[seqfan] Re: discordant permutations

[William Orrick]

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