[seqfan] Re: discordant permutations

[Neil Sloane]

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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