Lecture hall partitions and the wreath products C k S n

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1 Lecture hall partitios ad the wreath products C k S Thomas W. Pesyl ad Carla D. Savage Departmet of Computer Sciece, North Carolia State Uiversity Raleigh, NC , USA twpesyl@csu.edu ad savage@csu.edu submitted: Jauary 31, 2012; revised: December 12, 2012; February 26, 2013 Abstract It is show that statistics o the wreath product groups, C k S, ca be iterpreted i terms of atural statistics o lecture hall partitios. Lecture hall theory is applied to prove distributio results for statistics o C k S. Fially, some ew statistics o C k S are itroduced, ispired by lecture hall theory, ad their distributios are derived. 1 Itroductio The purpose of this ote is to show that statistics o the wreath product C k S of a cyclic group C k, of order k, ad the symmetric group S, ca be iterpreted i terms of atural statistics o lecture hall partitios. We demostrate that lecture hall theory ca be used to prove results about the distributio of statistics o C k S. We itroduce some ew statistics o C k S, ispired by lecture hall partitios, icludig a quadratic versio of flag-major idex, ad prove distributio results for these statistics. The paper is orgaized as follows. I Sectio 2, we defie the s-lecture hall partitios ad state a few useful results. Sectio 3 is devoted to statistics of iterest o the wreath product groups ad a very brief discussio of what is kow. Sectio 4 itroduces s- iversio sequeces, which will be used to relate statistics o C k S to statistics o lecture hall partitios. Sectio 5 describes a bijectio betwee (k, 2k,..., k)-iversio sequeces ad C k S that allows statistics to be traslated from oe domai to aother. Sectio 6 reviews recet work of Savage-Schuster [13] relatig iversio sequeces to lecture hall partitios. This work was developed with the itetio of extedig work o permutatio statistics to a more geeral settig. 1

2 Sectio 7 is the heart of the paper. We prove there a theorem which allows us to apply the tools of Sectio 6 to C k S. This cotais our mai results relatig statistics such as descet, flag-major idex ad flag-iversio umber to statistics o lecture hall partitios, also provig a Euler-Mahoia distributio result. I Sectio 8 we defie a ew statistic lhall o C k S ad derive its surprisigly ice distributio. I Sectio 9, we are led to defie a distorted versio of the descet statistic o C k S, that reveals a eve closer coectio to lecture hall partitios. A few words about otatio: Z is the set of itegers, R the set of real umbers, S the set of permutatios of elemets; [ j ] {1, 2,..., j}, where [ 0 ] ; [ ] q (1 q )/(1 q); ad for x (x 1, x 2,..., x ), x x 1 + x x. 2 Lecture hall partitios For a sequece s {s i } i 1 of positive itegers, the s-lecture hall partitios are the elemets of the set { L (s) λ Z λ 1 0 λ 2 λ }. s 1 s 2 s For example, (0, 1, 3, 4) L (1,2,3,4), but (0, 1, 3, 4) L (1,3,5,7), sice 3/5 > 4/7. The origial lecture hall partitios L L (1,2,...,) ad Eriksso i [3], where they showed that λ L y λ were itroduced by Bousquet-Mélou 1. (1) 1 y2i 1 I [4] they proved the followig refiemet, which will be useful i the preset work. Theorem 1. The Refied Lecture Hall Theorem [4]: For ay oegative iteger, λ L q λ y λ where λ ( λ 1 /1, λ 2 /2,..., λ / ). 1 + qy i 1 q 2, (2) y+i If the largest part i a lecture hall partitio i L is costraied, we have the followig. Theorem 2. [8, 13] For itegers 1 ad t 0, q λ [ t + 1 ] q. (3) λ L ; λ t 2

3 For example, whe 3 ad t 1, the set {λ L 3 λ 3 3} has the eight elemets: {(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 0, 3), (0, 1, 2), (0, 1, 3), (0, 2, 3), (1, 2, 3)} ad λ L 3 ; λ 3 3 q λ 1/1 + λ 2 /2 + λ 3 / q + 3q 2 + q 3 [ 2 ] 3 q. 3 Statistics o C k S A elemet π S is a bijectio π : [ ] [ ] ad we write π (π 1,..., π ), to mea that π(i) π i. A descet i π S is a positio i [ 1 ] such that π i > π i+1. The set of all descets of π is Des π ad des π Des π. The iversio umber of π is iv π {(i, j) 1 i < j ad π i > π j }. For example, if π (5, 4, 1, 3, 2), the Des π {1, 2, 4}, des π 3 ad iv π 8. For positive itegers k ad, we view C k S combiatorially as a set of pairs (π, σ): C k S {(π, σ) π S, σ {0, 1,..., k 1} }. We use the otatio π σ to deote (π, σ) ad write π σ (π σ 1 1, πσ 2 2,..., πσ ) ((π 1,..., π ), (σ 1,..., σ )) (π, σ). Statistics o C k S (or k-colored permutatios or k-idexed permutatios) have bee studied by may, startig with Reier s work o siged permutatios [12], followed by idepedet work of Breti [5] ad Steigrímsso [14] o the more geeral wreath products. Pairs of (descet, major idex) statistics have bee foud, satisfyig relatios like Carlitz s q-euleria polyomials, startig with work of Adi, Breti, ad Roichma [1]. There have very recetly bee may ew ad excitig discoveries, icludig [7, 10, 9, 2]. It is remarkable the may variatios i the defiitios of the statistics, eve whe they give the same distributio. We start with a fairly stadard defiitio of descet. The descet set of π σ C k S is Des π σ {i {0, 1,..., 1} σ i < σ i+1, or σ i σ i+1 ad π i > π i+1 }, (4) with the covetio that π 0 σ

4 We will cosider the followig statistics defied o C k S. des π σ Des π σ comaj π σ ( i) i Des π σ fmaj π σ k comaj π σ fiv π σ iv π + iσ i. As a example, for π σ (5 1, 4 1, 1 0, 3 0, 2 2 ) C 3 S 5, we have Des π σ {0, 1, 4}; des π σ 3; comaj π σ 10; fmaj π σ 26; ad fiv π σ 21. Note that this defiitio of fmaj differs a bit from those appearig elsewhere, eve amog those who defie the descet set as i (4) ([1, 7]). Usig lecture hall theory, we will show, amog other thigs: fmaj πσ fiv πσ, (5) π σ C k S q [ kt + 1 ] q xt t 0 λ L q λ x λ/(k) π σ C k S q σ i π σ C k S q fmaj πσ xdes πσ i0 (1, (6) xqki ) π σ C k S q fmaj πσ xdes πσ (1. (7) xqki ) Relatios of the form (6), for geeral k, have bee foud oly recetly, startig with Chow ad Masour [7] ad Hyatt [10], sometimes with slightly differet defiitios of Des or fmaj. Our itetio here is to highlight our methods, which are quite ovel, ad which allow us to prove ew results like (7). 4 Statistics o s-iversio sequeces The coectio betwee statistics o C k S ad statistics o lecture hall partitios will be made via statistics o iversio sequeces. Give a sequece s {s i } i 1 of positive itegers, ad positive iteger, the set I (s) s-iversio sequeces is defied by I (s) {(e 1,..., e ) Z 0 e i < s i for 1 i }. The familiar iversio sequeces associated with permutatios are the elemets of I (s) s (1, 2,..., ). 4 of for

5 The ascet set of a iversio sequece e I (s) Asc e { i {0, 1,..., 1} is the set e i s i < e i+1 s i+1 with the covetio that e 0 0. For example, as a elemet of I (3,6,9,12,15) 5, the iversio sequece e (1, 3, 2, 2, 13) has the ascet set Asc e {0, 1, 4}. The followig statistics o I (s) were defied i [13]: asc e Asc e, amaj e ( i), e i Asc e e i, lhp e e + i Asc e (s i s ). For e (1, 3, 2, 2, 13) I (3,6,9,12,15) 5, we have asc e 3; amaj e 10; e 21; ad lhp e 81. I this paper, our focus is the sequece s (k, 2k,..., k), where k is a positive iteger. Let I,k I (k,2k,...,k). We will require two ew statistics o I,k : N(e) j1 ej ; j Ifmaj e k amaj e N(e). For e (1, 3, 2, 2, 13) I (3,6,9,12,15) 5, N(e) 4 ad Ifmaj e 26. }, 5 From statistics o C k S to statistics o I,k We will make use of the followig bijectio betwee S ad I,1 which was proved i [13] to have the required properties. Lemma 1. For positive iteger, the mappig φ : S I,1 defied by φ(π) t (t 1, t 2,..., t ), where t i {j [i 1] π j > π i } is a bijectio satisfyig both Des π Asc t ad iv π t. For example, if π (5, 4, 1, 3, 2) the t φ(π) (0, 1, 2, 2, 3) I 5,1. statistics, Des π {1, 2, 4} Asc t ad iv π 8 t. Checkig the 5

6 Notig that, as sets, I,k ad C k S have the same cardiality, we set up a bijectio which traslates statistics from oe domai to the other i a useful way. Theorem 3. For each pair of itegers (, k) with 1, k 1, there is a bijectio Θ : C k S I,k with the followig properties. If Θ(π σ ) e (e 1,..., e ) the Proof. Defie Θ by Asc e Des π σ (8) N(e) σ i (9) Ifmaj e fmaj π σ (10) e (σ + 1) π (11) e iv π + iσ i fiv π σ. (12) e Θ(π σ 1 1, πσ 2 2,..., πσ ) (σ 1 + t 1, 2σ 2 + t 2,..., σ + t ), where (t 1, t 2,..., t ) φ(π), as i Lemma 1. For example, for π σ (5 1, 4 1, 1 0, 3 0, 2 2 ) C 3 S 5, t φ(5, 4, 1, 3, 2) (0, 1, 2, 2, 3), so we get e Θ(π σ ) (1, 3, 2, 2, 13). Note that properties (8) through (12) hold for this example: Asc e {0, 1, 4} Des π σ N(e) σ Ifmaj e 26 fmaj π σ e (σ 5 + 1) π 5 e 21 fiv π σ. Clearly, Θ(π σ ) I,k. Sice C k S ad I,k have the same cardiality, to show that Θ is a bijectio, it suffices to show that Θ is oto. Let e (e 1,..., e ) I,k. Defie σ (σ 1,... σ ) by σ i e i /i. The σ {0, 1,..., k 1}. Defie t (t 1,... t ) by t i e i iσ i. The t I,1. Fially, let π φ 1 (t) S. The π σ C k S ad Θ 1 (e) π σ. To prove properties (8) through (12), observe first that t π, so property (11) holds. It is clear from the defiitio of Θ that (12) is true. Also, ote that e i /i σ i sice 0 t i < i ad property (9) holds. So property (10) will follow oce we prove (8). By Lemma 1, sice t φ(π), we kow that Asc t Des π, so it remais to show Asc e Des π σ. 6

7 Note first that e 1 σ 1 + t 1 σ 1, sice t 1 0. So, 0 Des π σ σ 1 > 0 e 1 > 0 0 Asc e. For 1 i, i Asc e if ad oly if where 0 < e i+1 k(i + 1) e i ki So, i Asc e if ad oly if i > 0. If σ i σ i+1 the (i + 1)σ i+1 + t i+1 k(i + 1) iσ i + t i ki i(i + 1)(σ i+1 σ i ) + it i+1 (i + 1)t i ki(i + 1) i ki(i + 1), i i(i + 1)(σ i+1 σ i ) + it i+1 (i + 1)t i. i > 0 it i+1 (i + 1)t i > 0 i Asc t i Des π i Des π σ. For the remaiig cases, ote that sice 0 t i+1 i ad 0 t i i 1, i(i + 1)(σ i+1 σ i ) i i i(i + 1)(σ i+1 σ i ) + i 2. If σ i σ i+1, the i Des π σ if ad oly if σ i < σ i+1. But if σ i < σ i+1, the i i(i + 1) i i + 1 > 0, so i Asc e. Ad if σ i > σ i+1 the i i(i + 1) + i 2 i 0 ad i Asc e. This completes the proof. 6 Lecture hall polytopes ad s-iversio sequeces The s-lecture hall polytope was itroduced i [13], for a arbitrary sequece s {s i } i 1 of positive itegers, as { P (s) λ R λ 1 0 λ 2 λ } 1. s 1 s 2 s P (s) is a covex, simplicial polytope with the + 1 vertices: (0, 0,..., 0), (s 1, s 2,..., s ), (0, s 2,..., s ), (0, 0, s 3,..., s ),..., (0, 0,..., 0, s ), 7

8 all with iteger coordiates. The t-th dilatio of P (s) is the polytope { tp (s) λ R λ 1 0 λ 2 λ } t. s 1 s 2 s A multivariate fuctio, f (s) (t; q, y, z), was used i [13] to eumerate lattice poits i tp (s) accordig to statistics sigificat i the theory of lecture hall partitios: where λ s ɛ + (λ) f (s) (t; q, y, z), s 1 ( λ1 s 1 s 1 ( λ1 λ2 s 2,..., q λ tp (s) Z λ s λ 1, s 2 λ2 s 2 λ s y λ z ɛ+ (λ), ), (13) ) λ λ 2,..., s λ. (14) s The followig theorems show the coectio betwee statistics o s-iversio sequeces ad statistics o s-lecture hall partitios. Theorem 4. ([13]) For ay sequece s of positive itegers, ad ay positive iteger, f (s) (t; q, y, z) x t e I (s) x asc e q amaj e y lhp e z e i0 (1 xq i y s i+1+ +s ). (15) t 0 Theorem 5. ([13]) For ay sequece s of positive itegers, ad ay positive iteger, q λ s y λ z ɛ+ (λ) x λ/s e I (s) x asc e q amaj e y lhp e z e 1 i0 (1 xq i y s i+1+ +s ). (16) λ L (s) 7 Lecture hall partitios ad the iversio sequeces I,k I order to apply the results of the previous sectio to the problem of iterest, we eed a aalog of Ifmaj o I,k for lecture hall partitios. First observe that the followig sets of lecture hall partitios are all the same: L L (1,2,...,) L (2,4,...,2) L (3,6,...,3).... However, the lecture hall polytopes P,k defied by { P,k λ R λ 1 0 k λ 2 2k λ } k 1 8

9 are differet for differet k. O the other had, the followig dilatios are the same a fact we will exploit. Furthermore, t P,k kt P,1, (17) kt P,1 Z {λ L λ kt}. Sice the defiitios (13) ad (14) deped o the sequece s (k, 2k,..., k), we will make the depedece explicit i the otatio. For λ L ad k 1, let: ( ) λ1 λ2 λ λ k,,..., ; (18) k 2k k ) ɛ + k (k (λ) λ1 λ2 λ λ 1, 2k λ 2,..., k λ ; (19) k 2k k Note: for λ L, η k (λ) k λ k λ. (20) where λ was defied i Theorem 1. λ 1 λ, We ow show that the ew statistic η k o L correspods to the statistic N o I,k. Theorem 6. For positive itegers, k, let f,k (t; q, y, z, w) q λ k y λ z + ɛk (λ) w η k(λ). (21) Z The f,k (t; q, y, z, w) x t t 0 λ tp,k e I,k x asc e q amaj e y lhp e z e w N(e) i0 (1 xq i y k((+1) i(i+1))/2 ). (22) Proof. If w 1, this is just the case s (k, 2k,..., k) of Theorem 4. To iclude w, we appeal to the combiatorial proof of (15) i Theorem 4 that was preseted i [13]. I that proof, λ (tp,k Z ) is associated with the iversio sequece ɛ + k (λ), which, by defiitio, is i I,k. It suffices to check that η k (λ) N(ɛ + k (λ)): ik N(ɛ + k (λ)) λi /(ik) λ i i k λ i /(ik) λ i /i (k λ i /(ik) λ i /i ) k λ k λ 1 η k (λ). 9

10 The Ifmaj statistic is obtaied by settig q q k ad w q 1 i Theorem 6. Corollary 1. For positive itegers, k, t 0 q λ y λ z + ɛk (λ) x t λ ktp,1 Z e I,k x asc e q Ifmaj e y lhp e z e i0 (1 xqk( i) y k((+1) i(i+1))/2 ). (23) Proof. With q q k ad w q 1, the umerator i the right-had side of (22) becomes x asc e q k amaj e N(e) y lhp e z e x asc e q Ifmaj e y lhp e z e. From (21), the left-had side summad of (22) becomes by (18)-(20) ad by (17). f,k (t; q k, y, z, q 1 ) Corollary 2. For positive itegers, k, q k λ k ηk(λ) y λ z ɛ λ tp,k Z q λ y λ z + ɛk (λ) λ ktp,1 Z + k (λ) q λ y λ z + ɛk (λ) x λ/(k) λ L e I,k x asc e q Ifmaj e y lhp e z e 1 i0 (1 xqk( i) y k((+1) i(i+1))/2 ). (24) Proof. For t > 0, let H(t) λ ktp,1 Z q λ 1 y λ z ɛ+ k (λ) from (23), with H(0) 1. The for t > 0, sice { } { } { } λ λ λ λ L ; t λ L ; t λ L ; t 1 k k k (ktp,1 Z ) (k(t 1)P,1 Z ), 10

11 we have λ L q λ y λ z ɛ + k (λ) x λ/(k) t 0 x t λ L ; λ k t q λ y λ z ɛ+ k (λ) 1 + (H(t) H(t 1))x t t t 1 H(t)x t t 1 H(t 1)x t t 0 H(t)x t x t 0 H(t)x t (1 x) t 0 H(t)x t. But t 0 H(t)xt is the left-had side of (23), so we simply multiply the right-had side of (23) by (1 x) to complete the proof. We ca ow apply these results to the wreath product groups. expected result that the pair (des, fmaj ) is Euler-Mahoia. First, we have the Theorem 7. For positive itegers, k, [ kt + 1 ] q xt t 0 π σ C k S q fmaj πσ xdes πσ i0 (1. xqki ) Proof. Set y z 1 i (23). O the left-had side, i the summad, we get λ ktp,1 Z q λ. Sice kt P,1 Z {λ L λ kt}, by Theorem 2, For the right-had side, we get λ ktp,1 Z q λ [ kt + 1 ] q. e I,k x asc e q Ifmaj e i0 (1 xqk( i) ). Reidex the product i the deomiator ad for the umerator, use the fact that by Theorem 3, the distributio of (des, fmaj ) o C k S is the same as the distributio of (asc, Ifmaj ) o I,k. 11

12 Now, to iterpret the distributio (des, fmaj, fiv ) o C k S i terms of lecture hall partitios, set y 1 i (24) ad use Theorem 3. Theorem 8. For positive itegers, k, q λ z + ɛk (λ) x λ/(k) λ L π σ C k S q fmaj πσ x des πσ zfiv πσ (1. xqki ) The implicatio of Theorem 8 for z 1 is quite iterestig. We have π q λ x λ/(k) σ C k S q fmaj πσ xdes πσ λ L (1. (25) xqki ) I the left-had side of (25), the oly depedece o k is i the expoet of x, i a statistic ivolvig oly the last part of λ. We take this further i Sectio 9. 8 A lecture hall statistic o C k S From the poit of view of partitio theory, the most importat statistic for a lecture hall partitio λ is the umber λ λ 1 + +λ beig partitioed. So, what does λ correspod to o C k S? I [6], a quadratic versio of the major idex was defied o S by bi π i Des π I that spirit, we defie cobi o C k S by cobi π σ (( ) ( )) + 1 i i Des π σ Now defie the statistic lhall o C k S by lhall π σ k cobi π σ fiv π σ. ( i+1 ) 2. Observe that uder the bijectio Θ of Theorem 3, if e Θ(π σ ) the lhall π σ lhp e. This ca be see as follows, sice e fiv π σ ad Asc e Des e: lhp e e + (k(i + 1) + + k) i Asc e e + k (( ) ( )) + 1 i i Asc e fiv π σ + k (( ) ( )) + 1 i i Des e fiv π σ + kcobi π σ lhall π σ. The joit distributio of (lhall, fmaj ) o C k S has the followig form. 12

13 Theorem 9. For positive itegers, k, lhall πσ y π σ C k S q fmaj πσ (1 + qy i )(1 q k(+1 i) y k(i+ +) /2 1 q 2 y +i [ k(2i 1) ] qy +1 i /2 ([ 2 ] qy i [ ki ] q 2 y 2( i)+1) Proof. Uder the bijectio Θ of Theorem 3, if e Θ(π σ ) the lhall π σ lhp e ad fmaj π σ Ifmaj e. So, lhall πσ q fmaj πσ y lhp e q Ifmaj e. e I,k π σ C k S y So, by Corollary 2 with x z 1. π σ C k S y lhall πσ qfmaj πσ 1 i0 (1 qk( i) y k((+1) i(i+1))/2 ) Now apply Theorem 1 to get So, lhall πσ y π σ C k S π σ C k S y lhall πσ qfmaj πσ 1 i0 (1 qk( i) y k((+1) i(i+1))/2 ) q fmaj πσ which, after simplificatio, gives the theorem. Settig y 1 i Theorem 9 ad simplifyig, we get e I,k y lhp e q Ifmaj e 1 i0 (1 qk( i) y k((+1) i(i+1))/2 ) λ L y λ q λ. 1 + qy i 1 q 2 y +i. (1 q k( i+1) y k((+1) i(i+1))/2 1 + qy i ) 1 q 2 y +i, fmaj πσ q π σ C k S [ ki ] q, the same distributio as fiv, Ifmaj, ad e, as expected. But the statistic lhall itself also has a surprisigly simple distributio: Theorem 10. For positive itegers, k, lhall πσ q π σ C k S [ ki ] q 2( i)+1. 13

14 Proof. Set q 1 ad y q i the proof of the Theorem 9, but apply (1) istead of (2) to get: e I (1,2,...,) q lhp e /2 1 q k(i+ +) 1 q 2i 1 1 q k(2i 1)( i+1) 1 q 2i 1 [ ki ] q 2( i)+1. /2 1 q ki(2( i)+1) 1 q 2( i)+1 9 Iflated Euleria polyomials for C k S We showed i [11] how to obtai more refied iformatio about the s-lecture hall partitios by cosiderig the ratioal lecture hall polytope R (s) : { R (s) λ R λ 1 0 λ 2 λ } ad λ 1. s 1 s 2 s R (s) is a covex simplicial polytope, whose vertices are ( s1, s 2,..., s ) (, 0, s 2,..., s ), s s s s s (0, 0,..., 0), ( 0, 0, s 3 s,..., s s ) (,..., 0, 0,..., 0, s ), s with ratioal (but ot ecessarily iteger) coordiates. Let g (s) λ (t; q, y, z) k y λ z ɛ+ k (λ). (26) q λ tr (s) Z The followig theorems were proved i [11]. These are aalogs of Theorems 4 ad 5. Theorem 11. ([11]) For ay sequece s of positive itegers, ad positive iteger, g (s) (t; q, y, z)x t e I (s) q amaj e y lhp e z e xsasc e e (1 x) 1 i0 (1 xs q i y s i+1+ +s ). t 0 Theorem 12. ([11]) For ay sequece s of positive itegers, ad positive iteger, q λ k y λ z ɛ+ k (λ) x λ e I (s) q amaj e y lhp e z e xsasc e e 1 i0 (1. xs q i y s i+1+ +s ) λ L (s) 14

15 We ca specialize Theorems 11 ad 12 to s (k, 2k,..., k) ad modify to track Ifmaj as i Theorem 6 ad its corollaries. We should expect somethig iterestig because R (1,2,...,) R (2,4,...,2) R (3,6,...,3).... We get the followig theorem, which is a aalog of Theorem 6. aalogous to that of Theorem 6, is omitted. The proof, which is Theorem 13. For positive itegers, k, let g,k (t; q, y, z, w) q λ k y λ z + ɛk (λ) w η k(λ). (27) λ tr Z The g,k (t; q, y, z, w) x t t 0 e I,k x kasc e e q amaj e y lhp e z e w N(e) (1 x) 1 i0 (1 xk q i y k((+1) i(i+1))/2 ). (28) The followig corollaries of Theorem 13 are aalogs of Corollaries 1 ad 2 with y z 1. Note that i the right-had sides of the equatios there is o depedece o k. Corollary 3. For positive itegers, k, t 0 λ tr Z q λ x t e I,k x kasc e e q Ifmaj e (1 x) 1 i0 (1 xk q k( i) ). Corollary 4. For positive itegers, k, λ L q λ x λ e I,k x kasc e e q Ifmaj e 1 i0 (1. xk q k( i) ) Makig use of Theorem 3 givig the correspodece betwee statistics o I,k ad o C k S, we have the followig aalogs of Theorems 7 ad 8. First, We eed a result from [8]: Lemma 2. ([8]) For itegers t 0 ad > 0, let j ad i be the uique itegers satisfyig t j + i where j 0 ad 0 i <. The [ j + 1 ] i q [ j + 2 ] i q. λ tr Z q λ 15

16 Theorem 14. For positive itegers, k, 1 [ j + 1 ] i q [ j + 2 ] i q xj+i j 0 i0 π σ C k S q fmaj πσ x (k des πσ 1 σ )+π (1 x) (1 xk q ki. ) Proof. By Lemma 2, 1 [ j + 1 ] i q j 0 i0 [ j + 2 ] i q xj+i j 0 1 i0 λ (j+i)r Z q λ x j+i. Sice every t 0 ca be writte uiquely as t j + i for oegative itegers j ad i with i <, the last expressio ca be rewritte as q λ x t, t 0 λ tr Z which, by Corollary 3, is equal to e I,k x kasc e e q Ifmaj e i0 (1 xk q k( i). ) Uder the bijectio Θ of Theorem 3, if e Θ(π σ ) the Ifmaj e fmaj π σ, asc e des π σ, ad e (σ + 1) π. The result follows the, sice k asc e e k des π σ (σ + 1) + π. Theorem 15. For ay positive itegers, k, q λ x λ π σ C k S q fmaj πσ x (k des πσ 1 σ )+π λ L (1 xk q ki. ) Proof. Start from Corollary 4 ad apply Theorem 3. (Note: There is o depedece o k i the left-had side). Let Q,k (x) be the q 1 specializatio: Q,k (x) x (k des πσ 1 σ)+π. π σ C k S The Q,k (x) are referred to as iflated Euleria polyomials i [11]. For cotrast the usual, Euleria polyomials for C k S are E,k (x) des πσ. π σ C k S x It is iterestig that Q,k (x) is self-reciprocal, but i geeral E,k (x) is ot whe k > 2. 16

17 10 Cocludig remarks It is iterestig from the results i Sectios 7-9 that for fixed, statistics o C k S such as descet, flag-major idex, ad flag-iversio umber appear aturally i the geometry of the same simplicial coe, R, idepedet of k. It would be iterestig to see to what extet other statistics o C k S ca be iterpreted i terms of lecture hall partitios. Differet orderigs o C k S ad differet bijectios C k S I,k would give differet results. Lecture hall partitios were discovered i the settig of affie Coxeter groups, ad Theorem 1 was ispired by Bott s formula. It should be possible to trace through backwards to discover the algebraic sigificace of the statistic lhall, at least i the Coxeter groups A C 1 S or B C 2 S but we have ot see how to do this. Ackowledgemets. The secod author would like to thak Matthias Beck, Bejami Brau, ad Mattias Koeppe for discussios o Ehrhart theory ad siged permutatios. Thaks also to the America Istitute of Mathematics where some of those discussios were hosted. We are grateful to the referee for a careful readig of the mauscript ad for may helpful suggestios to improve the presetatio. Refereces [1] Ro M. Adi, Fracesco Breti, ad Yuval Roichma. Descet umbers ad major idices for the hyperoctahedral group. Adv. i Appl. Math., 27(2-3): , Special issue i hoor of Domiique Foata s 65th birthday (Philadelphia, PA, 2000). [2] Riccardo Biagioli ad Jiag Zeg. Eumeratig wreath products via Garsia-Gessel bijectios. Europea J. Combi., 32(4): , [3] Mireille Bousquet-Mélou ad Kimmo Eriksso. Lecture hall partitios. Ramauja J., 1(1): , [4] Mireille Bousquet-Mélou ad Kimmo Eriksso. A refiemet of the lecture hall theorem. J. Combi. Theory Ser. A, 86(1):63 84, [5] Fracesco Breti. q-euleria polyomials arisig from Coxeter groups. Eur. J. Comb., 15: , September [6] Katie L. Bright ad Carla D. Savage. The geometry of lecture hall partitios ad quadratic permutatio statistics. DMTCS Proceedigs, 22d Iteratioal Coferece o Formal Power Series ad Algebraic Combiatorics (FPSAC 2010), Sa Fracisco, (AN):569,580,

18 [7] Chak-O Chow ad Toufik Masour. A Carlitz idetity for the wreath product C r S. Adv. i Appl. Math., 47(2): , [8] Sylvie Corteel, Suyoug Lee, ad Carla D. Savage. Eumeratio of sequeces costraied by the ratio of cosecutive parts. Sém. Lothar. Combi., 54A:Art. B54Aa, 12 pp. (electroic), 2005/07. [9] Hilario L. M. Faliharimalala ad Arthur Radriaarivoy. Flag-major idex ad flagiversio umber o colored words ad wreath product. Sém. Lothar. Combi., 62:Art. B62c, 10, 2009/10. [10] Matthew Hyatt. Quasisymmetric fuctios ad permutatio statistics for Coxeter groups ad wreath product groups Ph.D. Thesis, Uiversity of Miami. [11] Thomas W. Pesyl ad Carla D. Savage. Ratioal lecture hall polytopes ad iflated Euleria polyomials. Ramauja J., DOI: /s , to appear. [12] Victor Reier. Siged permutatio statistics. Europea J. Combi., 14(6): , [13] Carla D. Savage ad Michael J. Schuster. Ehrhart series of lecture hall polytopes ad Euleria polyomials for iversio sequeces. J. Combi. Theory Ser. A, 119: , [14] Eiar Steigrímsso. Permutatio statistics of idexed permutatios. Europea J. Combi., 15(2): ,