# [seqfan] Re: discordant permutations

Neil Sloane njasloane at gmail.com
Tue Jul 21 06:51:54 CEST 2020

By the way, there is a simple relation between the menage numbers A000179
in the Kaplansky-Riordan 1946 Scripta Math paper
and A000270, namely A000270(n) = A000179(n+1)+A000179.(n)+A000179(n-1), for
n>=2.  E.g. 95 = 80+13+2.
I will use this as the new definition.

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.
Email: njasloane at gmail.com

On Mon, Jul 20, 2020 at 11:21 PM Neil Sloane <njasloane at gmail.com> 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.
> 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
>>
>> 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.
>> > 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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>> >>
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>