Corteel Involution in Permutations

• Corteel involution is an involutive bijection on S_n defined through the Foata–Zeilberger labelled-Motzkin-path model that exchanges crossing and nesting statistics.
• It preserves various cycle-type properties, including cycle peaks and fixed points, and supports a fixed-point enumeration of 2^(n-1) for n ≥ 1.
• The continued-fraction formulation links generating functions for crossings and nestings, offering analytic insights into the symmetry and cyclic sieving phenomenon.

The Corteel involution is the involutive bijection on $S_n$ obtained by conjugating a label-complementation map on the Foata–Zeilberger labelled-Motzkin-path model through the bijection ${\rm FZ}_n$. In the formulation used in "Cyclic sieving phenomena via combinatorics of continued fractions" (Deb, 19 Aug 2025), it is a permutation involution that exchanges the statistics $\mathrm{cross}$ and $\mathrm{nest}$, preserves several other cycle-type statistics, admits a fixed-point enumeration equal to $2^{n-1}$ for $n\ge 1$, and yields an instance of the cyclic sieving phenomenon for the action of $\langle \iota\rangle\cong \mathbb Z/2\mathbb Z$ on $S_n$. The same source states that this involution was first studied by Adams, Elder, Lafrenière, McNicholas, Striker and Welch in arXiv 2024, and that several of their results are reproved there by means of continued fractions, together with two conjectures (Deb, 19 Aug 2025).

1. Definition through the Foata–Zeilberger model

The starting point is the classical Foata–Zeilberger bijection

$${\rm FZ}_n:\;S_n\;\longleftrightarrow\; \{\,( \omega,\xi)\colon \omega\text{ a Motzkin path of length }n,\; \xi\text{ a compatible label-sequence}\,\}.$$

Here $\omega$ is a Motzkin path of length ${\rm FZ}_n$0 with steps ${\rm FZ}_n$1, ${\rm FZ}_n$2, and ${\rm FZ}_n$3, staying in the nonnegative quadrant and ending at height ${\rm FZ}_n$4. If ${\rm FZ}_n$5 is the path-height just before step ${\rm FZ}_n$6, then the labels satisfy

${\rm FZ}_n$7

The formulation explicitly notes that the full detail of the three level-step types is not needed there; what matters is that the Foata–Zeilberger model encodes the usual crossing and nesting statistics in its labels (Deb, 19 Aug 2025).

Let ${\rm FZ}_n$8 denote the set of Foata–Zeilberger labelled paths of length ${\rm FZ}_n$9. The path-level involution is

$\mathrm{cross}$0

defined by complementing each label $\mathrm{cross}$1 within its allowed range. Concretely, if step $\mathrm{cross}$2 starts at height $\mathrm{cross}$3, then

$\mathrm{cross}$4

so that $\mathrm{cross}$5. The source states that one checks easily that $\mathrm{cross}$6 is a bijection of order $\mathrm{cross}$7 on $\mathrm{cross}$8 (Deb, 19 Aug 2025).

The genuine permutation involution is then

$\mathrm{cross}$9

and therefore satisfies $\mathrm{nest}$0. In this formulation, the Corteel involution is not defined directly on one-line notation or cycle notation; it is defined by transport of structure from the Foata–Zeilberger labelled-path model.

2. Exchange of crossings and nestings

The central structural property is the exchange of $\mathrm{nest}$1 and $\mathrm{nest}$2. Corteel’s original motivation, as quoted in the supplied exposition, was to show the symmetry

$\mathrm{nest}$3

In the notation of the continued-fraction paper, this is expressed by the proposition that if $\mathrm{nest}$4, then

$\mathrm{nest}$5

(Deb, 19 Aug 2025).

The same proposition states that all the other cycle-type statistics listed there are preserved by $\mathrm{nest}$6: cycle peaks, cycle valleys, double rises/falls, and fixed points. This places the involution in a specific class of statistic-exchanging bijections: it is not merely sign-reversing or orbit-pairing, but rather a symmetry that interchanges two statistics while leaving a broader cycle-type profile unchanged.

The broader abstract of the paper identifies a common feature shared by several involutions treated there: other than the CDDSY involution, they are constructed via bijections to weighted lattice paths, and they exchange crossings and nestings on their respective objects (Deb, 19 Aug 2025). In the case of permutations, the Corteel involution is the corresponding instance arising from the Foata–Zeilberger labelled-Motzkin-path representation.

3. Continued-fraction formulation

For crossings on permutations, the ordinary generating polynomial is defined by

$\mathrm{nest}$7

The supplied text states that Sokal–Zeng proved the continued-fraction identity

$\mathrm{nest}$8

with explicitly known coefficients

$\mathrm{nest}$9

Within this framework, nestings satisfy the same continued fraction with $2^{n-1}$0 up to a simple prefactor, and the exposition concludes that one sees directly the symmetry “cross $2^{n-1}$1 nest” under $2^{n-1}$2 (Deb, 19 Aug 2025). This is the analytic counterpart of the bijective exchange implemented by $2^{n-1}$3.

A plausible implication is that the Corteel involution occupies a dual role: it is simultaneously a combinatorial involution on $2^{n-1}$4 and a mechanism explaining a functional symmetry visible in continued fractions. The data explicitly supports this connection by stating that the permutation instances of cyclic sieving in the paper involve the Corteel involution and that several of the previously known results are reproved using the setting of continued fractions (Deb, 19 Aug 2025).

4. Fixed points and their enumeration

Lemma 3.3, as quoted in the supplied material and attributed originally to Adams–Elder–Lafrenière–McNicholas–Striker–Welch (Adams et al., 2024), gives the fixed-point count: $2^{n-1}$5 Equivalently, the generating function for fixed points is

$2^{n-1}$6

which is described there as a finite-depth Jacobi-type continued fraction (Deb, 19 Aug 2025).

This fixed-point formula is one of the most concrete enumerative consequences attached to the involution in the supplied text. It does more than count invariant permutations: it also provides the numerical value needed for the $2^{n-1}$7 specialization of the sieving polynomial discussed below. Because the same source stresses the continued-fraction machinery, the fixed-point series is not presented as an isolated enumeration, but as part of the same analytic-combinatorial apparatus governing crossings.

The attribution to Adams et al. also matters historically. The abstract of (Deb, 19 Aug 2025) states that the Corteel involution was first studied by Adams, Elder, Lafrenière, McNicholas, Striker and Welch, and that several of their results are reproved in the continued-fraction setting, with two conjectures proved there as well. This places the fixed-point enumeration in a line of work linking explicit involutions, orbit structure, and cyclic sieving (Adams et al., 2024).

5. Role in the cyclic sieving phenomenon

The cyclic sieving instance is formulated by taking

$2^{n-1}$8

Because $2^{n-1}$9 exchanges crossings and nestings, the exposition states that

$n\ge 1$0

while $n\ge 1$1 (Deb, 19 Aug 2025).

The same source then identifies this as an instance of the cyclic sieving phenomenon in the $n\ge 1$2 or “$n\ge 1$3” case, using Stembridge’s two-element formulation: $n\ge 1$4 Here the polynomial is $n\ge 1$5 and the generator of $n\ge 1$6 acts by $n\ge 1$7. Thus the Corteel involution furnishes the group action required for the sieving triple.

The continued-fraction aspect reappears immediately. The paper notes that

$n\ge 1$8

which exactly matches the continued fraction for the fixed-point series in the preceding section. In that sense, the cyclic sieving verification is made manifest by agreement of two series: one obtained from a specialization of the crossing polynomial, the other from the enumeration of fixed points under $n\ge 1$9 (Deb, 19 Aug 2025).

6. Example, interpretation, and a noted discrepancy

For $\langle \iota\rangle\cong \mathbb Z/2\mathbb Z$0, the supplied example lists the six permutations of $\langle \iota\rangle\cong \mathbb Z/2\mathbb Z$1 together with values of $\langle \iota\rangle\cong \mathbb Z/2\mathbb Z$2, $\langle \iota\rangle\cong \mathbb Z/2\mathbb Z$3, and $\langle \iota\rangle\cong \mathbb Z/2\mathbb Z$4:

$\langle \iota\rangle\cong \mathbb Z/2\mathbb Z$5	$\langle \iota\rangle\cong \mathbb Z/2\mathbb Z$6, $\langle \iota\rangle\cong \mathbb Z/2\mathbb Z$7	$\langle \iota\rangle\cong \mathbb Z/2\mathbb Z$8
$\langle \iota\rangle\cong \mathbb Z/2\mathbb Z$9	$S_n$0	$S_n$1
$S_n$2	$S_n$3	$S_n$4
$S_n$5	$S_n$6	$S_n$7
$S_n$8	$S_n$9	$${\rm FZ}_n:\;S_n\;\longleftrightarrow\; \{\,( \omega,\xi)\colon \omega\text{ a Motzkin path of length }n,\; \xi\text{ a compatible label-sequence}\,\}.$$0
$${\rm FZ}_n:\;S_n\;\longleftrightarrow\; \{\,( \omega,\xi)\colon \omega\text{ a Motzkin path of length }n,\; \xi\text{ a compatible label-sequence}\,\}.$$1	$${\rm FZ}_n:\;S_n\;\longleftrightarrow\; \{\,( \omega,\xi)\colon \omega\text{ a Motzkin path of length }n,\; \xi\text{ a compatible label-sequence}\,\}.$$2	$${\rm FZ}_n:\;S_n\;\longleftrightarrow\; \{\,( \omega,\xi)\colon \omega\text{ a Motzkin path of length }n,\; \xi\text{ a compatible label-sequence}\,\}.$$3
$${\rm FZ}_n:\;S_n\;\longleftrightarrow\; \{\,( \omega,\xi)\colon \omega\text{ a Motzkin path of length }n,\; \xi\text{ a compatible label-sequence}\,\}.$$4	$${\rm FZ}_n:\;S_n\;\longleftrightarrow\; \{\,( \omega,\xi)\colon \omega\text{ a Motzkin path of length }n,\; \xi\text{ a compatible label-sequence}\,\}.$$5	$${\rm FZ}_n:\;S_n\;\longleftrightarrow\; \{\,( \omega,\xi)\colon \omega\text{ a Motzkin path of length }n,\; \xi\text{ a compatible label-sequence}\,\}.$$6

The accompanying text states that one checks by drawing the Foata–Zeilberger path and complementing labels that

$${\rm FZ}_n:\;S_n\;\longleftrightarrow\; \{\,( \omega,\xi)\colon \omega\text{ a Motzkin path of length }n,\; \xi\text{ a compatible label-sequence}\,\}.$$7

It also states that $${\rm FZ}_n:\;S_n\;\longleftrightarrow\; \{\,( \omega,\xi)\colon \omega\text{ a Motzkin path of length }n,\; \xi\text{ a compatible label-sequence}\,\}.$$8 and $${\rm FZ}_n:\;S_n\;\longleftrightarrow\; \{\,( \omega,\xi)\colon \omega\text{ a Motzkin path of length }n,\; \xi\text{ a compatible label-sequence}\,\}.$$9 are the only two fixed points when $\omega$0, and then immediately remarks that this is inconsistent with $\omega$1 (Deb, 19 Aug 2025).

That inconsistency is addressed explicitly in the supplied material itself: “one must check carefully the precise definition of ‘cross’ vs. ‘nest.’” The parenthetical clarification adds that the full table in the paper gives the correct four-element fixed set for the particular crossing/nesting definitions in use. This is the principal cautionary point surrounding informal presentations of the involution: small examples may depend sensitively on the exact version of the crossing and nesting statistics being used.

The same example gives the sieving polynomial

$\omega$2

and computes

$\omega$3

In the toy table, that value matches the two fixed points $\omega$4 and $\omega$5. However, the note quoted above indicates that the definitive fixed-point count for the intended definitions is four, in accordance with the general lemma. The discrepancy is therefore not presented as a contradiction in the theory, but as a warning that the toy table does not fully encode the precise conventions used in the paper (Deb, 19 Aug 2025).