A New Integer Sequence from New Graphical Representation of Permutations?

[Марко Радуловић]

I have defined a new permutation statistic that gives rise to an integer sequence. The construction starts from a graphical representation of permutations on an n×n grid. For every odd position i of a permutation σ∈Sn, all cells in column i from the first row up to σ(i) row are shaded. For every even position j, all cells in row j from the first column down to σ(j) column are shaded. A cell may therefore be shaded once or twice.

For a permutation σ, I define f(σ) to be the number of doubly shaded cells. This produces a nonnegative integer associated with every permutation. I then define S(n)=∑σ∈Snf(σ), the total number of doubly shaded cells over all permutations of size n. The first values of this sequence are 0,1,5,34,246,2064,18960,194400,2167200,…

The sequence appears to have interesting combinatorial properties. For example, numerical computations suggest that S(n) is even for every n>4. I am also interested in finding a closed formula, a recurrence relation, a generating function, asymptotic estimates, or a combinatorial interpretation. The statistic arises naturally from the graphical construction, but I have not found an equivalent description in the existing literature.

Question. Is S(n) or, equivalently, the permutation statistic f(σ) genuinely new, or is it already known under another name in the mathematical literature?

[Pontus von Brömssen]

An equivalent definition of your f(σ) is: the number of odd indices with σ(i)>=i (excedances) times the number of even indices with σ(i)>=i. I don't know if there's any literature about this.

Cheers,

Pontus

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[Марко Радуловић]

OK,what to do now? I cannot upload my latex file about this sequence,because OEIS website is old?

[Марко Радуловић]

What is your advice in this situation? 

[Sean A. Irvine]

Hi,

Pontus was responding to your question about prior work or what others might call this, not about the OEIS per se.

Since your sequence does not appear in the OEIS, you could submit it for consideration. I could not see any account under your name, so the first step would be to register for an OEIS account, which can be done here:

https://oeis.org/wiki/Special:RequestAccount

Sean.

To view this discussion visit https://groups.google.com/d/msgid/seqfan/c6640c7e-b6f3-44c3-8460-791419994e41n%40googlegroups.com.

[Марко Радуловић]

Hi,

I have an account on OEIS.

But I don't know how to exactly describe my sequence in their form

Marko Radulovic 

To view this discussion visit https://groups.google.com/d/msgid/seqfan/CAL0FETmiN5Vhukqfzx_7%3D21QBFkB5vrDp3L1YmB8ERLUwO9yxw%40mail.gmail.com.

[Марко Радуловић]

Can anyone help me with OEIS? Please

[Sean A. Irvine]

Start here (assuming you are logged in):

https://oeis.org/Submit.html

Fill in as many fields as you can. Each field is described in the Style sheet:

https://oeis.org/wiki/Style_Sheet

You can save and edit the proposal multiple times.

When you have done the best you can, click "These changes are ready for review by an OEIS Editor", then wait for the fireworks -- you'll likely get emails about problems that need to be corrected.

The overall contribution process is described here:

https://oeis.org/wiki/Overview_of_the_contribution_process

To view this discussion visit https://groups.google.com/d/msgid/seqfan/8784fba8-f474-487f-8823-74fe1dc144f1n%40googlegroups.com.

[Марко Радуловић]

I made a draft of this sequence. It is number A395245. How long does it take to verify the originality of this series?

[Joerg Arndt]

This is unlikely in any literature because of the multiplication part ("times").
Using "plus" instead of "times" we of cause get the statistics "number of excedances".
I doubt the suggested sequence is useful as a measure/statistic.

Best, jj

>> *Question.* Is S(n) or, equivalently, the permutation statistic f(σ) genuinely

>> new, or is it already known under another name in the mathematical
>> literature?
>>
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>> "SeqFan" group.
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>>