I will outlay a few observations applying to the harmonic numbers that may be interesting to prove (if it hasn't already been proven).

From the Online Encyclopedia of Positive Integers we have:

$a(n)$ is the number of permutations $p$ of $\{1,\ldots,n\}$ such that the minimum number of block interchanges required to sort the permutation $p$ to the identity permutation is maximized.

$1, 1, 5, 8, 84, 180, 3044, 8064, 193248, 604800,\ldots$ (https://oeis.org/A260695)

Consider the following harmonic numbers:

$$1 + 1/2 = (1 + 2)\cdot 1/2!$$ $$1 + 1/2 + 1/3 + 1/4 = (1 + 2 + 3 + 4)\cdot 5/4!$$ $$1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 = (1 + 2 + 3 + 4 + 5 + 6)\cdot 84/6!$$ $$1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)\cdot 3044/8!$$

and so on.

Therefore, the following generalization suggests itself:

$$1 + 1/2 + 1/3 + 1/4 + \cdots + 1/2n = (2n^2 + n)\cdot a(2n - 1)/(2n)!$$

If this hasn't been proven, then I will leave it as a conjecture.

Cheers,

Robert