How many total order relations on a set $A$?

Question

Let's define a set $T_A$ which is the set of all total order relations on $A$.

This set is a subset of the set of all $2$-adic relations on $A$: $$T_A \subset \mathcal P(A^2) $$

1-Which is the cardinality of $T_A$?

2-Which is the standard notation for the set $T_A$?

Answer 1

Hint: Note if $<$ is a total order on $A$ then for every permutation of $A$, $\pi$, the relation $\{\langle \pi a,\pi b\rangle\mid a<b\}$ is a total order on $A$ as well. In fact it is isomorphic to $<$.

Additional Hint: If $A$ is finite, then there is only one linear order of $A$ (up to isomorphism); if $A$ is infinite then there cannot be more orders than $2^A$, and there are $2^A$ permutations.

Answer 2

You can do this question yourself. There has to be a unique minimum in the order, then a unique element above that, and so on. So you get a bijection, form a standard chain $I_n$ on $n=\#A$ elements to $A$, which preserves order (you can take $I_n=\{1,\ldots,n\}$ for instance, then $i$ maps to the $i$-th smallest element). Moreover every bijection $f:I_n\to A$ defines a unique total ordering $\leq_f$ on $A$, by the rule $f(x)\leq_f f(y)$ whenever $x\leq y$ in $I_n$.

How many bijections $f:I_n\to A$ are there?