Permutations/ Same Sequence Of Signs As Inverses

[Leroy Quet]

First, thanks to Max Alekseyev for the work he
did regarding my last post to seq.fan about
permutations.

Second, sorry for the confusing title to this
email.

--

Consider the permutations
(p(1),p(2),p(3),...p(n)) of (1,2,3,...n).
We can list the (n-1) signs (+ or -) of the
differences between consecutive elements of the
permutations.

So, for example, if we have this permutation of
the first 7 positive integers: (3,2,1,7,5,6,4),
the signs would be --+-+-.
In other words, the kth sign is the sign of
p(k+1)-p(k).

So, what is the number of permutations of
(1,2,3,...n) where each permutation and its
inverse permutation both have the same sequence
of signs?

For example, for n = 6, we can have the
permutation and its inverse:
4,2,6,3,5,1
6,2,4,1,5,3

Both permutations have the sequence of signs:
-+-+-

Of course, if a permutation is its own inverse,
then it has the same sequence of signs as its
inverse.

Could someone please calculate more terms of the
sequence where the nth term equals the number of
such permutations of the first n positive
integers?
I get the sequence beginning 1,2,4,10,... (The
permutations for n = 1 through 4 or 5 that fit
the conditions are all inverses of themselves.
The case is different for 6, obviously, given the
example above.)

Is this sequence in the EIS already?

I have to admit that I do not know the vocabulary
of permutations as well I as I should. (Well,
that's a shock to you all!)
So this whole thing may be well studied, but
described using terminology I am unaware of.

Thanks,
Leroy Quet




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