[seqfan] Re: discordant permutations

Brendan McKay Brendan.McKay at anu.edu.au
Tue Jul 21 05:03:54 CEST 2020

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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