[seqfan] discordant permutations

William Orrick will.orrick at gmail.com
Mon Jul 20 16:37:01 CEST 2020

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

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.

Will Orrick

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