin{array}{cc}2& \begin{array}{cc}3& \begin{array}{cc}1& \begin{array}{cc}-1& \begin{array}{cc}-3& \begin{array}{cc}-2& -1\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right\}$