# [seqfan] Re: b-files for apparently matching sequences?

Maximilian Hasler maximilian.hasler at gmail.com
Tue Jun 29 18:08:01 CEST 2010

```In many other examples the definition is either the same or rather
obviously equivalent.

" of length n within distance D."
means the same as
" of 1..n with i-D<=p(i)<=i+D"

so

> %N A002524 Number of permutations of length n within distance 2
> %N xxxxxxx Number of permutations of 1..n with i-2<=p(i)<=i+2

> %N A002526 Number of permutations of length n within distance 3
> %N xxxxxxx Number of permutations of 1..n with i-3<=p(i)<=i+3

> %N A072856 Number of permutations satisfying i-4<=p(i)<=i+4, i=1..n (permutations of length n within distance 4).
> %N xxxxxxx Number of permutations of 1..n with i-4<=p(i)<=i+4

> %N A154654 Number of permutations of length n within distance 5
> %N xxxxxxx Number of permutations of 1..n with i-5<=p(i)<=i+5

> %N A154656 Number of permutations of length n within distance 7
> %N xxxxxxx Number of permutations of 1..n with i-7<=p(i)<=i+7

> %N A154657 Number of permutations of length n within distance 8
> %N xxxxxxx Number of permutations of 1..n with i-8<=p(i)<=i+8

> %N A154655 Number of permutations of length n within distance 6
> %N xxxxxxx Number of permutations of 1..n with i-6<=p(i)<=i+6

> %N A154658 Number of permutations of length n within distance 9
> %N xxxxxxx Number of permutations of 1..n with i-9<=p(i)<=i+9

are all obvious.
>From symmetry considerations it is clear that

> %N A072827 Number of permutations satisfying i-2<=p(i)<=i+3, i=1..n.

and

> %N xxxxxxx Number of permutations of 1..n with i-3<=p(i)<=i+2

also defines the same sequence, and similar in the other cases:

> %N A072850 Number of permutations satisfying i-2<=p(i)<=i+4, i=1..n.
> %N xxxxxxx Number of permutations of 1..n with i-4<=p(i)<=i+2

> %N A072854 Number of permutations satisfying i-3<=p(i)<=i+4, i=1..n.
> %N xxxxxxx Number of permutations of 1..n with i-4<=p(i)<=i+3

> %N A072852 Number of permutations satisfying i-2<=p(i)<=i+5, i=1..n.
> %N xxxxxxx Number of permutations of 1..n with i-5<=p(i)<=i+2

> %N A072855 Number of permutations satisfying i-3<=p(i)<=i+5, i=1..n.
> %N xxxxxxx Number of permutations of 1..n with i-5<=p(i)<=i+3

> %N A072853 Number of permutations satisfying i-2<=p(i)<=i+6, i=1..n.
> %N xxxxxxx Number of permutations of 1..n with i-6<=p(i)<=i+2

I'm not sure if it is useful, but you can add a comment about these
equalities in the respective sequences.
(and maybe explain the meaning of "distance" with the more explicit
formula (inequality) in the former cases.

M.

```