[seqfan] Re: discordant permutations

[Brendan McKay]

Note that A000183 is also "discordant permutations", this time with
a definition.

I thought that A000270 might count permutations that are derangements
of both (1,2,...,n,n+1) and (n+1,1,2,...,n).  However, the 1/n term in the
asymptotic expansion doesn't seem to match.

I ordered a copy of the Touchard paper and should have it in a few days.

Brendan.

On 21/7/20 1:21 pm, Neil Sloane wrote:
> I'm puzzled too.  We definitely need to locate a copy of the Touchard
> paper. It appeared in
> Scripta Math., 1953. Rutgers Library has the bound volumes in the Annex,
> but that collection is closed for the duration.
>
> I did  manage to locate a copy of the Kaplansky-Riordan 1946 Scripta Math
> this afternoon, and annotated it and scanned it. See A000166 or A000179.
>
> 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 Mon, Jul 20, 2020 at 11:04 PM Brendan McKay <Brendan.McKay at anu.edu.au>
> wrote:
>
>> Hi Neil,
>>
>> I'm a bit puzzled by A000270.  First, I don't think that "discordant
>> permutations" is a well-understood phrase, so I suggest that a definition
>> be added.
>>
>> Second, what is n?  I assumed that permutations of {1,...,n} are
>> considered,
>> but that seems to be incorrect as the values are greater than n!. Are they
>> permutations of {1,...,n+1} perhaps?
>>
>> Best wishes, Brendan.
>>
>> On 21/7/20 2:53 am, Neil Sloane wrote:
>>> Concerning A000270:  In the binder where there would normally be a copy
>> of
>>> the source, the Touchard paper, all I have is a note saying that T's
>> paper
>>> is very similar to the paper of Kaplansky and Riordan.
>>>
>>> There are actually two relevant papers by K and R, but only one appeared
>> in
>>> Scripta Math., in vol 12 1946, 113-124.
>>>
>>> I have the Math Sci Net reviews of  the 3 papers, in pdf format, if
>> anyone
>>> wants to see them
>>>
>>> I just found a copy of the K and R Scripta Math paper, in an old (really
>>> old, the earliest is a typescript from 1934) binder of Riordan's papers.
>>> These are a mixture of Bell Labs typescripts and offprints of the
>> published
>>> papers.
>>>
>>>
>>>
>>> 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 Mon, Jul 20, 2020 at 10:37 AM William Orrick <will.orrick at gmail.com>
>>> wrote:
>>>
>>>> Dear SeqFans,
>>>>
>>>> The following paper seems difficult to obtain, but is the basis for
>>>> A000270:
>>>>
>>>> J. Touchard, Permutations discordant with two given permutations,
>>>> Scripta Math.,
>>>> 19 (1953), 109-119.
>>>>
>>>> Does anyone have a copy?
>>>>
>>>> The subject of the paper would seem to be the same as that of,
>>>>
>>>> J. Touchard, Sur un problème de permutations
>>>> <https://gallica.bnf.fr/ark:/12148/bpt6k31506/f631.item.zoom>. Comptes
>>>> Rendus Acad. Sci. Paris, 198 (1934) 631-633,
>>>>
>>>> written two decades earlier. I'm trying to understand two things.
>>>>
>>>> 1) Looking the earlier paper, A000270 seems to correspond to phi(1;h),
>>>> which Touchard defines, for n >=2, to be the number of permutations of
>>>> {1,2,...,n+1} discordant with both of two permutations whose relative
>>>> permutation consists of one 1-cycle and one n-cycle. Can someone verify
>>>> that this is right? I cross reference A000270 in the proposed sequence
>>>> A335391, but would feel more comfortable having confirmation of the
>>>> definition.
>>>>
>>>> 2) I'm trying to understand why the zeroth term of A000270 is 1. If the
>>>> same definition is being used as in Touchard's earier paper, I think
>> this
>>>> term should be 0.
>>>>
>>>> Best,
>>>> Will Orrick
>>>>
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