The composer Arnold Schönberg introduced in the 3rd decade of the 20th century the composition with 12-tone rows. Many composers after him used the 12-tone system, each one in his own way: Anton Webern, Alban Berg, Igor Stravinsky are some of the most prominent.

We will denote a 12-tone row as a permutation of the set S={0,...9,A,B} (we will use A for 10 and B for 11). But each 12-tone row has 12 transpositions (S+i, i=0,...11) and 4 forms: the original form, the inversion (the permutation with the complementary intervals), the retrograde form, and the retrograde inversion. Each form has 12 transpositions, thus we have 48 permutations. For example, the main row that uses Berg in opera Lulu is: 4 8 9 6 B 1 A 0 3 2 7 5. All 12-tone-permutations of it are listed in the following table:

Thus we can code all these 48 permutations as (0 4 5 2 7 9 6 8 B A 3 1) and denote the prime forms with P_i (where i is the first element of the P_i-form), the retrogrades of  P_i as R_i, the inversion of P_i as I_i and the retrograde inversions of P_i as RI_i, where i=0,...11. Thus all 48 permutations can be coded as one form of the prime one. This is an information compression. These 48 permutations can be used in a 12-tone work, that is based on a 12-tone row (say P₄).

But not all rows have 48 permutations. For some of them the retrograde form has the same interval sequence as the prime one. And for some, the retrograde inversion has the same interval sequence as the prime one. These rows have only 24 permutations.

In some musicological books I read, that all possible rows are 12!. This is not correct, because —as we saw above— we can reduce 48 permutations (or 24, in the case of the symmetrical rows) into just 1 row. And here comes the first question: how many rows are there? If we reduce them sufficiently, then we could index them, which would be very convienient for the musicological reference to the rows.

The second question is a bit more difficult. Some composers use the permutations in rotated form. Thus we could reduce the indexing further. E.g. we could denote (4 5 2 7 9 6 8 B A 3 1 0) as the 1st rotation of (0 4 5 2 7 9 6 8 B A 3 1) and not as a different row. But there are rows, that have symmetries with reference to rotation. If we respect rotation as well, how many rows do we have?