Name of similar elements of symmetric group

Question

For permutations of length $n$ of the symmetric group $S_n$, they can be arranged as follows (for example, $S_4$):

(1234) (2341) (3412) (4123)
(1243) (2431) (3124) (4312)
(1324) (2413) (3241) (4132)
(1342) (2134) (3421) (4213)
(1423) (2314) (3142) (4231)
(1432) (2143) (3214) (4321)

Is there a name for the rows? For the columns?

Answer 1

Assuming this is cycle notation and you mean cycles of length $n$:

Each row is simply four way of writing down the same permutation.

I'm not aware of any nice name for the columns; they would usually be considered unimportant groupings based on accidents of your notation for permutations.

Answer 2

If these are the permutations of $S_4$, denoted using the second row of two-line notation, then we have:

• The column $j$ contains the permutations $\sigma \in S_4$ for which $\sigma(1)=j$.

• The rows list the permutations $\sigma \in S_4$ composed with powers of the $4$-cycle $(1234)$ [in cycle notation]. That is $$\sigma,\sigma \circ (1234),\sigma \circ (1234)^2,\sigma \circ (1234)^3.$$