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THE k-EXTENSION OF A MAHONIAN STATISTIC

BY

Guo-Niu HAN ()

ABSTRACT. — Clarke and Foata have recently studied thek-extension of several Mahonian statistics. There is an alternate definition for the k- Denert statistic that is derived in the present paper.

RESUM ´´ E. — R´ecemment, Clarke et Foata ont ´etudi´e la k-extension de plusieurs statistiques mahoniennes. Dans cet article, on introduit une autre d´efinition pour la k-statistique de Denert.

1. Introduction Let

(a;q)n =

1, if n= 0 ;

(1−a)(1−aq). . .(1−aqn−1), if n≥1 ;

denote theq-ascending factorialand for each sequence c= (c1, c2, . . . , cr) of non-negative integers, of sum m, let

m

c1, c2, . . . , cr

= (q;q)m

(q;q)c1(q;q)c2. . .(q;q)cr

denote the q-multinomial coefficient. Also denote by R(c) the class of all m!/(c1!c2! . . . cr!) rearrangements of the (non-decreasing) word 1c12c2. . . rcr. By Mahonian statistic it is meant an integer-valued map- ping “stat” defined on each class R(c) that satisfies the identity

m

c1, c2, . . . , cr

=X

w

qstatw (w∈ R(c)).

The inversion number “inv” and the major index “maj” are classical examples of Mahonian statistics (see [M]). Another example of such a statistic is provided with the Denert statistic, “den”, introduced recently (see [Den]) in the study of hereditary orders in central simple algebras.

The statistics “maj” and “den” have two definitions, one “natural” or () Supported in part by a grant of the European Community Programme

on Algebraic Combinatorics, 1994-95.

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more direct, the other one involving the notion of cyclic interval. The natural definitions refer to descents and excedances, two other statistics that have ben extensively studied since MacMahon (see [L, chap. 10]). We first recall the definitions of k-descent and of k-excedance, as they were introduced in the work by Clarke and Foata [CF1]).

Let r, l, k be three non-negative integers such that 1 ≤ r and l +k = r. Let X denote the alphabet [r] = {1,2, . . . , r} equipped with the natural order of the integers. Call the letters 1,2, . . . , l small and the letters l + 1, l + 2, . . . , l +k = r large. Also let S = Sk = {1,2, . . . , l}

and L = Lk = {l + 1, l + 2, . . . , r}. Let w = x1x2. . . xm be a word in the alphabet X and denote by w = y1y2. . . ym its non-decreasing rearrangement. Let 1 ≤ i ≤ m; say that i is a place of k-descent (resp.

place of k-excedance) for w, if i 6=m and xi > xi+1 or xi =xi+1 ≥l+ 1, or if i = m and xi ≥ l + 1 (resp. if xi > yi or xi = yi ≥ l + 1). The number of places of k-descent (resp. of places of k-excedance) in w is denoted by “deskw” (resp. “exckw”). Clarke and Foata [CF1] proved that

“desk” and “exck” wereequidistributed on each rearrangement classR(c).

In their second paper [CF2] they constructed a bijection ρ of R(c) onto itself satisfying

(desk,majk)(w) = (exck,denk)(ρ(w)),

where “majk” and “denk” are two k-extensions of Mahonian statistics in the following sense. Let SPDkw (resp. SPEkw) be the sum of the places of k-descents (resp. of k-excedances) inw. Clarke and Foata [CF2] defined the k-major index, “majkw,” of w as

majkw = SPDkw.

It would have seemed natural to define “denkw” as denkw= SPEkw, but SPDk is not equidistributed with majk on each class R(c). However, the following inequality

denkw≥SPEkw

holds. Therefore, the natural question arises : can well-defined predicates be found to characterize the (non-negative) difference “denk−SPEk” and then provide the “natural” definition of “denk” ? For k = 0 the question had been answered successfully by Han [H] (see the solutions by [FZ] and [C] in the case of words without repetitions of letters). It is the purpose of this paper to provide the predicates for an arbitrary k.

2. The k-extensions

Recall the definition of ak-cyclic interval, a notion introduced in [CF2].

First, for a, b∈X thecyclic interval ]]a, b]] is defined by (see [H]) : ]]a, b]] =

]a, b], ifa ≤b; X\ ]b, a], otherwise.

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Notice that ]]a, a]] =∅. Secondly, the k-cyclic interval a, b

k is defined as

a, b

k =













a, b

, ifa, b∈S;

a, b

∪ {a}, ifa ∈L, b∈S;

a, b

\ {b}, ifa ∈S, b∈L;

a, b

∪ {a} \ {b}, ifa, b∈L, a 6=b;

X, ifa =b∈L.

Letw= x1x2. . . xm be a word in the alphabetX. For i= 1, . . . , mdenote by Factiw=x1. . . xi−1 itsleft factor of length (i−1). For each subset B of X let Factiw∩B be thesubword of Factiw consisting of all the letters belonging to B and let|Factiw∩B| be the length of that subword.

We still denote by w = y1y2. . . ym the non-decreasing rearrangement of w. The denk-codingof w is defined to be the sequence (si)1≤i≤m+1 (see the example at the end of this section), where

si = Factiw∩ xi, yi

k

, if 1≤i≤m;

|w∩L|, if i=m+ 1 ; and the statistic “denkw” to be

denkw=

m+1

X

i=1

si.

Furthermore, the inversion numberand the weak inversion number are k-updated as follows :

invkw =#{1≤i < j ≤m|xi > xj ou xi=xj ≥l+ 1}

+ #{1≤i≤m|xi ≥l+ 1} ;

imvkw =#{1≤i < j ≤m|xi > xj ou xi=xj ≤l} .

The definition “invk” was introduced in an unpublished note by Foata [F].

Clearly, inv0 = inv (the usual number of inversions) and imv0 = imv (the number of (weak) inversions, as already used in [H]).

Let i1 < i2 < · · · < ie be the increasing sequence of the places of k- excedance in w and j1 < j2 < · · · < jm−e the complementary sequence, i.e., the increasing sequence of the places of non-k-excedance. Form the two subwords : Exckw = xi1xi2· · ·xie and Nexckw = xj1xj2· · ·xjme. The main result of this paper is the following result, a true k-extension of Theorem 2.1 (ii) in [H].

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Theorem 1. — For each word w in the alphabet X we have :

denkw = SPEkw+ imvk(Exckw) + invk(Nexckw).

For example, let r = 5, k = 2, l = r − k = 3, so that X = {1,2,3,4,5} with the small letters 1, 2, 3 and the large letters 4 and 5.

First, invk(21321144) = 3 + 0 + 3 + 2 + 0 + 0 + 1 + 0 + 2 = 11 and imvk(3253345445) = 3 + 0 + 5 + 1 + 0 + 0 + 2 + 0 + 0 + 0 = 11. With the word w = 325323143251441454 and w = 111222333344444555 we obtain the following table, where the values of k-excedance are printed in bold face on the third row.

id = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

w= 1 1 1 2 2 2 3 3 3 3 4 4 4 4 4 5 5 5

w= 3 2 5 3 2 3 1 4 3 2 5 1 4 4 1 4 5 4

s= 0 1 0 2 0 3 5 7 0 4 9 7 12 13 7 3 16 4 8

Hence denk = P

isi = 101. On the other hand, SPEkw = 79, imvk(Exckw) = 11 and invk(Nexckw) = 11.

3. Cardinality properties

It will be convenient to make use of abridged notations for cardinalities of sets that are associated with biwords. Those notations speak for themselves. They all apply to biwords (ww) = (xy11xy22···y···xmm), where w = x1x2· · ·xm is a word in the alphabet X and where w = y1y2· · ·ym is its non-decreasing rearrangement. If z is a letter in X, define :

h· z

i :=#{1≤i≤m|xi =z};

hz 6=

z z

i :=#{1≤i < j ≤m|xi =xj =yi =z, yj 6=z};

h · z

> z

i :=#{1≤i < j ≤m|xi > z, xj =yj =z}.

The following lemma already appears in Dumont [D].

Lemma 2. — Let w be a permutation of X having the fixed point z.

Then, the following identity holds :

h · z

> z

i =hz · z <

i.

Proof. — It suffices to consider the graph of the permutation w represented in a square r ×r (as shown in Fig. 1). The above identity

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simply says that rectangles B and C contain the same numbers of points of the graph.

1 z r

r

z

1

B D

A C

Fig. 1

This result can be generalized to arbitrary words as follows.

Lemma 3. — Let w be an arbitrary word in the alphabet X and z a letter of that alphabet. Then

h · z

> z

i=hz · z <

i+hz 6=

z z

i−hz z z 6=

i.

Proof. — For each integer j such that 1≤j ≤m let h · z

> z i

j :=#{i < j |xi > z, xj =yj =z};

hz 6=

z z

i

j :=#{j < i|xi =z, yi 6=z, xj =yj =z};

hz z z · i

j :=#{j < i|yi =z, xj =yj =z};

· · ·

so that, for instance, h · z

> z i

j is zero if (yj, xj)6= (z, z). The biword [ww] can be represented as the graph of the mapping i7→xi(1 ≤i≤m). Such a graph is contained in an rectangle m×r as shown in Fig. 2.

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1 j−1 j j+1 m

B D

A C

r z+ 1

z •

1

Fig. 2

Keep the same notations for the word w ∈ R(c) and its non-decreasing rearrangement w = 1c12c2· · ·rcr and let (yj, xj) = (z, z). We then have j ≤c1+c2+· · ·+cz. Let d =c1+c2+· · ·+cz −j = [z zz ·]j. Now as the number of points of the graph contained in rectangle B is equal to the number of points of the same graph contained in C, minus d, we get :

h · z

> z i

j =hz ·

z ≤

i

j −hz z z · i

j

=hz · z <

i

j +hz · z z

i

j −hz z z · i

j

=hz · z <

i

j +hz 6=

z z

i

j−hz z z 6=

i

j . Summing over all j yields the identity of the lemma.

Lemma 4. — Let w be a word and z a letter. Then h<

z

i+h6= 6=

z z

i−h6= z z 6=

i−h> >

z z

i+h< <

z z

i = 0.

Proof. — As there is no biletter (zz) on the left-hand side of the previous relation, it suffices to prove the identity when the biword (ww) contains no biletter (zz). In other words, we add the condition :

hz z

i= 0.

Under this condition the biword (ww) has a unique factorization w

w

=α α

·β β

·γ γ

,

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where α, α, β, β, γ, γ are words having the following properties : (i) all the letters inβ are equal to z;

(ii) there is no occurrence of the letter z in the words α, γ and β. Let a (resp. b) be the number of occurrences of z in β (resp. in α). The number of occurrences of z in γ is then a−b. Therefore, the left-hand side of the relation in the statement of the Lemma is equal to :

b+a(a−1)/2−ab−(a−b)(a−b−1)/2 +b(b−1)/2 = 0.

4. Proof of Theorem 1

By induction on k. For k= 0 it is true, as shown in [H]. Let k≥1 and k =k−1. It then suffices to prove

∆ = ∆ den−∆ SPE−∆ imv−∆ inv = 0 ; where

∆ den = denkw−denkw;

∆ SPE = SPEkw−SPEkw;

∆ imv = imvk(Exckw)−imvk(Exckw) ;

∆ inv = invk(Nexckw)−invk(Nexckw).

Again letw = x1x2· · ·xm and w =y1y2· · ·ym. Also let z = l+ 1 and denote byLk ={z, z+1,· · ·, z+k−1 =r}andLk ={z+1, z+2,· · ·, z+ k =r} the sets of the large letters associated with k and k, respectively.

The basic fact is that the letter z is large in thek-extension, but small in the k-extension.

4.1. Calculation of∆ den. — First, reproduce the expressions of “denk” and “denk” :

si =

|Factiw∩]]xi, yi]]k|, if 1 ≤i≤m;

|w∩Lk|, if i=m+ 1 ; denkw=s1+s2+· · ·+sm+1;

si =

|Factiw∩]]xi, yi]]k|, if 1 ≤i≤m;

|w∩Lk|, if i=m+ 1 ; denkw=s1+s2+· · ·+sm+1.

As ∆ den = denkw −denkw = P

i(si −si), we have to calculate the difference si−si for each i, and therefore the difference of the two cyclic intervals I = ]]xi, yi]]k and I = ]]xi, yi]]k. Using the notation C =A+B to mean that A∩B = ∅ and C =A∪B we see that there are five cases to consider :

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(1) —if xi =yi=z, then I =X, I =∅; (2) —if xi =z, yi 6=z, then I =I+{z}; (3) —if xi 6=z, yi =z, then I =I+{z}; (4) —if xi 6=z, yi 6=z, then I =I;

(5) —for i=m+ 1 we haveLk =Lk +{z}.

We then get :

∆ den =X

i

(si−si) =h· z

· z

i+h · 6=

z z

i−h · z z 6=

i+h· z

i.

4.2. Calculation of ∆ SPE. — We have

∆ SPE = SPEkw−SPEkw = X

xi=yi=z

i=h· z

· z

i+hz z

i.

4.3. Calculation of ∆ inv and ∆ imv. — Call biword of k-excedance the subword of (ww) formed with all the biletters whose bottom letters are k-excedances. Also denote by Exckw that subword. Define thebiword of non-k-excedance in an equivalent manner and denote it by Nexckw.

Those biwords as well as the corresponding biwords defined for k can be represented as

Exckw=w1zz· · ·zw2

w3zz· · ·zw4

; Exckw=w1w2

w3w4

; Nexckw=w5w6

w7w8

; Nexckw=w5zz· · ·zw6

w7zz· · ·zw8

;

where |w1| = |w3|, |w2| = |w4|, |w5| = |w7| and |w6| = |w8|. As [> z· z] = 0 for the biword Nexckw and [zz <·] = 0 for the biword Exckw, we derive the following relations :

∆ inv = invkNexckw−invkNexckw=−hz · z <

i+h> >

z z

i+h>

z i;

∆ imv = imvkExckw−imvkExckw =h · z

> z

i−h< <

z z

i.

4.4. Calculation of ∆. — Putting all the above relations together and noting that the term [·· zz] occurs both in ∆ den and in ∆ SPE, we obtain :

∆ = ∆ den−∆ SPE−∆ imv−∆ inv

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=h· 6=

z z

i−h · z z 6=

i+h · z

i−hz z i

+hz · z <

i−h> >

z z

i−h>

z

i−h · z

> z i

+h< <

z z

i

=h<

z

i−h> >

z z

i+h< <

z z

i

+h · 6=

z z

i−h · z z 6=

i+hz · z <

i−h · z

> z i

=h<

z

i−h> >

z z

i+h< <

z z

i

+h · 6=

z z

i−h · z z 6=

i+hz z z 6=

i−hz 6=

z z

i [Lemma 3]

=h<

z

i−h> >

z z

i+h< <

z z

i

+h6= 6=

z z

i−h6= z z 6=

i

= 0 . [Lemma 4]

This completes the proof of Theorem 1.

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REFERENCES

[C] Robert J. Clarke. — A short proof of a result of Foata and Zeilberger, to appear inAdvances in Appl. Math.

[CF1] Robert J. Clarke and Dominique Foata. — Eulerian Calculus, I : univariable statistics,Europ. J. Combinatorics, vol.15,, p. 345–362.

[CF2] Robert J. Clarke and Dominique Foata. — Eulerian Calculus, II : an extension of Han’s fundamental transformation, to appear inEurop. J. Combinatorics,.

[D] Dominique Dumont. — Interpr´etations Combinatoires des nombres de Genocchi, Duke Math. J., vol.41,, p. 305-317.

[Den] Marlene Denert. — The genus zeta function of hereditary orders in central simple algebras over global fields,Math. Comp., vol.54,, p. 449–465.

[F] Dominique Foata. — Unpublished Note, .

[FZ] Dominique Foata and Doron Zeilberger. — Denert’s permutation statistic is indeed Euler-Mahonian,Studies in Appl. Math., vol.83,, p. 31–59.

[H] Guo-Niu Han. — Une transformation fondamentale sur les r´earrangements de mots, Advances in Math., vol.105,, p. 26–41.

(Encyclopedia of Math. and its Appl.,17).

[M] (Major) P.A. MacMahon. — Combinatory Analysis, vol. 1. — Cambridge, Cambridge Univ. Press, (Reprinted by Chelsea, New York, ).

Guo-NiuHan,

Institut de Recherche Math´ematique Avanc´ee, ULP et CNRS,

7, rue Ren´e-Descartes, F-67084 Strasbourg, France.

email : guoniu@math.u-strasbg.fr

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