Séminaire Lotharingien de Combinatoire 62 (2010), Article B62c

FLAG-MAJOR INDEX AND FLAG-INVERSION NUMBER ON COLORED WORDS AND WREATH PRODUCT

Hilarion L. M. Faliharimalala^{1} and Arthur Randrianarivony^{2}

1Université de Lyon, Université Lyon 1, Institut Camille Jordan, UMR 5208 du CNRS, F-69622, Villeurbanne Cedex, France

                           E-mail: heritianamihanta@yahoo.fr

2Département de Mathématiques, Université d’Antananarivo BP 906 Antananarivo Madagascar

                          WWW: arthur@univ-antananarivo.mg

Abstract. In [Proc. Amer. Math. Soc. 19 (1968), 236–240], Dominique Foata constructed a map Φ, called second fundamental transformation, exchanging the integervalued statistics inversion number “inv” and major index “maj” on words whose letters are integers. Later, Foata and Han introduced the flag-inversion number “finv” and extended Φ on signed words and permutations, showing that the flag major index “fmaj” and “finv” were equidistributed. In this paper we give an extension of Φ to ℓ-colored words. Using this extension, we show that the bistatistics (fmaj, des^{∗}) and (finv, pcol) are equidistributed, where “pcol” is the sum of color powers and “des^{∗}” is a new statistic derived from “des”.

1. Introduction

The second fundamental transformation, denoted by Φ and described in [5] by Foata, is defined on finite words whose letters are integers. If m = (m_{1}, ⋯, m_{r}) is a sequence of nonnegative integers, let R_{m} be the set of all rearrangements w = x_{1} x_{2} ⋯ x_{m} of the sequence 1^{m_{1}}2^{m_{2}} … n^{m_{r}} where m = m_{1} + m_{2} + ⋯ + m_{r}. The transformation Φ maps each word w to another word Φ(w) and has the following properties:

(1) maj w = inv Φ(w);

(2) Φ(w) is a rearrangement of w and the restriction of Φ to R_{m} is a bijection of R_{m}

onto itself. Further properties were proved later on by Foata and Schützenberger [7], and by Björner and Wachs [3], in particular, when the transformation is restricted to act on the symmetric group S_{r}.

The purpose of this paper is to extend the transformation Φ to ℓ-colored words.

Let C_{ℓ} be the ℓ-cyclic group generated by ζ = e^{2iπ/ℓ}. By an ℓ-colored word, we understand a pair (ε, x), where ε ∈ (C_{ℓ})^{m} and x is a word of length m whose letters are nonnegative integers. For reasons which will appear later, if w := (ε, x) is an ℓ-colored word where ε = (ε_{1}, ε_{2}, …, ε_{m}) and x = x_{1} x_{2} ⋯ x_{m}, we write w := w_{1}w_{2} ⋯ w_{m} where w_{j} = ε_{j}x_{j} (1 ≤ j ≤ m). For any j with 1 ≤ j ≤ m, ε_{j} is called the color of w_{j} and, if ε_{j} = ζ^{k_{j}}, k_{j} is the power of this color. For small values of ℓ, we shall use k_{j} bars over x_{j} instead of ζ^{k_{j}}x_{j}.

For example, if w = ζ^{2}3 ζ^{2}1 ζ^{0}4 ζ1 ζ^{2}3, then we write w = ¯ ¯3¯¯14¯1¯¯3.

Any ℓ-colored word can be considered as a finite word over the alphabet Σ_{ℓ} := {ξj; ξ ∈ C_{ℓ}, j ≥ 1}.

Let w := w_{1}w_{2} ⋯ w_{m} := ε_{1}x_{1} ε_{2}x_{2} ⋯ ε_{m}x_{m} be an ℓ-colored word. We write

\begin{aligned}|w_{i}| & := x_{i},  1 ≤ i ≤ m; \\ |w| & := |w_{1}||w_{2}| ⋯ |w_{m}|; \end{aligned}  (1.1)

and we define the statistic power-color “pcol” by

pcol_{i}w := ∑_{0≤j≤ℓ−1} jχ(ε_{i} = ζ^{j}),  1 ≤ i ≤ m;
pcol w := ∑_{1≤i≤m} pcol_{i} w.

(1.2)

If m = (m_{1}, ⋯, m_{r}) is a sequence of nonnegative integers such that m_{1}+⋯+m_{r} = m, let G_{ℓ,m} be the set of all ℓ-colored words w = w_{1} w_{2} ⋯ w_{m} such that |w| ∈ R_{m}. The class G_{ℓ,m} contains ℓ^{m}\binom{m}{m_{1},m_{2},...,m_{r}} ℓ-colored words. When m_{1} = m_{2} = ⋯ = m_{r} = 1, the class G_{ℓ,m} is the wreath product C_{ℓ} ≀ S_{r} denoted by G_{ℓ,r}. We define an order relation on Σ_{ℓ} as follows:

ζ^{j}i > ζ^{j′}i′ ⟺ [j < j′] or [(j = j′) and (i > i′)].  (1.3)

The restriction of this order to the class of ordinary words (with nonnegative letters) is the usual order.

As in [6], the statistics “inv” and “maj” must be adapted to ℓ-colored words and correspond to classical statistics when applied to ordinary words. Let

(ω; q)_{n} := \begin{cases}1 & if n = 0; \\ (1 − ω)(1 − ωq) ⋯ (1 − ωq^{n−1}) & if n ≥ 1; \end{cases}

denote the usual q-shifted factorial, and let

[\begin{matrix}m_{1} + m_{2} + ⋯ + m_{r} \\ m_{1}, m_{2}, …, m_{r} \end{matrix}]_{q} := \frac{(q; q)_{m_{1}+m_{2}+···+m_{r}}}{(q; q)_{m_{1}}(q; q)_{m_{2}} ⋯ (q; q)_{m_{r}}}

be the q-multinomial coefficient.

With the order relation defined in (1.3), the natural extensions of the flag-major index “fmaj” and the flag-inversion number “finv” introduced by Foata and Han [6] to ℓ-colored

FLAG-MAJOR INDEX AND FLAG-INVERSION NUMBER ON COLORED WORDS 3

words are defined as follows: for all ℓ-colored word w := w_{1}w_{2} ⋯ w_{m},

fmaj w := ℓ ∑_{i=1}^{m−1} iχ(w_{i} > w_{i+1}) + pcol w;
finv w := _{∑_{1≤i<j≤m} ; ξ∈C_{ℓ}}χ(ξw_{i} > w_{j}) + pcol w.

Foata and Han defined (−q; q)_{m} [ \begin{matrix}m \\ m_{1},m_{2},...,m_{r} \end{matrix}]_{q} as a q-analog of 2^{m} ( \begin{matrix}m \\ m_{1},m_{2},...,m_{r} \end{matrix}). By anal-

ogy, \frac{(q^{ℓ};q^{ℓ})_{m}}{(q;q)_{m}} [ \begin{matrix}m \\ m_{1},m_{2},...,m_{r} \end{matrix}]_{q} is a natural q-analog of ℓ^{m} ( \begin{matrix}m \\ m_{1},m_{2},...,m_{r} \end{matrix}).

We claim that

\frac{(q^{ℓ}; q^{ℓ})_{m}}{(q; q)_{m}} [\begin{matrix}m \\ m_{1}, m_{2}, …, m_{r} \end{matrix}]_{q} = ∑_{w∈G_{ℓ,m}} q^{finv w}.  (1.4)

This can be established by induction on m. Indeed, let us consider the bijective transformation

φ : G_{ℓ,m} ⟶ {0, 1, …, ℓ − 1} × ⋃_{k=1}^{r} G_{ℓ,m-1_{k}},
w := w_{1}w_{2} ⋯ w_{m} ⟼ (s, w′) := (pcol_{m} w, w_{1}w_{2} ⋯ w_{m−1})

where m-1_{k} = (m_{1}, m_{2}, …, m_{k−1}, m_{k} − 1, m_{k+1}, …, m_{r}). We have k = |w_{m}| and

finv w = finv w′ + s +_{∑_{1≤i≤m−1} ; 0≤j≤ℓ−1}χ(ζ^{j}|w_{i}| > w_{m})
= finv w′ + ms + (m_{k+1} + ⋯ + m_{r})χ(k < r).

So,

∑_{w∈G_{ℓ,m}} q^{finv w} = ∑_{0≤s≤ℓ−1} q^{ms} ( ∑_{w′∈G_{ℓ,m-1r}} q^{finv w′} + ∑_{1≤k≤r−1} q^{m_{k+1}+···+m_{r}} ∑_{w′∈G_{ℓ,m-1_{k}}} q^{finv w′} )

= \frac{1 − q^{ℓm}}{1 − q^{m}} ( ∑_{w′∈G_{ℓ,m-1r}} q^{finv w′} + ∑_{1≤k≤r−1} q^{m_{k+1}+···+m_{r}} ∑_{w′∈G_{ℓ,m-1_{k}}} q^{finv w′} ) .

By induction, for each 1 ≤ k ≤ r, we have

∑_{w′∈G_{ℓ,m-1_{k}}} q^{finv w′} = \frac{(q^{ℓ}; q^{ℓ})_{m−1}}{(q; q)_{m−1}} \frac{(q; q)_{m−1}}{(q; q)_{m_{1}} ⋯ (q; q)_{m_{k−1}}(q; q)_{m_{k}−1}(q; q)_{m_{k+1}} ⋯ (q; q)_{m_{r}}}

= \frac{(q^{ℓ}; q^{ℓ})_{m−1}(1 − q^{m_{k}})}{(q; q)_{m}} \frac{(q; q)_{m}}{(q; q)_{m_{1}} ⋯ (q; q)_{m_{k}} ⋯ (q; q)_{m_{r}}} .

Thus,

∑_{1≤k≤r−1}q^{m_{k+1}+···+m_{r}} ∑_{w′∈G_{ℓ,m-1_{k}}} q^{finv w′}

= \frac{(q^{ℓ}; q^{ℓ})_{m−1}}{(q; q)_{m}} \frac{(q; q)_{m}}{(q; q)_{m_{1}} ⋯ (q; q)_{m_{k}} ⋯ (q; q)_{m_{r}}} ∑_{1≤k≤r−1} q^{m_{k+1}+···+m_{r}}(1 − q^{m_{k}})

= \frac{(q^{ℓ}; q^{ℓ})_{m−1}(q^{m_{r}} − q^{m})}{(q; q)_{m}} \frac{(q; q)_{m}}{(q; q)_{m_{1}} ⋯ (q; q)_{m_{k}} ⋯ (q; q)_{m_{r}}}

and

∑_{w∈G_{ℓ,m}} q^{finv w} = \frac{1 − q^{ℓm}}{1 − q^{m}} (\frac{(q^{ℓ}; q^{ℓ})_{m−1}(1 − q^{m_{r}})}{(q; q)_{m}} \frac{(q; q)_{m}}{(q; q)_{m_{1}} ⋯ (q; q)_{m_{k}} ⋯ (q; q)_{m_{r}}}

+ \frac{(q^{ℓ}; q^{ℓ})_{m−1}(q^{m_{r}} − q^{m})}{(q; q)_{m}} \frac{(q; q)_{m}}{(q; q)_{m_{1}} ⋯ (q; q)_{m_{k}} ⋯ (q; q)_{m_{r}}} )

= \frac{(q^{ℓ}; q^{ℓ})_{m}}{(q; q)_{m}} \frac{(q; q)_{m}}{(q; q)_{m_{1}} ⋯ (q; q)_{m_{k}} ⋯ (q; q)_{m_{r}}}

= \frac{(q^{ℓ}; q^{ℓ})_{m}}{(q; q)_{m}} [\begin{matrix}m \\ m_{1}, m_{2}, …, m_{r} \end{matrix}]_{q} .

This concludes the proof of the claim in (1.4).

We construct the extension Φ̂ of the second fundamental transformation Φ to ℓ-colored words in the next section. Define

des^{∗} w = ℓ des w − des |w| + pcol_{1} w,  (1.5)

where des w := ∑_{i=1}^{m−1} χ(w_{i} > w_{i+1}).

The main purpose of this paper is to prove the following theorem.

Theorem 1.1. The transformation Φ̂ constructed in Section 2 has the following properties

(1) For every ℓ-colored word w, (fmaj, des^{∗}) w = (finv, pcol) Φ̂(w) ;

(2) The restriction of Φ̂ to each class G_{ℓ,m} is a bijection of G_{ℓ,m} onto itself.

Corollary 1.2. For each m = (m_{1}, m_{2}, …, m_{r}), the bistatistics (fmaj, des^{∗}) and (finv, pcol) are equidistributed on G_{ℓ,m}.

Example 1.3. Let us consider the hyperoctahedral group of order 2.

w  1 2 ¯1 2 1 ¯2 ¯1 ¯2 2 1 ¯2 1 2 ¯1 ¯2 ¯1
fmaj w 0 1 3 2 2 1 3 4
des^{∗} w 0 1 2 1 1 0 1 2
finv w 0 1 2 3 1 2 3 4
pcol w 0 1 1 2 0 1 1 2

Now consider the statistic Rfinv defined on the hyperoctahedral group of order n as follows:

Rfinv w = inv w + ∑_{i=1}^{n} |w_{i}|χ(w_{i} < 0).

If one uses the natural order relation on [−n, n] given by

−n < −(n − 1) < ⋯ < −1 < 1 < ⋯ < (n − 1) < n,  (1.6)

Brenti [4] shows that finv coincides with the traditional length function, and Adin and Roichman [1] proved that Rfinv and fmaj are equidistributed on the hyperoctahedral group.

Back to the order relation (1.3) on [−n, n], i.e.,

−1 < ⋯ < −(n − 1) < −n < 1 < ⋯ < (n − 1) < n

one has length function ≠ Rfinv and finv ≠ Rfinv,

but we observe that Rfinv remains equidistributed with fmaj, and we prove that its extension to the wreath product is also Mahonian. We have the following theorem.

Theorem 1.4. The statistic Rfinv defined on the wreath product C_{ℓ} ≀ S_{n} by

Rfinv w = inv w + ∑_{i=1}^{n} |w_{i}| pcol_{i} w  (1.7)

is Mahonian.

By a result of Haglund, Loehr and Remmel [8], we obtain the following corollary.

Corollary 1.5. We have

∑_{σ∈G_{ℓ,n}} q^{Rfinv σ} = ∑_{σ∈G_{ℓ,n}} q^{finv σ} = \frac{(q^{ℓ}; q^{ℓ})_{n}}{(1 − q)^{n}} .  (1.8)

2. The construction of the transformation Φ̂

Let us recall the second fundamental transformation Φ from [5]. First, for each integer x, we recall the transformation γ_{x}. Let w = x_{1} x_{2} ⋯ x_{m} be a word with positive letters. The first (respectively last) letter x_{1} (respectively x_{m}) is denoted by F (w) (respectively L(w)). If L(w) ≤ x (respectively L(w) > x), w admits the unique factorization

(u_{1}y_{1}, u_{2}y_{2}, ⋯, u_{p}y_{p})

called its x-right-to-left factorization having the following properties:

(1) each y_{i} (1 ≤ i ≤ p) is a letter verifying y_{i} ≤ x (respectively y_{i} > x);

(2) each u_{i} (1 ≤ i ≤ p) is a factor which is either empty or has all its letters greater than (respectively smaller than or equal to) x.

Then, the bijective transformation γ_{x} maps w = u_{1}y_{1}u_{2}y_{2} … u_{p}y_{p} to the word

γ_{x}(w) = y_{1}u_{1}y_{2}u_{2} ⋯ y_{p}u_{p}.

Foata defined Φ(w) by induction on the length of w. If w has length one, then Φ(w) = w. If it has more than one letter, write the word as vx where x is the last letter and define Φ(vx) to be the juxtaposition product

Φ(vx) := γ_{x}(Φ(v))x.  (2.1)

We now define Φ̂ as follows. For each word u = x_{1} x_{2} ⋯ x_{m} with nonnegative letters and each element ϵ := (ϵ_{1}, ⋯, ϵ_{m}) of (C_{ℓ})^{m}, we denote by Ψ_{u}(ϵ) the element ϵ′ = (ϵ′_{1}, ⋯, ϵ′_{m}) of (C_{ℓ})^{m} defined as follows:

\begin{cases}ϵ′_{i} = \frac{ϵ_{i}}{ϵ_{i+1}} ζ^{−χ(x_{i}>x_{i+1})} & if i < m, \\ ϵ′_{m} = ϵ_{m} & if i = m. \end{cases}  (2.2)

Let w := (ϵ, u) be an ℓ-colored word (u = |w|). Define

Φ̂(w) = (Ψ_{u}(ϵ), Φ(u)).  (2.3)

Example 2.1. Let us take ℓ = 4 and w = ¯ ¯3¯14¯1¯¯3. We have

w = ((ζ^{2}, ζ, 1, ζ, ζ^{2}), 31413).

By construction of Φ (relation (2.1)), we have

Φ(3) = 3,
Φ(31) = γ_{1}(Φ(3))1 = γ_{1}(3)1 = 31,
Φ(314) = γ_{4}(Φ(31))4 = γ_{4}(31)4 = 314,
Φ(3141) = γ_{1}(Φ(314))1 = γ_{1}(314)1 = 3411,
Φ(31413) = γ_{3}(Φ(3141))3 = γ_{3}(3411)3 = 31413.

In the other hand,

ϵ′_{1} = \frac{ζ^{2}}{ζ} ζ^{−1} = 1, ϵ′_{2} = \frac{ζ}{1} = ζ, ϵ′_{3} = \frac{1}{ζ^{1}} ζ^{−1} = ζ^{2}, ϵ′_{4} = \frac{ζ^{1}}{ζ^{2}} = ζ^{3}, ϵ′_{5} = ζ^{2}.

Therefore,

w′ = Φ̂(¯¯3¯14¯1¯¯3) = 3¯1¯¯4¯¯¯1¯¯3.

We have (fmaj, des^{∗})w = (finv, pcol)w′ = (34, 8).

3. Proof of Theorem 1.1

Lemma 3.1. Let w := w_{1}w_{2} ⋯ w_{m} be an ℓ-colored word. With notation in relation (1.2), we have

finv w = inv |w| + ∑_{i=1}^{m} i pcol_{i} w.  (3.1)

FLAG-MAJOR INDEX AND FLAG-INVERSION NUMBER ON COLORED WORDS 7

Proof. For all integers i, j, k such that 1 ≤ i < j ≤ m and 0 ≤ k ≤ ℓ − 1, one has:

χ(ζ^{k}|w_{i}| > w_{j}) = χ(k < pcol_{j} w) + χ(k = pcol_{j} w)χ(|w_{i}| > |w_{j}|).

Thus,

finv w = ∑_{j=1}^{m}(j − 1) pcol_{j} w + inv |w| + ∑_{j=1}^{m} pcol_{j} w = inv |w| + ∑_{j=1}^{m} j pcol_{j} w.

□

Now, slightly abusing notation, let w = (ϵ, |w|) be an ℓ-colored word of length m and w′ = (ϵ′, |w′|) := Φ̂(w). For each i such that 1 ≤ i ≤ m − 1, we have

– if |w_{i}| ≤ |w_{i+1}|, then ϵ′_{i} = \frac{ϵ_{i}}{ϵ_{i+1}} = ζ^{pcol_{i} w−pcol_{i+1} w};

– if |w_{i}| > |w_{i+1}|, then ϵ′_{i} = \frac{ϵ_{i}}{ϵ_{i+1}} ζ^{−1} = ζ^{pcol_{i} w−pcol_{i+1} w−1}.

So,

pcol_{i} w′ = [pcol_{i} w − pcol_{i+1} w + ℓχ(pcol_{i} w < pcol_{i+1} w)]χ(|w_{i}| ≤ |w_{i+1}|)
+ [pcol_{i} w − pcol_{i+1} w − 1 + ℓχ(pcol_{i} w ≤ pcol_{i+1} w)]χ(|w_{i}| > |w_{i+1}|)
= pcol_{i} w − pcol_{i+1} w + ℓχ(pcol_{i} w < pcol_{i+1} w)
+ ℓχ(pcol_{i} w = pcol_{i+1} w)χ(|w_{i}| > |w_{i+1}|) − χ(|w_{i}| > |w_{i+1}|).

Thus, we have

pcol w′ = ∑_{i=1}^{m} pcol_{i} w′

= ∑_{i=1}^{m} pcol_{i} w − ∑_{i=2}^{m} pcol_{i} w + ℓ ∑_{i=1}^{m−1}[χ(pcol_{i} w < pcol_{i+1} w)

+ χ(pcol_{i} w = pcol_{i+1} w)χ(|w_{i}| > |w_{i+1}|)] − ∑_{i=1}^{m−1} χ(|w_{i}| > |w_{i+1}|)

= pcol_{1} w + ℓ des w − des |w|

= des^{∗} w,

and, by Φ,

finv w′ = inv |w′| + ∑_{i=1}^{m} i pcol_{i} w′

= maj |w| + ∑_{i=1}^{m} i pcol_{i} w − ∑_{i=1}^{m}(i − 1) pcol_{i} w

+ ℓ ∑_{i=1}^{m−1} i[χ(pcol_{i} w < pcol_{i+1} w) + χ(pcol_{i} w = pcol_{i+1} w)χ(|w_{i}| > |w_{i+1}|)]

− ∑_{i=1}^{m−1} iχ(|w_{i}| > |w_{i+1}|)

= maj |w| + ∑_{i=1}^{m} pcol_{i} w + ℓ ∑_{i=1}^{m−1} iχ(w_{i} > w_{i+1}) − maj |w|

= ℓ ∑_{i=1}^{m−1} iχ(w_{i} > w_{i+1}) + pcol w

= fmaj w.

Finally, we show that Φ̂ is a bijection of G_{ℓ,m} onto itself. Indeed, let w′ := (ϵ′, u) be an element of G_{ℓ,m}. By the relation (2.2), if w := (ϵ, u′) is an element of G_{ℓ,m} such that Φ̂(w) = w′, then u = Φ^{−1}(u′), ϵ_{m} = ϵ′_{m} and, for i < m,

ϵ_{i} = ϵ′_{m} ∏_{i≤j≤m−1} ϵ′_{j}ζ^{χ(x_{j}>x_{j+1})},

where u := x_{1}x_{2} ⋯ x_{m}.

This concludes the proof of Theorem 1.1.

4. Proof of Theorem 1.4

Consider the following transformations:

▷ Transformation ρ

ρ : G_{ℓ,n} ⟶ G_{ℓ,n}
w = (ϵ, |w|) ⟼ ρ(w) = w′ = (ϵ′, |w′|),

where

|w′| = |w|^{−1} and ϵ′_{i} = ϵ_{|w|^{−1}(i)};

▷ Transformation τ For each α = (α_{1}, α_{2}, …, α_{n}) in [0, ℓ − 1]^{n}, put Σ_{α} = {ζ^{α_{1}}, ζ^{α_{2}}2, …, ζ^{α_{n}}n}, and let G_{α} be the class of ℓ-colored permutations whose letters are in Σ_{α}, i.e.,

G_{α} = {w = w_{1}w_{2} ⋯ w_{n} ∈ G_{ℓ,n} : pcol_{i} w = α_{i} for all i ∈ [n]}.

Note that ♯G_{α} = n!. We denote by I_{α} the increasing bijection from [n] to Σ_{α}, and we define τ for each class G_{α} by τ(w) = w′_{1}w′_{2} … w′_{n}, where

w′_{i} = I_{α}(|w_{i}|).

Lemma 4.1. For all w ∈ G_{ℓ,n}, we have

finv w = Rfinv τ ∘ ρ(w).

Proof of Lemma 4.1. Let w = w_{1}w_{2} ⋯ w_{n} ∈ G_{ℓ,n}. Consider the auxiliary statistics

℘(w) := ∑_{i=1}^{n} i pcol_{i} w;

ℑ(w) := ∑_{i=1}^{n} |w_{i}| pcol_{i} w;
|inv| w := inv |w|.

It is easy to see that ρ is an involution preserving |inv| and transforming ℘ into ℑ and vice versa:

(|inv|, ℘)w = (|inv|, ℑ)ρ(w),

and τ preserves ℑ and transforms |inv| into inv, i.e.,

(|inv|, ℑ)w = (inv, ℑ)ρ(w).

By Lemma 3.1, we have

finv w = |inv| w + ℘(w)
= |inv|ρ(w) + ℑ ρ(w) = inv τ ∘ ρ(w) + ℑ τ ∘ ρ(w)
= Rfinv τ ∘ ρ(w).
□

Example 4.2. w = 5¯¯3¯12¯4, finv w = 18; ρ(w) = ¯34¯¯2¯51. Let α = (0, 2, 1, 0, 1). I_{α} is defined as follows:

\begin{matrix}i & 1 & 2 & 3 & 4 & 5 \\ I_{α}(i) & ¯¯2 & ¯3 & ¯5 & 1 & 4 \end{matrix}

So

τ ∘ ρ(w) = w′ = ¯51¯34¯¯2 and Rfinv w′ = 18.

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