**$d$-regularity**

Define $d$-regularity of a permutation as the minimum over all possible differences within cycles of the permutation. It is convenient to let the identity permutation in $S_n$ has $d$-regularity equal to $n$.

This gives the following table (column $d$ count number of $d$-regular permutations):
$$
\begin{array}{ccccccc}
1 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\
1 & 1 & \text{} & \text{} & \text{} & \text{} & \text{} \\
4 & 1 & 1 & \text{} & \text{} & \text{} & \text{} \\
19 & 3 & 1 & 1 & \text{} & \text{} & \text{} \\
103 & 12 & 3 & 1 & 1 & \text{} & \text{} \\
651 & 54 & 10 & 3 & 1 & 1 & \text{} \\
4702 & 281 & 42 & 10 & 3 & 1 & 1 \\
\end{array}
$$
The first column gives no hit in the OEIS (which here seems like the most natural to test).

Accumulating the rows give
$$
\begin{array}{ccccccc}
1 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\
1 & 2 & \text{} & \text{} & \text{} & \text{} & \text{} \\
4 & 5 & 6 & \text{} & \text{} & \text{} & \text{} \\
19 & 22 & 23 & 24 & \text{} & \text{} & \text{} \\
103 & 115 & 118 & 119 & 120 & \text{} & \text{} \\
651 & 705 & 715 & 718 & 719 & 720 & \text{} \\
4702 & 4983 & 5025 & 5035 & 5038 & 5039 & 5040 \\
\end{array}
$$

**$d$-irregularity**

We can do a similar calculation,
where we compute the maximum over all possible differences in the cycles,
and then take the minimum over all such values.
We can perhaps call this the $d$-irregullarity of a permutation.
$$
\begin{array}{ccccccc}
1 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\
1 & 1 & \text{} & \text{} & \text{} & \text{} & \text{} \\
1 & 2 & 3 & \text{} & \text{} & \text{} & \text{} \\
1 & 4 & 7 & 12 & \text{} & \text{} & \text{} \\
1 & 7 & 17 & 35 & 60 & \text{} & \text{} \\
1 & 12 & 44 & 93 & 210 & 360 & \text{} \\
1 & 20 & 103 & 275 & 651 & 1470 & 2520 \\
\end{array}$$
that is, for permutations of $\{1,2,3,4,5,6\}$, there are 44 permutations which are strictly $2$-irregular, i.e., maximal difference within at least one block is exactly $2$.

Reading of the coefficients and accumulating, we obtain
$$
\begin{array}{ccccccc}
1 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\
1 & 2 & \text{} & \text{} & \text{} & \text{} & \text{} \\
1 & 3 & 6 & \text{} & \text{} & \text{} & \text{} \\
1 & 5 & 12 & 24 & \text{} & \text{} & \text{} \\
1 & 8 & 25 & 60 & 120 & \text{} & \text{} \\
1 & 13 & 57 & 150 & 360 & 720 & \text{} \\
1 & 21 & 124 & 399 & 1050 & 2520 & 5040 \\
\end{array}
$$
For example, there are 399 $3$-irregular permutations of $1...7$.

I find no hit in the OEIS.

The $d$-irregularity of a permutation can be seen as a statistic on permutations,
and this is now available as [St000209] in FindStat.

EDIT:
Here is the non-accumulated table for d-regularity, cyclic difference.
Columns give no hits in OEIS.

$$
\begin{array}{ccccccc}
1 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\
1 & 1 & \text{} & \text{} & \text{} & \text{} & \text{} \\
5 & 0 & 1 & \text{} & \text{} & \text{} & \text{} \\
20 & 3 & 0 & 1 & \text{} & \text{} & \text{} \\
109 & 10 & 0 & 0 & 1 & \text{} & \text{} \\
668 & 44 & 7 & 0 & 0 & 1 & \text{} \\
4801 & 210 & 28 & 0 & 0 & 0 & 1 \\
\end{array}
$$