Taking your corrections into account, I think we can conjecture that $$\lim_{k' \to k}x^{-k'}I_{k'}(x)= (x-1) \left(\sum_{i=0}^k \frac{H_{k+1} - H_i}{x^{i+1}}\right) - 2 \tanh^{-1}(1-2x)$$

The coefficients in the polynomials seem to correspond to OEIS A027446.

$c_k$ looks suspiciously like $2H_{k+1}$, which probably counts as a closed form for this purpose.

This isn't just a matching-like problem. It's asking for the largest monocolour matching in a complete undirected edge-coloured graph.

And, in the same way that Fibonacci number calculation by matrix powers can be optimised, this can be optimised with the recurrences $T_{2n} = 2 T_n{}^2 - 1$ and $T_{2n+1} = 2 T_n T_{n+1} - x$

The answer to the question in your comment is still no. $0, 0, 0, 16, 110, 492, 1680, 4160, 8640, \ldots$.

I confirm the first table. The next two rows are $2 + 1590q + 112874q^2 + 234014q^3 + 14400q^4$ and $2 + 2442q + 388126q^2 + 2622844q^3 + 615386q^4$.

The permutations with $D(\pi) = 0$ are the identity permutation and the reverse permutation. The number of permutations with $D(\pi) = 1$ starts $0,0,4,22,82,240,520,960,1590, 2442$ (from $n=1$ to $n=10$) and is not in OEIS.

@user144684, I'm not sure what misfire of the brain caused me to see that as an exponential relationship. Trying to find the appropriate term to correct it to.

@user144684, that's $f(0), f(1), f(2), \ldots, f(6)$.

gist.github.com/pjt33/42a3bd6eaae04b8f74163825021a9d03

Hmm. These examples are all exponential (specifically, $f(2x) = 2f(x)$). I should be able to enumerate exponential functions much more efficiently, so I can check whether this class extends to $GF(127)$...

Incidentally, since there's no use of multiplication in the definition of these functions, it just occurred to me to try searching for suitable functions over $Z_{15}$, and there is one pair, with representative $0(1,4)(2,8)(3,14)(5,10)(6,13)(7,9)(11,12)$.

I think there must be a sign error in the last line. $|A(\mathbf{F}_{41})| = 1 + 41 + (-7) = 35$ ?