Different patterns in Sn modulo 2, is it A001405?

[Max Alekseyev]

On Dec 19, 2007 12:38 AM, Ivica Kolar <telpro at kvid.hr> wrote:

> Sequence definition:
> Different binary patterns in the set of all permutations Sn taken modulo 2.
> Example: 0,1,2,3,4 modulo 2 = 0,1,0,1,0
> Short calculation gives sequence:
> ;N             0  1  2  3  4   5   6   7   8    9
> ;                  1, 2, 3, 6, 10, 20, 35, 70, 126
> ;A001405: 1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462, 924, 1716
>
> Recently, I thought that such binary patterns are of use in  the problem I'm
> researching.
> That assumption turned to be wrong so I had no reason to go any deeper into
> that sequence.
> However, I'm fairly confident that the sequence in question is A001405
> indeed.
>
> What is your opinion, is it realy A001405?

It is. Note that among integers 1,2,...,n there are exactly floor(n/2)
even numbers and ceil(n/2) odd numbers. Then any binary pattern
consists of floor(n/2) zeros and ceil(n/2) ones and the total number
of such patterns equal to the number of ways to choose floor(n/2)
positions from n, that is binomial(n,floor(n/2))=A001405(n).

> Reason why I'm posting this is:
> 1. I see no explicite corellation with permutations there, or I've
> overlooked something.

See above.

> 2. Sequence definition over Sn seems simplest than definition(s) given in
> OEIS,
>    from my point of view at least.

I disagree. The current definition of A001405 is simple, elegant, and
self-contained.

%N A001405 Central binomial coefficients: C(n,floor(n/2)).

> Is it worth of noting it there?

Not sure.

Max