Some combinatorics question concerning symmetric groups Let $n = ht$ where $n, h ,t $ are all positive integers. I want to count $\omega \in S_t$ satisfying the following two properties:

*

*$\omega(t+1 - \omega(i)) = t+1 - i$.


*$\sum_{i: i \geq \omega(i)} (h + i - \omega(i)) > n$.
Does anyone have some ideas on how to do that?
 A: Not a full solution, but some observations. Perhaps they spark or qualify as 'ideas.'
The first condition is equivalent to asking that
$$\omega(i) + \omega^{-1}(t+1-i) = t+1$$
In other words, if $\omega(i) = t+1-j$ then $\omega(j) = t+1-i$ (and visa versa!). So we may enumerate permutations satisfying the first condition fairly easily. Greedily choose from the remaining unused inputs $i,j$ assign $\omega(i)=t+1-j$ and $\omega(j) = t+1-i,$ and repeat. Note one can choose $i=j,$ where we get the same equation twice.
This also gives a nice recurrence relation for the number of permutations satisfying the first constraint,
$$a_{t+1} = t a_{t-1} + a_{t}$$
with $a_1 =1, a_2=2.$ (Indeed, the enumeration gives a bijection with involutions, $\omega^2=1$, many of the comments here may be applicable. If we pair $i,j$ in the above construction, then the corresponding involution swaps $i$ and $j.$).
The sum in the second condition increases with our choices - the pair $i < j$ contributes $0$ if $i+j < t+1$ and $2(h+i+j-t-1)$ otherwise, while small fixed points ($i=j \leq \frac {t+1}2$) contribute $h.$ Call this sum the weight of a permutation.
If $h$ is large compared to $t$ (for example, $h > t^2$) then the only way to satisfy the second condition is to have each summand be nonzero. So every $i$ is either fixed, or paired with something at least as big as $t+1-i.$
Likewise, assuming $h$ is large, we only get a weight of exactly $n$ if each $i$ is paired with $t+1-i.$ So there is just one permutation (the identity) with exactly weight $n.$
For the first few values of $t$, in this large $h$ case, I find the sequence:
$$1, 1, 2, 2, 5, 7, 17, 29, 74, 136$$
for those satisfying weights $\geq n,$ so the sequence for weights strictly greater than $n$ is:
$$0, 0, 1, 1, 4, 6, 16, 28, 73, 135$$
These do not appear on OEIS.
