Inspired by Dominic van der Zypen's question and Ilya Bogdanov's answer I wonder what to make of two cardinal characteristics. To define them, let $e_U$ be the increasing enumeration of an infinite set $U$ of natural numbers as before. Furthermore, if $\alpha$ is a function into the natural numbers whose domain contains $U$, let $\alpha_U$ be an abbreviation for the function $e^{-1}_{\alpha[U]} \circ \alpha \circ e_U$. Furthermore let $B = \left\{b | b : \omega \longleftrightarrow \omega\right\}$, i.e. the family of all permutations of $\omega$.
$$ \begin{aligned} \mathfrak{n} = & \min\left(\left\{|F| : F \subset B \wedge \forall \alpha \in B \exists \varphi \in F \forall U \in [\omega]^\omega : \varphi \ne \alpha_U\right\}\right),\\ \mathfrak{o} = & \min\left(\left\{|A| : A \subset B \wedge \forall b \in B \exists \alpha \in A \exists U \in [\omega]^\omega : b = \alpha_U \right\}\right). \end{aligned} $$
I hope that these letters are not yet taken. Given Ilya Bogdanov's answer, I suspect that we have $\mathfrak{o} \geqslant \mathfrak{b}$. Can we say anything else? Have these or similar cardinal characteristics been investigated in the past?
Edit:
Recall that $\mathfrak{b}$ denotes the unbounding number, cf. e.g. Andreas Blass's chapter in the handbook of set theory, or Eric van Douwen's chapter in the handbook of set-theoretic topology, or Lorenz Halbeisen's book, i.e. $$ \begin{aligned} \mathfrak{b} & = \min\left(\left\{|F| : F \subset {}^\omega \omega \wedge \nexists g \in {}^\omega \omega \forall f \in F\left(f \leqslant^* g\right) \right\}\right),\\ \text{where } f & \leqslant^* g \Leftrightarrow \{n | n < \omega \wedge g(n) < f(n)\} \text{ is finite.} \end{aligned} $$
