[seqfan] Re: discordant permutations

[Brendan McKay]

I now have Touchard's 1953 paper from Scripta Math.  In an moment
I'll send it to Neil, Will and Frank.  Anyone else, feel free to ask.
(I'm guessing this mailing list doesn't allow attachments, right?)

A000270 appears at the top of page 118.  Five minutes is not enough
for me to figure out what it means, but please note that it cannot be
a count of permutations of {1,...,n} because a(8) > 8!.  I suspect that
studying the whole paper is necessary.

Cheers, Brendan.

On 21/7/20 11:10 pm, Neil Sloane wrote:
> Will :
>
> good suggestion, about making a(1) = 1
>
> i will take care of it
>
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>
>
> On Tue, Jul 21, 2020 at 1:54 AM William Orrick <will.orrick at gmail.com>
> wrote:
>
>> Dear Neil,
>>
>> I just noticed your changes to A000270. I agree with changing a(0) to 0,
>> but you might consider keeping a(1)=1, rather than changing it to 0.
>> There's a sequence A102761 that is the same as A000179 except for the first
>> term. If you use A102761 instead of A000179 to generate A000270, and keep
>> a(1)=1 in A000270, then the relation you now use to define A000270 works
>> for all n, even negative n, with the convention a(-n) = a(n). I don't see a
>> clear combinatorial meaning for a(0) and a(1), or for a(n) with n negative,
>> but Touchard needs these to have the values implied by A000179 in order for
>> equation (1) in his 1934 paper to work. (This is the general formula for
>> the number of permutations discordant with two given permutations, and
>> seems to correspond to equation (22) in Kaplansky and Riordan.)
>>
>> Best,
>> Will
>>
>> On Tue, Jul 21, 2020 at 12:33 AM William Orrick <will.orrick at gmail.com>
>> wrote:
>>
>>> Dear SeqFans:
>>>
>>> Thanks Neil for posting the annotated copy of Kaplansky and Riordan.  Is
>>> the other Kaplanasky and Riordan paper you mentioned this one:
>>>
>>>   The problem of the rooks and its applications. Duke Math. J. 13 (1946)
>>> 259-268?
>>>
>>> I would be interested in seeing the MathSciNet reviews you mentioned if
>>> it's easy to send them.
>>>
>>> Brendan: in the original post in this thread I suggested that A000270 is
>>> the number of permutations of {1,2,...,n+1} discordant with both the
>>> identity permutation and with a permutation consisting of a 1-cycle and
>> an
>>> n-cycle.
>>>
>>> I have a new proposed sequence, A335391, not yet approved that is based
>> on
>>> Touchard's earlier paper of 1934. I believe A000270 is the second row of
>>> the square array in that sequence. Only the element in the first column
>>> disagrees. The new sequence contains a link to a post on
>> math.stackexchange
>>> where some of the statements in Touchard's paper are proved. The relation
>>> Neil mentioned with the menage numbers is also proved there.
>>>
>>> There are quite a few sequences in the OEIS with the title "discordant
>>> permutations" or similar.  Many of these are related to permutations
>>> discordant with three given permutations, rather than with two given
>>> permutations as is the case here.
>>>
>>> Best,
>>> Will Orrick
>>>
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