Up/Down-Symmetric Permutations

[Olivier Gerard]

On Feb 10, 2008 6:58 PM, Leroy Quet <q1qq2qqq3qqqq at yahoo.com> wrote:
> This post deals with two kinds of permutations.
>
>
> The signs of the differences between adjacent
> terms forms the sequence: ++--++, which has
> reflective symmetry.
>
> Is the sequence, where a(n) equals the number of
> such permutations for n, in the EIS already?
>

The beginning of this sequence is

1, 2, 2, 12, 24, 200, 540, 6160

and is not in the OEIS

> --
>
>
> The signs of the differences between adjacent
> terms forms the sequence: ++-+--, which is the
> negative of its reversal.
>
> Is the sequence, where a(n) equals the number of
> such permutations for n, in the EIS already?
>
>
> Note: a(2n) = 0 for all n, in regards to the
> latter sequence, since there are an odd number of
> signs, and the middle sign can't equal its
> negative.
>
1, 0, 4, 0, 44, 0, 1028, 0, ...

Neither is in the OEIS even without the zeros.

>
> If one or both of these sequences isn't in the
> EIS already, could someone please calculate and
> submit it/them?
> Thanks.
>

Leroy, why not make yourself the submissions ?

There are also in the same spirit:

Signs taken between p and the reversal of p
1, 0, 2, 4, 24, 52, 540, 1712
1, 2, 0, 12, 0, 196, 0, 6064

Signs taken between p and p^2
1, 1, 3, 1, 11, 47, 177, 839
1, 2, 2, 10, 10, 124, 124, 2968

Signs taken between p and identity (anti symmetric case)
1, 2, 2, 10, 10, 150, 150, 4942

None of them are in the OEIS.

Olivier