Recall that we number positions in permutations starting from 0,
but the $n$-permutations consist of numbers from $\{1,2,\ldots,n\}$.
We have $\alpha_3=12121$ and the generation of $\{1,2,3\}$ is:
$$123\Arrow{1} 213 \Arrow{2} 312\Arrow{1} 132\Arrow{2} 231 \Arrow{1} 321$$
The generation of all 24 permutations of $\{1,2,3,4\}$ has the following structure.
$$1234\Arrow{\alpha_3} 3214\Arrow{3}4123 \Arrow{\alpha_3} 2143\Arrow{3}$$
$$ 3412 \Arrow{\alpha_3} 1432 \Arrow{3} 2341 \Arrow{\alpha_3} 4321$$
Usually the permutation generating algorithms produce each successive permutation by applying some kind of a basic operation. The sequence of this operations, corresponding to the permutation ordering, is called the generating sequence. It is treated as a word over the alphabet consisting of names of basic operations.
Assume the permutation of numbers $\{1,2,\ldots,n\}$ is stored in the table $\pi$, with positions numbered from zero. Our alphabet of basic operations is $\Sigma_n=\{1,2,\ldots,n-1\}$.
The symbol $i$ corresponds to the basic operation: reverse the prefix $\pi[0\dd i]$ of permutation $\pi$.
We define the sequence $\alpha_n$ as a prefix of size $n!-1$ of the following sequence $\rho\,=\,(\rho_1,\rho_2,\rho_3,\ldots )$, where $$\rho_k= \max \lbrace j \mid j! \ \mbox{is a divisor of}\ k \rbrace, \ \mbox{for}\ k\geq 1.$$ We have $\rho\,=\, 1\,2\,1\,2\,1\,3\,1\,2\,1\,2\,1\,3\, 1\,2\, \ldots$, $\rho_{12} = 3$, $\rho_{44} = 2$, $\rho_{13} = 1$.
$\alpha_n$ is a generating sequence: starting from the id-permutation $\pi=(\pi_0,\pi_1,\pi_2,\ldots,\pi_{n-1})\,=\, (1,2,\ldots,n),$ and consecutively applying operations from $\alpha_n$ all $n$-permutations are generated, each exactly once.
We consider compression in terms of straight-line programs. A straight-line program, briefly SLP, is a context-free grammar that produces a single word $w$ over a given alphabet $\Sigma$.
An SLP can be also defined as a sequence of recurrences (equations), using operations of concatenation of words. Compression by straight-line programs is also called Grammar-Based Compression.