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Let $J$ be a permutation of the first $N$ integers (1, 2, ..., $N$), so that the permuted sequence reads $(J(1),J(2),...,J(N))$. The function $J$ must of course be a bijection. Additionally, suppose that $J$ is its own inverse: $J(J(i))=i$, and that $J$ has, at most, one fixed point. That is, there is either none or one value of $i$ such that $J(i)=i$. It isn't hard to see that there is a single fixed point if and only if $N$ is odd, and there is none if $N$ is even.

A permutation $J$ with the above properties establishes a pairing of the integers from 1 to $N$, where $i$ is paired with $J(i)$ (except if $N$ is odd, in which case the fixed point is not paired).

Is there a name for this type of permutation?

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Since "pairing" could also apply to any permutation which is the product of disjoint swaps, but where the number of fixed points is not 0 or 1, perhaps "complete pairing" would be better? Or even "matching", since the swaps define a bipartite graph that is as close to a matching as possible? – András Salamon Apr 25 '13 at 22:52
@AndrásSalamon I like all the terms you suggest. "Matching" is particularly good. But it would be nice if there was already an established name in the literature. – becko Apr 25 '13 at 23:59

I would call this a fixed-point free involution, respectively involution with one fixed point.

After some searching I found the term "maximum matching of the complete graph on $N$ points" that unambiguously describes the set of edges that bijectively corresponds to a permutation of this type (you could also say "maximal" instead of "maximum", as this is the same thing for a complete graph). This avoids the even/odd dichotomy, but has the drawback of only somewhat indirectly describing a set of permutations.

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-1 That's two names, one for $N$ even (fixed-point free involution) and one for $N$ odd (involution with one fixe poitn). I think "pairing" is more appropriate and descriptive, but I suppose no one uses it. – becko Apr 25 '13 at 22:03
The term "pairing" has a connotation with bilinear forms, which is why I would not advise using it. – Marc van Leeuwen Apr 26 '13 at 9:32

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