# parity of integer sequences

Olivier Gerard ogerard at ext.jussieu.fr
Thu Aug 1 22:16:59 CEST 2002

```This message forwarded from Frank Ruskey.

Olivier Gérard

From: Frank Ruskey <fruskey at cs.uvic.ca>
To: ogerard at ext.jussieu.fr
Subject: Re: parity of integer sequences

Generally speaking, I am against the "automatic" generation of new
sequences from existing sequences.  More useful, but perhaps
in line with your idea of using parity, would be to "split" existing
sequences according to some natural parity assignment to the
objects being counted.

Some examples:
(a) There are many sequences in the database that
count various types of permutations.  Classify these
permutations according to the parity of the permutation,
or classify them according to the parity of the position
of n, or the parity of the first element of the permutation.
(b) There are many sequences in the database that count
various types of words (e.g., Lyndon words).
Classify these according to
the parity of the sum of the characters in the word, or
classify them according to the parity of the number of
non-zero elements that occur in odd positions, etc.

These contributions to the database would be more likely
of future use.  Moreover, I suspect that some of these
"splittings" would give sequences already in the database,
but counting some other type of combinatorial object.
Then you could have fun looking for bijections.

Regards,
Frank R.

jeremy.gardiner at btinternet.com wrote:
>
> I was thinking about how to categorise integer sequences and I wonder how useful it would be if the on-line encyclopaedia were to contain a reference to the parity representation for each sequence?
>
> Many of the entries would be all odd, all even or simply alternating parity but there are also many more interesting examples such as 1010010001 ... (A023531), however there is no easy way of data mining these without calculating parity for each sequence.
>
> Of course parity is not the only way to categorise integer sequences but it is an important property.
>
> Perhaps other people have a view on this?

--
----------------------
Frank Ruskey                     e-mail: fruskey at cs.uvic.ca <- NEW!
Dept. of Computer Science        fax:    250-721-7292
University of Victoria           office: 250-721-7232
Victoria, B.C. V8W 3P6 CANADA    WWW: http://www.cs.uvic.ca/~fruskey

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