ANTI-LECTURE HALL COMPOSITIONS AND ANDREWS GENERALIZATION OF THE WATSON-WHIPPLE TRANSFORMATION

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1 ANTI-LECTURE HALL COMPOSITIONS AND ANDREWS GENERALIZATION OF THE WATSON-WHIPPLE TRANSFORMATION SYLVIE CORTEEL, JEREMY LOVEJOY AND CARLA SAVAGE Abstract. For fixed ad k, we fid a three-variable geeratig fuctio for the set of sequeces (λ 1,..., λ ) satisfyig k λ1 λ2... λ 0, a 1 a 2 a where a := (a 1,..., a ) = (1, 2,..., ) or (, 1,..., 1). Whe k we recover the refied ati-lecture hall ad lecture hall theorems. Whe a = (1, 2,..., ) ad, we obtai a refiemet of a recet result of Che, Sag ad Shi. The mai tools are elemetary combiatorics ad Adrews geeralizatio of the Watso-Whipple trasformatio. 1. Itroductio ad mai result A lecture hall partitio of legth is a partitio λ = (λ 1,..., λ ) such that λ 1 λ λ 1 0. Let L deote the set of such partitios. These were itroduced ad extesively studied i a series of three papers by Bousquet-Mélou ad Eriksso 3, 4, 5]. I their third paper they established the three-variable geeratig fuctio L (u, v, q) := u λ v o( λ ) q λ = ( uvq) (u 2 q +1, (1.1) ) λ L where λ = ( λ 1 /,..., λ /1 ), λ = λ λ, 1 (a) := (a; q) = (1 aq i ), (1.2) ad o(w) is the umber of odd parts i a sequece w = (w 1,..., w ). Subsequetly the first ad last author itroduced the set A of ati-lecture hall compositios 7]. These are compositios λ = (λ 1,..., λ ) satisfyig i=0 λ 1 1 λ λ 0. Date: March 6, The first two authors are partially fuded by ANR-08-JCJC The third author is partially fuded by Simos Foudatio Grat

2 2 SYLVIE CORTEEL, JEREMY LOVEJOY AND CARLA SAVAGE The aalogue of (1.1) for ati-lecture hall compositios is 7] where A (u, v, q) := u λ v o( λ ) q λ = ( uvq) (u 2 q 2, (1.3) ) λ A λ = ( λ 1 /1,..., λ / ). I a recet paper, Che, Sag ad Shi 6] studied ati-lecture hall compositios i A satisfyig λ 1 k. Let A,k deote the set of these compositios, ad let A,k (u, v, q) deote the geeratig fuctio A,k (u, v, q) = u λ v o( λ ) q λ. λ A,k The mai result i 6] is the idetity where the otatio i (1.2) is exteded to A,k (1, 1, q) = ( q) (q, q k+1, q k+2 ; q k+2 ) (q), (1.4) 1 (a 1,..., a j ) = (a 1 ; q) (a j ; q) = (1 a 1 q i ) (1 a j q i ). (1.5) To prove (1.4), Che, Sag ad Shi first used two log ad ivolved combiatorial argumets, motivated by costructios of Bousquet-Mélou ad Eriksso 5] ad depedig o the parity of k, to express A,k (1, 1, q) as a q-hypergeometric multisum. The they applied Adrews geeralizatio of the Watso-Whipple trasformatio (see (2.3)) to covert the multisum to a ifiite product. For example, i the eve case their combiatorial argumet gives A,2k 2 (1, 1, q) = k 1 0 i=0 q ( )+2( )+...+2( k ) ( q)1 (q) (q) k 2 k 1 (q) k 1, (1.6) ad the a appropriate specializatio of (2.3) turs (1.6) ito (1.4). (For other combiatorial iterpretatios of the multisum i (1.6), see 11].) Here we observe that a elemetary recurrece combied with a differet applicatio of Adrews trasformatio leads swiftly ad eatly to the followig geeratig fuctio for A,k (u, v, q), ad as a cosequece, equatios (1.3) ad (1.4). We employ the q-biomial coefficiet, (q) :=, (1.7) (q) m (q) m which we ote (for later use) is the geeratig fuctio for partitios ito at most m parts, each less tha or equal to m. Theorem 1.1. If k is odd the A,k (u, v, q) = ( uvq) (1 u 2 q 2m+1 ) (u 2 q) m ( 1) m q k(m+1 2 )+m 2 u (k+1)m (u 2 q +2 ) m, (1.8)

3 ad if k is eve the A,k (u, v, q) = ( uvq) ANTI-LECTURE HALL COMPOSITIONS AND ANDREWS TRANSFORMATION 3 (1 u 2 q 2m+1 ) (u 2 q) m ( uq/v) m ( 1) m q k(m+1 2 )+m 2 u (k+1)m v m (u 2 q +2 ) m ( uvq) m. (1.9) Note that if v = 1 the equatios (1.8) ad (1.9) are idetical. Namely, for all k, A,k (u, 1, q) = ( uq) (1 u 2 q 2m+1 ) (u 2 q) m ( 1) m q k(m+1 2 )+m 2 u (k+1)m (u 2 q +2. (1.10) ) m Also ote that if k i (1.8) or (1.9), oly the m = 0 term survives ad we obtai (1.3). If ad u = v = 1, the the sums o the right-had sides reduce to m 0 ( 1) m (1 q 2m+1 )q k(m+1 2 )+m 2 1 q = m Z ( 1) m q k(m+1 2 )+m 2 1 q = (q, qk+1, q k+2 ; q k+2 ) 1 q by the triple product idetity, z q (+1 2 ) = ( 1/z, zq, q; q), (1.11) Z ad this gives (1.4). I fact, the sum o the right-had side of (1.8) is idepedet of v, so we have the followig refiemet of (1.4) for odd k: Corollary 1.2. If k 1 the A,2k 1 (1, v, q) = ( vq) (q, q 2k, q 2k+1 ; q 2k+1 ) (q). We preset the proof of Theorem 1.1 i Sectio 2, ad i Sectio 3 we show how Theorem 1.1 implies a similar result for lecture hall partitios. I Sectio 4 we also show how to exted the results to the trucated ati-lecture hall compositios ad trucated lecture hall partitios. We coclude with some remarks. 2. Proof of Theorem 1.1 We use a decompositio of ati-lecture hall compositios give i 9, Propositio 7], amely: Lemma 2.1. For k > 0, we have Moreover A,0 (u, v, q) = 1. A,k (u, v, q) = A m,k 1 (u, 1/v, q)u m v m q (m+1 2 ). (2.1) Proof. The proof is straightforward. Give a ati-lecture hall compositio λ i A,k, let m be the largest idex such that λ i i. The (λ 1 1, λ 2 2,..., λ m m) is i A m,k 1 ad (λ m+1, λ m+2,..., λ ) is a partitio ito m o egative parts which are at most m. Iteratig Lemma 2.1 gives the followig geeratig fuctio.

4 4 SYLVIE CORTEEL, JEREMY LOVEJOY AND CARLA SAVAGE Propositio 2.2. We have A,k (u, v, q) = k k u k i=1 i v k i=1 ( 1)k i i q k i=1 ( i +1 2 ) (q) (q) k... (q) 2 1 (q) 1. (2.2) To fiish the proof of Theorem 1.1, we eed Adrews trasformatio 2], N (1 aq 2m ) (a, b 1, c 1,..., b k, c k, q N ) m ( a k q k+n ) m (1 a) (q, aq/b 1, aq/c 1,..., aq/b k, aq/c k, aq N+1 ) m b 1 c 1 b k c k = (aq, aq/b kc k ) N (aq/b k, aq/c k ) N N k (b k, c k ) k 1 (b 2, c 2 ) 1 (aq) k q k 1 (q) k 1 k 2 (q) 2 1 (q) 1 (b k 1 c k 1 ) k 2 (b2 c 2 ) 1 (q N ) k 1 (aq/b k 1 c k 1 ) k 1 k 2 (aq/b 2 c 2 ) 2 1 (aq/b 1 c 1 ) 1 (b k c k q N. /a) k 1 (aq/b k 1, aq/c k 1 ) k 1 (aq/b 1, aq/c 1 ) 1 (2.3) I this trasformatio we replace k by k + 1 ad N by, set a = u 2 q, b i = uqv ( 1)i+k, ad let c i. Simplifyig usig stadard q-series limits ad the idetity we obtai equatios (1.8) ad (1.9). (q ) j = (q) (q) j ( 1) j q (j 2) j, 3. Applicatio to Lecture Hall partitios We ca use Theorem 1.1 to compute similar geeratig fuctios for lecture hall partitios with largest part less tha or equal to k, i.e., sequeces λ = (λ 1,..., λ ) such that k λ 1 λ λ 1 0. Let L,k be the set of such partitios ad write L,k (u, v, q) = u λ u o( λ ) q λ. λ L,k From 8, Corollary 3], we kow that if k is odd, the ad if k is eve, the L,k (u, v, q) = u k v q k(+1 2 ) A,k (1/u, 1/v, 1/q), (3.1) L,k (u, v, q) = u k q k(+1 2 ) A,k (1/u, v, 1/q). (3.2) Usig Theorem 1.1 together with (3.1) ad (3.2), we obtai the followig: Theorem 3.1. If k is odd the L,k (u, v, q) = ( uvq) (1 u 2 q 2m+1 ) (u 2 q) m ( 1) +m q k((+1 2 ) ( m+1 2 ))+ m u (k+1)( m) (u 2 q +2 ) m, (3.3)

5 ad if k is eve the ANTI-LECTURE HALL COMPOSITIONS AND ANDREWS TRANSFORMATION 5 L,k (u, v, q) = ( uq/v) (1 u 2 q 2m+1 ) (u 2 q) m ( uvq) m ( 1) +m q k((+1 2 ) ( m+1 2 ))+ m u (k+1)( m) v m (u 2 q +2 ) m ( uq/v) m. Whe k, oly the term m = i the sum survives ad we recover (1.1). Whe q 1, applyig the classical biomial theorem gives ( (1 + uv)(1 u k+1 ) ) L,k (u, v, 1) = 1 u 2 (3.5) for k odd ad ( 1 + uv u k+1 v u k+2 ) L,k (u, v, 1) = 1 u 2 (3.6) for k eve. Settig v = 1 i either, we recover the Mahoia geeratig fuctio from 12, Corollary 1]: ( ) 1 u k+1 L,k (u, 1, 1) =. (3.7) 1 u 4. Trucated objects Let A,j,k be the set of sequeces (λ 1,..., λ j ) such that k λ 1 j + 1 λ 2 j λ j 0. These are called trucated ati-lecture hall compositios 9]. Let A,j,k (u, v, q) = u λ v o( λ ) q λ λ A,j,k where λ = ( λ 1 /( j + 1),..., λ j / ). Arguig i the spirit of Lemma 2.1 (see also Propositio 8 of 9]), oe obtais the followig recurrece : A,j,k (u, v, q) = A j] j,k 1 (uq j, v, q) + v odd(k) u k q k( j+1) A,j 1,k (u, v, q) (4.1) if j > 0 ad k > 0, ad A,0,k (u, v, q) = A,j,0 (u, v, q) = 1. Here odd(k) = 1 if k is odd ad 0 otherwise. This gives j A,j,k (u, v, q) = A m,k 1 (uq m, v, q)v odd(k)(j m) u k(j m) q k(j m)( j+1)+(j m 2 ), (4.2) ad a applicatio of Theorem 1.1 the gives a double sum formula for A,j,k (u, v, q). Whe k, oly the term j = m i (4.2) survives ad we recover Theorem 2 of 9], ] ( uvq j+1 ) j A,j, = j (u 2 q 2( j+1). (4.3) ) j (3.4)

6 6 SYLVIE CORTEEL, JEREMY LOVEJOY AND CARLA SAVAGE Next let L,j,k be the set of sequeces (λ 1,..., λ j ) such that k λ 1 λ λ j j These are called trucated lecture hall partitios 9]. Let L,j,k (u, v, q) = u λ v o( λ ) q λ, λ L,j,k where λ = ( λ 1 /,..., λ j /( j + 1) ). As usual, we ca treat the lecture hall case by settig q = 1/q i the ati-lecture hall case. Namely, as with equatios (3.1) ad (3.2), we have if k is odd ad L,j,k (u, v, q) = u kj v j q k(j+1 2 )+k( j)j A,j,k (1/u, 1/v, 1/q) (4.4) L,j,k (u, v, q) = u kj q k(j+1 2 )+k( j)j A,j,k (1/u, v, 1/q) (4.5) if k is eve. This gives a double sum formula for L,j,k (u, v, q), ad whe k we recover Theorem 1 of 9]. We leave the details to the reader. 5. Cocludig remarks The results i this paper raise a umber of iterestig combiatorial prospects. For example, if we set v = 0 i Corollary 1.2, we obtai a relatio betwee ati-lecture hall compositios λ with λ cotaiig oly eve parts ad certai of the Adrews-Gordo idetities (for k = 2 this is the secod Rogers Ramauja idetity). Aother promisig lie of research would be to ivestigate the relatioship betwee q-series idetities like (2.3) ad Ehrhart theory, followig up o the coectios betwee lecture hall objects ad Ehrhart theory made by the third author 10, 12]. 6. Ackowledgemets The authors are idebted to the referees for their careful readig of the paper ad suggestios for its improvemet. The first author would like to thak the Simos Foudatio, who fuded her trip to NCSU where part of this work was doe. Refereces 1] George E. Adrews, The theory of partitios. Reprit of the 1976 origial. Cambridge Mathematical Library. Cambridge Uiversity Press, Cambridge, xvi+255 pp. 2] George E. Adrews, Problems ad prospects for basic hypergeometric series, i: R. Askey, Theory ad Applicatio of Special Fuctios, Academic Press, New York 1975, ] Mireille Bousquet-Mélou ad Kimmo Eriksso, Lecture hall partitios, Ramauja J. 1 No.1 (1997) ] Mireille Bousquet-Mélou ad Kimmo Eriksso, Lecture hall partitios 2, Ramauja J. 1 No.2 (1997) ] Mireille Bousquet-Mélou ad Kimmo Eriksso, A refiemet of the lecture hall theorem. J. Combi. Theory Ser. A 86 (1999), o. 1, ] William Y. C. Che, Doris D.M. Sag ad Diae Y. H. Shi, Ati-lecture hall compositios ad overpartitios. J. Combi. Theory Ser. A 118 (2011), o. 4, ] Sylvie Corteel ad Carla D. Savage, Ati-lecture hall compositios. Discrete Math. 263 (2003), o. 1-3,

7 ANTI-LECTURE HALL COMPOSITIONS AND ANDREWS TRANSFORMATION 7 8] Sylvie Corteel, Suyoug Lee, ad Carla D. Savage, Eumeratio of sequeces costraied by the ratio of cosecutive parts. Sém. Lothar. Combi. 54A (2005/07), Art. B54Aa, 12 pp. 9] Sylvie Corteel ad Carla D. Savage, Lecture hall theorems, q-series ad trucated objects. J. Combi. Theory Ser. A 108 (2004), o. 2, ] Fu Liu ad Richard Staley, The lecture hall parallelepiped, A. Comb., 18 (2014), o. 3, ] Jeremy Lovejoy ad Olivier Mallet, Overpartitio pairs ad two classes of basic hypergeometric series, Adv. Math. 217 (2008), ] Carla D. Savage ad Michael Schuster, Ehrhart series of lecture hall polytopes ad Euleria polyomials for iversio sequeces. J. Combi. Theory Ser. A 119 (2012), o. 4, LIAFA, CNRS et Uiversité Paris Diderot, Paris, Frace address: corteel@liafa.uiv-paris-diderot.fr LIAFA, CNRS et Uiversité Paris Diderot, Paris, Frace address: lovejoy@math.crs.fr Departmet of Computer Sciece, NCSU, Raleigh NC, USA address: savage@csu.edu