Combinatorial study of the Dellac configurations and the q-extended normalized median Genocchi numbers

In two recent papers (\textit{Mathematical Research Letters,18(6):1163--1178,2011} and \textit{European J. Combin.,33(8):1913--1918,2012}), Feigin proved that the Poincar\'e polynomials of the degenerate flag varieties have a combinatorial interpretation through the Dellac configurations, and related them to the $q$-extended normalized median Genocchi numbers $\bar{c}_n(q)$ introduced by Han and Zeng, mainly by geometric considerations. In this paper, we give combinatorial proofs of these results by constructing statistic-preserving bijections between the Dellac configurations and two other combinatorial models of $\bar{c}_n(q)$.

. . .  It is well known that H 2n+1 is divisible by 2 n (see [1]) for all n ≥ 0. The normalized median Genocchi numbers (h n ) n≥0 = (1, 1, 2, 7, 38, . . .) are the positive integers defined by Dumont [3] gave several combinatorial models of the Genocchi numbers and the median Genocchi numbers, among which are the Dumont permutations. We denote by S n the set of permutations of the set [n] := {1, 2, . . . , n}, and by inv(σ) the number of inversions of a permutation σ ∈ S n , i.e., the quantity of pairs (i, j) ∈ [n] 2 with i < j and σ(i) > σ(j). Broadly speaking, the number of inversions inv(w) of a word w = l 1 l 2 . . . l n with n letters in the alphabet N is the quantity of pairs (i, j) ∈ [n] 2 such that i < j and l i > l j . In particular, the number inv(σ) associated with a permutation σ ∈ S n is the quantity inv(w) associated with the word w = σ(1)σ (2) . . . σ(n).
Definition 1.1. A Dumont permutation of order 2n is a permutation σ ∈ S 2n such that σ(2i) < 2i and σ(2i − 1) > 2i − 1 for all i. We denote by D n the set of these permutations.
It is well-known (see [3]) that H 2n+1 = |D n+1 | for all n ≥ 0. In [9], Han and Zeng introduced the set G n of normalized Genocchi permutations, which consist of permutations σ ∈ D n such that for all j ∈ [n − 1], the two integers σ −1 (2j) and σ −1 (2j + 1) have the same parity if and only if σ −1 (2j) < σ −1 (2j + 1), and they proved that h n = |G n+1 | for all n ≥ 0. The number h n also counts the Dellac configurations of size n (see [6]). Definition 1.2. A Dellac configuration of size n is a tableau of width n and height 2n which contains 2n dots between the lines y = x and y = n+x, such that each row contains exactly one dot and each column contains exactly two dots. Let DC(n) be the set of Dellac configurations of size n. An inversion of C ∈ DC(n) is a pair (d 1 , d 2 ) of dots whose Cartesian coordinates in C are respectively (j 1 , i 1 ) and (j 2 , i 2 ) such that j 1 < j 2 and i 1 > i 2 . We denote by inv(C) the number of inversions of C. For example, the tableau depicted in Figure 2 is a Dellac configuration C ∈ DC(3) with inv(C) = 2 inversions (represented by two segments). In [9,10], Han and Zeng defined the q-Gandhi polynomials of the second kind (C n (x, q)) n≥1 by C 1 (x, q) = 1 and C n+1 (x, q) = (1 + qx)∆ q (xC n (x, q)), where ∆ q P (x) = (P (1 + qx) − P (x))/(1 + qx − x) for all polynomial P (x). They proved that the polynomials C n (1, q) are q-analogs of the median Genocchi numbers (C n (1, 1) = H 2n−1 ). Furthermore, they gave a combinatorial interpretation of C n (1, q) through D n . Theorem 1.1 (Han and Zeng, 1997). Let n ≥ 1. For all σ ∈ D n , we define st(σ) as the quantity where σ o and σ e are the two words σ(1)σ(3) . . . σ(2n − 1) and σ(2)σ(4) . . . σ(2n) respectively. Then, the polynomial C n (1, q) has the following combinatorial interpretation: By introducing the subset G n ⊂ D n of normalized Genocchi permutations and using the combinatorial interpretation provided by Theorem 1.1, Han and Zeng proved combinatorially that the polynomial (1 + q) n−1 divides C n (1, q), which gives birth to polynomials (c n (q)) n≥1 defined byc This divisibility had previously been proved in the same paper with a continued fraction approach, as a corollary of the following theorem and a well-known result on continued fractions (see [8]).
Theorem 1.2 (Han and Zeng, 1997). The generating function of the sequence (c n+1 (q)) n≥0 is where the sequence (λ n ) n≥1 is defined by The polynomials (c n (q)) n≥1 are q-refinements of normalized median Genocchi numbers: c n (1) = h n−1 for all n ≥ 1. They are named q-extended normalized median Genocchi numbers. In §2.1, we give a combinatorial interpretation ofc n (q) by slightly adjusting the definition of normalized Genocchi permutations. In [6,7], Feigin introduced a q-analog of the normalized median Genocchi number h n with the Poincaré polynomial P F a n (q) of the degenate flag variety F a n (whose Euler characteristic is P F a n (1) = h n ), and gave a combinatorial interpretation of P F a n (q) through Dellac configurations. Theorem 1.3 (Feigin, 2012). For all n ≥ 0, the polynomial P F a n (q) is generated by DC(n): The degree of the polynomial P F a n (q) being n(n + 1) (for algebraic considerations, or because every Dellac configuration C ∈ DC(n) has at most n 2 inversions, see §2.1), Feigin introduced the following q-analog of h n : and proved the following theorem by using the geometry of quiver Grassmannians (see [11]) and Flajolet's theory of continued fractions [8].
This raises two questions.
(1) Prove combinatorially Corollary 1.5 by constructing a bijection between Dellac configurations and some appropriate model ofc n (q) which preserves the statistics. (2) Prove combinatorially Theorem 1.4 within the framework of Flajolet's theory of continued fractions by defining a combinatorial model ofh n (q) related to Dyck paths (see [8]), and constructing a statistic-preserving bijection between Dellac configurations and that new model. The aim of this paper is to answer above two questions. We answer the first one in §2. In §2.1, we define a combinatorial model ofc n (q) through normalized Dumont permutations, and we provide general results about Dellac configurations. In §2.2, we enounce and prove Theorem 2.2, which connects Dellac configurations to normalized Dumont permutations through a stastisticpreserving bijection, and implies immediatly Corollary 1.5. We answer the second question in §3. In §3.1, we recall the definition of a Dyck path and some results of Flajolet's theory of continued fractions. In §3.2, we define Dellac histories, which consist of Dyck paths weighted with pairs of integers, and we show that their generating function has the continued fraction expansionn of Formula (4). In §3.3, we enounce and prove Theorem 3.3, which connects Dellac configurations to Dellac histories through a statisticpreserving bijection, thence proving Theorem 1.4 combinatorially.

Connection between Dellac configurations and Dumont permutations
In §2.1, we define normalized Dumont permutations of order 2n, whose set is denoted by D n , and we prove that they generatec n (q) with respect to the statistic st defined in Formula (1), then we define the label of a Dellac configuration and a switching transformation on the set DC(n). In §2.2, we enounce Theorem 2.2 and we intend to demonstrate it. To do so, we first give two algorithms φ : DC(n) → D n+1 and ϕ : D n+1 → DC(n), and we prove that φ and ϕ |D n+1 are inverse maps. Then, we show that Equation (6) is true for all C ∈ DC(n), by showing that it is true for some particular C 0 ∈ DC(n), then by connecting C 0 to every other C ∈ DC(n) thanks to the switching transformation, which happens to preserve Equation (6).

Label of a Dellac configuration.
Definition 2.2. Let C ∈ DC(n). For all i ∈ [n], the dot of the i-th line of C (from bottom to top) is labeled by the integer e i = 2i + 2, and the dot of the (n + i)-th line is labeled by the integer e n+i = 2i − 1 (see Figure 3 for an example). >From now on, we will assimilate each dot of a Dellac configuration into its label.  . For all j ∈ [n], we define i C 1 (j) < i C 2 (j) such that the two dots of the j-th column of C (from left to right) are e i C 1 (j) and e i C 2 (j) . When there is no ambiguity, we write e i 1 (j) and e i 2 (j) instead of e i C 1 (j) and e i C 2 (j) . Finally, for all i ∈ [n], we define the integers p C (i) and q C (i) such that e p C (i) and e n+q C (i) are respectively the i-th even dot and i-th odd dot of the sequence e i 1 (1) , e i 2 (1) , e i 1 (2) , e i 2 (2) , . . . , e i 1 (n) , e i 2 (n) .  Likewise, the last n − j + 1 columns of C always contain the n − j + 1 odd dots e n+j , e n+j+1 , . . . , e 2n , and the only even dots they may contain are e j , e j+1 , e j+2 , . . . , e n .
It is obvious that C 0 (n) is the unique Dellac configuration of size n with 0 inversion, and that inv(C 1 (n)) = n 2 . We can also prove by induction on n ≥ 1 that every Dellac configuration C ∈ DC(n) has at most n 2 inversions with equality if and only if C = C 1 (n).

Refinements of the inv statistic on DC(n).
Definition 2.5. Let C ∈ DC(n) and i ∈ [2n]. We define the quantity l C (e i ) (resp. r C (e i )) as the number of inversions of C between the dot e i and any dot e i with i > i (resp. i < i).

Switching of a Dellac configuration.
In the following definition, we provide a tool which transforms a Dellac configuration of DC(n) into a slightly modified tableau, which is not necessarily a Dellac configuration and consequently brings the notion of switchability.
Definition 2.6. Let C ∈ DC(n) and i ∈ [2n − 1]. We denote by Sw i (C) the tableau obtained by switching the two consecutive dots e i and e i+1 (i.e., inserting e i in e i+1 's column and e i+1 in e i 's column). If the tableau Sw i (C) is still a Dellac configuration, we say that C is switchable at i. In Figure 5, we give an example C ∈ DC(3) switchable at 2.
It is easy to verify the following assertions.
if and only if C and i satisfy one of the two following conditions: (1) i ≤ n and if e i+1 is in the j i+1 -th column of C, then j i+1 < i + 1.
(2) i > n and if e i is in the j i -th column of C, then j i > i − n.
In particular : is still switchable at i and Sw i (Sw i (C)) = C.
Fact 2.4. If e i and e i+1 are in the same column of C, then C is switchable at i and C = Sw i (C).
is an inversion of C, then C is switchable at i and inv(Sw i (C)) = inv(C) − 1 (like in Figure 5).
Fact 2.6. A Dellac configuration C ∈ DC(n) is always switchable at n.

2.2.
Construction of a statistic-preserving bijection. In this part, we intend to prove the following result.
There exists a bijection φ : DC(n) → D n+1 such that the equality is true for all C ∈ DC(n).
In the following, we define φ : DC(n) → D n+1 and in order to prove that it is bijective, we construct ϕ : D n+1 → DC(n) such that φ and ϕ |D n+1 are inverse maps.
2.2.1. Algorithms. Definition of φ. We define φ : DC(n) → S 2n+2 by mapping C ∈ DC(n) to the permutation φ(C) ∈ S 2n+2 defined as the inverse map of the permutation where we recall that e i 1 (j) and e i 2 (j) are respectively the lower and upper dots of the j-th column of C for all j ∈ [n]. (3) is the Dellac configuration depicted in Figure 6, we obtain φ(C) −1 = 2 84 16 537.  Proof. Let σ be φ(C). It is a Dumont permutation : (σ(2), σ(2n + 1)) = (1, 2n + 2) and Definition of ϕ. Let T n be the set of tableaux of size n × 2n whose each row contains one dot and each column contains two dots. We define ϕ : D n+1 → T n by mapping σ ∈ D n+1 to the tableau ϕ(σ) ∈ T n whose j-th column contains the two dots labelled by σ −1 (2j) and σ −1 (2j + 1) for all j ∈ [n].
Proof. Recall that if the dot e i is located in the j-th column of C, then φ(C)(e i ) = 2j or 2j + 1. Consequently, since y i = i if i is even, and y i = i + 2 if i is odd, then τ C (i) = 2j or 2j − 1. Now let 1 ≤ p < q ≤ 2n, and let (j p , j q ) such that the dot e p (resp. e q ) is located in the j p -th column (resp. j q -th column) of C. If (e p , e q ) is an inversion of C, i.e., if j p > j q , then It means that e p and e q are the lower dot and the upper dot of the j-th column respectively, which translates into y τ C (p) = φ(C)(e p ) = 2j + 1 and y τ C (q) = φ(C)(e q ) = 2j. Consequently, we obtain τ C (p) = 2j − 1 and τ C (q) = 2j, which is in contradiction with τ C (p) > τ C (q). So j p > j q and (e p , e q ) is an inversion of C.
Proof of Lemma 2.6. >From Lemma 2.5, we know that So, the lemma follows from the well-known equality for all permutation π ∈ S m and for all integer m ≥ 1.

Switchability and Dumont permutations.
We have built a bijection φ : DC(n) → D n+1 . To demonstrate Formula 6, we will use the notion of switchability defined in §2.1, by showing that if Formula 6 is true for some particuliar configuration C 0 , and if C 1 is a configuration connected to C 0 by a switching transformation, then Formula 6 is also true for C 1 . We will also need Lemma 2.7 and Proposition 2.8 to prove (in Proposition 2.9) that any two Dellac configurations are connected by a sequence of switching transformations.
with the permutation σ. The Dellac configuration ϕ(σ) is switchable at i if and only if σ is still a Dumont permutation, and in that case ϕ(σ ) = Sw i (ϕ(σ)).
(1) ϕ(σ ) = ϕ(σ); (2) the two dots e i and e i+1 are not in the same column of ϕ(σ); Proposition 2.9. Let (C 1 , C 2 ) ∈ DC(n) 2 . There exists a finite sequence of switching transformations from C 1 to C 2 , i.e., a sequence (C 0 , C 1 , . . . , C m ) in DC(n) for some m ≥ 0 such that (C 0 , C m ) = (C 1 , C 2 ) and such that Proof. From Fact 2.3, it is sufficient to prove that for all C ∈ DC(n), there exists a finite sequence of switching transformations from C to C 0 (n), the unique Dellac configuration of size n with 0 inversion (see Definition 2.4). If C = C 0 (n), the statement is obvious. Else, let C 0 = C. >From Lemma 2.5, for all i ∈ [2n], the pair (e i , e i+1 ) is an inversion of C 0 if and only if the integer i is a descent of τ C 0 , i.e., if τ C 0 (i) > τ C 0 (i + 1). Now, from Proposition 2.6, the permutation τ C 0 (n) is the identity map Id of S 2n+2 . Consequently, since C 0 = C 0 (n), we have τ C 0 = Id S 2n , so τ C 0 has at least one descent. Let i 0 be one of those descents, and let Since (e i 0 , e i 0 +1 ) is an inversion of C 0 , in particular e i 0 and e i 0 +1 are not in the same column, so, from Proposition 2.8, we have φ( Iterating the process with C 1 , and by induction, we build a finite sequence of switching transformations (C 0 , C 1 , . . . , C m ) such that τ C m has no descent, i.e., such that τ C m = Id = τ C 0 (n) , which implies C m = C 0 (n).
Let C ∈ DC(n). From Lemma 2.9, there exists a finite sequence of switching transformations Since Formula (6) is true for C 1 (n), it will be true for C by induction if we show that for all k. We know that the quantity inv(C k ) − inv(C k+1 ) equals ±1. From Fact 2.3, we have Sw i k (C k+1 ) = C k . Then, provided that C k is replaced by Sw i k (C k ) = C k+1 , we can assume that the quantity inv(C k ) − inv(C k+1 ) equals 1, which means the pair (e i k , e i k+1 ) is an inversion of C k . Consequently, to achieve the proof of Theorem 2.2, it suffices to prove the equality under the hypothesis inv(C k ) − inv(C k+1 ) = 1. Let σ k = φ(C k ) and σ k+1 = φ(C k+1 ). Since e i k and e i k +1 are not in the same column of C k , we have σ k+1 = σ k • (e i k , e i k +1 ) in view of Proposition 2.8.
In view of Lemma 2.10, Equalities (9) and (10) >From y i = i + 1 − (−1) i for all i, we deduce the two following formulas.
Remark 2.4. In [9], the authors proved thatc n (q) is divisible by 1 + q if n is odd, but requested a combinatorial proof of this statement. Now, if n is odd, one can prove that every Dellac configuration C ∈ DC(n − 1) is switchable at some even integer, which yields a natural involution I on DC(n − 1) such that inv(I(C)) = inv(C) ± 1 for all C. This proves combinatorially the divisibility ofc n (q) by 1 + q in view of Theorem 2.2.

Dellac histories
3.1. Weighted Dyck paths. Recall (see [8]) that a Dyck path γ of length 2n is a sequence of points (p 0 , p 1 , . . . , p 2n ) in N 2 such that (p 0 , p 2n ) = ((0, 0), (2n, 0)), and for all i ∈ [2n], the step (p i−1 , p i ) is either an up step (1, 1) or a down step (1, −1). We denote by Γ(n) the set of Dyck paths of length 2n. Furthermore, let µ = (µ n ) n≥1 be a sequence of elements of a ring. A weighted Dyck path is a Dyck path γ = (p i ) 0≤i≤n ∈ Γ(n) whose each up step has been weighted by 1, and each down step (p i−1 , p i ) from height h (i.e., such that p i−1 = (i − 1, h)) has been weighted by µ h . The weight ω µ (γ) of the weighted Dyck path γ is the product of the weights of all steps.
where n u (i) and n d (i) are defined as the numbers of up steps and down steps on the left of p i respectively (in particular n u (i) + n d (i) = i). Consequently, since the final point of γ is p 2n = (2n, 0), the path γ has exactly n up steps and n down steps, and for all j ∈ [n], the points p 2j−1 and p 2j are at heights respectively odd and even.

Figure 8
The weight ω(γ, ξ) of the history (γ, ξ) is the product of the weights of all down steps. We denote by DH(n) the set of Dellac histories of length 2n.
Prior to connecting Dellac histories to weighted Dyck paths, one can easily verify the two following results.
Lemma 3.1. For all p ≥ 1, we have the equality Proposition 3.2. For all γ 0 ∈ Γ(n), we have the equality where ω λ has been defined in (22), and where λ = (λ n ) n≥1 is the sequence defined in Theorem 1.2.

3.3.
Proof of Theorem 3.3. In this part, we give preliminaries and connections between Dellac configurations and Dyck paths. Then, we define the algorithm Φ : DC(n) → DH(n) and we demonstrate the statistic preservation formula (23). Finally, we prove that Φ is bijective by giving an algorithm Ψ : DH(n) → DC(n) which happens to be Φ −1 .

Preliminaries on Dellac configurations.
Definition 3.3. Let C ∈ DC(n). If i ≤ n, we denote by l e C (e i ) the number of inversions of C between e i and any even dot e i ≤n with i > i. In the same way, if i > n, we denote by r o C (e i ) the number of inversions of C between e i and any odd dot e i >n with i < i. Definition 3.4. Let C ∈ DC(n) and j ∈ [n]. We define the height h(j) of the integer j as the number n e (j) − n o (j) where n e (j) (resp. n o (j)) is the number of even dots (resp. odd dots) in the first j − 1 columns of C (with n e (1) = n o (1) = 0). Remark 3.3. Since the first j − 1 columns of C contain exactly 2j − 2 dots and, from Remark 2.1, always contain the j − 1 even dots e 1 , e 2 , . . . , e j−1 , there exists k ∈ {0, 1, . . . , j − 1} such that n e (j) = j − 1 + k and n o (j) = j − 1 − k. In particular h(j) = 2k. Lemma 3.4. Let C ∈ DC(n), let j ∈ [n] and k ≥ 0 such that h(j) = 2k. If the j-th column of C contains two odd dots, there exists j < j such h(j + 1) = 2k and such that the j -th column of C contains two even dots.
(3) If n < i 1 (j), let i ∈ [n] such that i − 1 down steps have already been defined. We define (p 2j−2 , p 2j−1 ) and (p 2j−1 , p 2j ) as down steps (see Figure 8, (3)). Afterwards, let k ≥ 0 such that p 2j−2 = (2j − 2, 2k). Obviously, the number n u (2j − 2) = j − 1 + k of up steps (resp. the number n d (2j − 2) = j − 1 − k of down steps) that have already been defined is the number n e (j) of even dots (resp. the number n o (j) of odd dots) in the first j − 1 columns of C, thence h(j) = 2k. From Lemma 3.4, there exists j < j such that h(j + 1) = 2k (which means p 2j = (2j , 2k)) and such that the j -th column of C contains two even dots, which means (p 2j −2 , p 2j −1 ) and (p 2j −1 , p 2j ) are two consecutive up steps (see Figure 9). Now, we consider the maximum j m < j of Figure 9. Two consecutive up steps and down steps at the same level.
the integers j that verify this property, and we consider the two dots e i 1 (jm) and e i 2 (jm) (with i 1 (j m ) < i 2 (j m )) of the j m -th column of C. Finally, we define ξ i and ξ i+1 as   Figure 11. Ψ(C) ∈ DH(6).
Remark 3.4. If Φ(C) = (γ, ξ), there are as many up steps (resp. down steps) as even dots (resp. odd dots) in the first j columns of C. With precision, for all i ∈ [n], the even dot e p C (i) and the odd dot e n+q C (i) (see Definition 2.3) give birth to the i-th up step and the i-th down step of γ respectively. In particular, the path γ has n up steps and n down steps, so p 2n = (2n, 0). To prove that γ is a Dyck path, we still have to check that it never goes below the line y = 0.
Proposition 3.6. For all C ∈ DC(n), the data Φ(C) is a Dellac history of length 2n.
• If (p 2j−1 , p 2j ) is the down step s d i in the context (2)(a) of Definition 3.5, then ξ i = (n 1 , n 2 ) = (l e C (e i 1 (j) )), r o C (e i 2 (j) )) with l e C (e i 1 (j) ) > r o C (e i 2 (j) ). Here, the appropriate inequality to check is k ≥ n 1 > n 2 (this is the context (1) of Definition 3.2). Since the first j − 1 columns of C contain j − 1 + k even dots, including the j − 1 dots e 1 , e 2 , . . . , e j−1 (with j − 1 < i 1 (j)), there is no inversion between any of these dots and e i 1 (j) . Consequently, in the first j − 1 columns of C, there are at most (j − 1 + k) − (j − 1) = k even dots e i with n ≥ i > i 1 (j), thence n 1 = l e C (e i 1 (j) ) ≤ k. • Similarly, if (p 2j−2 , p 2j−1 ) is the down step s d i set in the context (2)(b) of Definition 3.5, then we have ξ i = (n 1 , n 2 ) = (l e C (e i 1 (j) )), r o C (e i 2 (j) )), with l e C (e i 1 (j) ) ≤ r o C (e i 2 (j) ). Now, the appropriate equality to check is n 1 ≤ n 2 ≤ k (this is the context (2) of Definition 3.2). The first j columns of C contain j − k odd dots and the i 2 (j) − n lines from the (n + 1)-th line to the i 2 (j)-th line contain i 2 (j) − n odd dots, so, in the n − j last columns, the number of odd dots e i with n < i < i 2 (j) is at most (i 2 (j) − n) − (j − k) = k + (i 2 (j) − j − n) ≤ k, thence n 2 = r o C (e i 2 (j) ) ≤ k. • Finally, if (p 2j−2 , p 2j−1 ) and (p 2j−1 , p 2j ) are two consecutive down steps s d i and s d i+1 in the context (3) of Definition 3.5, then ξ i = (l e C (e i 1 (jm) ), l e C (e i 2 (jm) )), ξ i+1 = (r o C (e i 1 (j) ), r o C (e i 2 (j) )) and the two inequalities to check (this is the context (3) of Definition 3.5) are: -Proof of (24): since i 1 (j m ) < i 2 (j m ), obviously l e C (e i 1 (jm) ) ≥ l e C (e i 2 (jm) ). Afterwards, since p 2jm−2 is at the level h(j m ) = 2k − 2, there are j m − 1 + (k − 1) = j m + k − 2 even dots in the first j m − 1 columns of C. Since the first j m − 1 rows of C contain the j m − 1 even dots e 1 , e 2 , . . . , e jm−1 , the first j m − 1 columns of C contain at most (j m + k − 2) − (j m − 1) = k − 1 even dots e i with n ≥ i > i 1 (j m ), thence l e C (e i 1 (jm) ) ≤ k − 1.
-Proof of (25): since i 1 (j) < i 2 (j), obviously r o C (e i 1 (j) ) ≤ r o C (e i 2 (j) ). Afterwards, since p 2j is at the level h(j + 1) = 2k − 2, there are j − (k − 1) = j − k + 1 odd dots in the first j columns of C. Since the j rows, from the (n + 1)-th row to the (n+j)-th row of C, contain j odd dots, the n−j last columns of C contain at most j − (j − k + 1) = k − 1 odd dots e i with n < i < i 2 (j m ), thence r o C (e i 2 (j) ) ≤ k − 1. So Φ(C) is a Dellac history of length n.
(26) Since p 2j−2 is at the level h(j) = 2k, the first j − 1 columns of C contain j − 1 − k odd dots. Consequently, following Definition 3.5, the step s d i is the (j − k)-th down step of γ, i.e., the integer i equals j − k. Also, since the first j columns of C contain j + k even dots, the last n − j columns of C (from the (j + 1)-th column to the n-th column) contain n − (j + k) = n − j − k = i − k even dots. As a result, we obtain the equality In view of (27), Equality (26) becomes ω i = q n−i−(l e C (ei 1 (j)) +r C (ei 2 (j))) . With the same reasoning, if s d i and s d i+1 are two consecutive down steps in the context (3) of Definition 3.2, then by commuting factors of ω i and ω i+1 , we obtain the equality ω i ω i+1 = q n−i−(l e C (ei 1 (jm)) +r C (ei 2 (jm))) q n−(i+1)−(l e C (ei 1 (j)) +r C (ei 2 (j))) .
Now, it is easy to see that inv(C) = i≤n l e C (e i ) + i>n r C (e i ). In view of the latter remark, Formula (28) becomes Formula (23).
(1) Insertion of the n odd dots e n+1 , e n+2 , . . . , e 2n . Let I o 0 = (1, 2, . . . , n). For i = 1 to n, consider j i ∈ [n] such that the i-th down step s d i of γ is one of the two steps (p 2j i −2 , p 2j i −1 ) or (p 2j i −1 , p 2j i ). If the set I o i−1 ⊂ I o 0 is defined, we denote by H(i) the hypothesis "I o i−1 has size n + 1 − i such that for all j ∈ {i, i + 1, . . . , n}, the (j − i + 1)-th element of I o i−1 is inferior to n + j". If the hypothesis H(i + 1) is true, then we iterate the algorithm to i + 1. At the beginning, I o 0 is defined and H(1) is obviously true so we can initiate the algorithm. (a) If s d i is a down step in the context (1) or (2) of Definition 3.2, let (n 1 , n 2 ) = ξ i . In particular, since n 2 ≤ k = j i − i (see Remark 3.2) and j i ≤ n, we have 1 + n 2 ≤ n − i + 1 so, from Hypothesis H(i), we can consider the (1 + n 2 )-th element of I o i−1 , say, the integer q. We insert the odd dot e n+q in the j i -th column of T . From Hypothesis H(i), the (j i − i + 1)-th element of I o i−1 is inferior to n + j i , and 1 + n 2 ≤ 1 + k = j i − i + 1. Consequently, the dot e n+q is between the lines y = x and y = x + n. Afterwards, we define I o i as the sequence I o i−1 from which we have removed q (by abusing the notation, we write I o i := I o i−1 \{q}). Thus, the set I o i has size n + 1 − (i + 1). Also, if j ∈ {i + 1, i + 2, . . . , n}, then following Hypothesis H(i), the (j − i)-th element of I o i−1 is inferior to n + j − 1, so the (j − (i + 1) + 1)-th Example 3.2. If S ∈ DH(6) is the Dellac history Φ(C) of Example 3.1, we obtain Ψ(S) = C.
Following Remark 3.6, it is easy to prove the following lemma by induction on i ∈ [n].
Lemma 3.7. Let S ∈ DH(n). We consider the two sequences (I o i ) and (I e i ) defined in the computation of C = Ψ(S) (see Definition 3.6). Then for all i ∈ [n], the integer q C (i) is the (1 + r o C (e n+q C (i) ))-th element of the sequence I o i−1 , and the integer p C (n + 1 − i) is the (1 + l e C (e p C (n+1−i) ))-th element of the sequence I e i−1 . Proposition 3.8. The maps Φ : DC(n) → DH(n) and Ψ : DH(n) → DC(n) are inverse maps.
This puts an end to the proof of Theorem 3.3. As an illustration of the entire paper, the table depicted in the next page (see Figure 12