[seqfan] Re: discordant permutations

[William Orrick]

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
>