Anti-Leture Hall Compositions Sylvie Corteel CNRS PRiSM, UVSQ 45 Avenue des Etats-Unis 78035 Versailles, Frane syl@prism.uvsq.fr Carla D. Savage Dept. of Computer Siene N. C. State University, Box 8206 Raleigh, NC 27695, USA savage@s.nsu.edu May 4, 2002 Abstrat We study the set A k of integer sequenes =( ;::: ; k ), dened by and show that the generating funtion is 2 2 ::: k k 0; 2Ak q jj = +q i, q i+ ; where jj = + + k.we establish this by giving a bijetive proof of the following renement: 2Ak q jj u jbj v o(b) = +uvq i, u 2 q i+ : To our knowledge this is a new result that omplements the family of the Leture Hall Theorems [3, 4, 5]. Introdution For a sequene =( ; 2 ;:::; k )ofintegers, dene the weight of to be + + k and all eah i a part of. If has all parts nonnegative, we allitaomposition and if, in addition, is a noninreasing sequene, we all it a partition. Researh supported by NSA grant MDA 904-0-0-0083
In [3], Bousquet-Melou and Eriksson onsidered leture hallpartitions, speially, the set L k of partitions,, into at most k parts satisfying k 2 k, ::: k, 2 k 0; and proved the following surprising result, known as the Leture HallTheorem: 2L k q jj = k, Y i=0 : (), q2i+ Subsequently, partition analysis was used by Andrews in [] to give an analytial proof (with ombinatorial aspets) and by Andrews, Paule, Riese, and Strehl, in [2], to onstrut a bijetion. An elegant bijetive proof of the Leture Hall Theorem, due to Yee, appears in [6]. A dierent approah of Bousquet-Melou and Eriksson in [5] led to the Rened Leture Hall Theorem: 2L k q jj u de v o(de) = +uvq i ; (2), u 2 qk+i where for a partition =( ;:::; k ), o() is the number of odd parts of and de denotes the partition (d =ke; d 2 =(k, )e;:::;d k, =2e; d k =e). Setting u = v = in (2) gives (). Yee gave also a beautiful bijetive proof [7] of this result. In this paper, we onsider a new twist on these results by studying the set A k of ompositions into at most k parts satisfying 2 2 ::: k k 0: We refer to these as anti-leture hall ompositions and show the following. Theorem (The Anti-leture Hall Theorem) 2A k q jj = In fat, we prove the following renement of(3): Theorem 2 (The Rened Anti-leture Hall Theorem) 2A k q jj u jbj v o(b) = +q i : (3), qi+ +uvq i ; (4), u 2 qi+ 2
where b =(b =; b 2 =2;:::;b k =k) and o() denotes the number of odd parts of a omposition. The bijetive proof we give follows the idea of Yee's proof [7] of the Rened Leture Hall Theorem (2). 2 Proof of Main Result Dene the generating funtion : H k (u; v; q)= 2A k q jj u jbj v o(b) ; where, as in Theorem 2, b =(b =; b 2 =2;:::;b k =k) ando() denotes the number of odd parts of the omposition. Given an anti-leture hall omposition 2 A k,we an write as ((l ;:::;l k ); (r ;:::;r k )) where i = il i + r i, with 0 r i i, for i k. Then (l ;:::;l k )=b. Note that 2 A k if and only if (i) l l 2 :::l k 0 and (ii) r i r i+ whenever l i = l i+. Moreover jj = P k r i + il i. Proof of Theorem 2. Let D k be the set of the partitions into distint parts less than or equal to k. Let E k be the subset of A k onsisting of those for whih every l i is even. To prove Theorem 2 we givetwo bijetions : A bijetion between A k and D k E k suh that if (; ) is the image of then jj+jj = jj, jbj + L() = jbj and L() = o(b), where L() is the number of positive parts of. A bijetion between E k and the set P k, of partitions into parts in the set f2; 3;:::;k+ g, suh that if is the image of then jj = jj and L() = P k l. The rst bijetion will show that : H k (u; v; q)= ( + uvq i )E k (u; q); where E k (u; q) = 2E k q jj u jbj : 3
The seond bijetion will show that : ompleting the proof of (4). E k (u; q)= k+ Y, u 2 q i ; The rst bijetion (between A k and D k E k ). Given 2 A k we onstrut 2 D k and 2 E k with the following algorithm : empty partition While one of the l i is odd do d i maxfi j l i oddg minfj d j j = k or l d >l j+ +orr d r j+ g l d l d, r d For j from d to i, do r i r j r j+, [ i (Note l d >l d+ byhoie of d.) Consider the j th iteration of the \while" loop. Let d j and i j the indies hosen during that iteration. Then l dj, the last remaining odd l i, is dereased by and no other l i is altered during this iteration. Thus d j >d j+ and eah iteration dereases by the number of odd l i. Iteration j preserves the property 2 A k, so at termination 2 E k. During iteration j, l dj is dereased by andi j, d j of the r i are dereased by,sothe weight of is dereased by d j +(i j, d j )=i j. Sine the part i j is added to, the weight jj = jj + jj is preserved. For eah iteration of the loop, we derease one odd part, l dj, by and add one part, i j,to, sojbj + L() =jbj and L() =o(b). Clearly any part i j added to satises i j k, but we must verify that the parts of are distint. We show i j+ <i j.atthe beginning of iteration j, ift<d j and l t is odd, then either (i) l t >l dj or (ii) l t = l dj and r t r dj =. Thenbyhoie of i j+, in ase (i), i j+ <d j i j. Sine r ij = at the end of iteration j, in ase (ii) we alsohave i j+ <i j. Example. Starting with =((7; 6; 4; 3; 3; 2); (0; ; ; 2; 2; 3)) we apply the rst bijetion and get empty and = ((7; 6; 4; 3; 3; 2); (0; ; ; 2; 2; 3)) (initially) 4
d =5;i=6, = (6) and =((7; 6; 4; 3; 2; 2); (0; ; ; 2; 2; 2)) (iteration ) d =4;i=4, =(6; 4) and =((7; 6; 4; 2; 2; 2); (0; ; ; 2; 2; 2)) (iteration 2) d =;, =(6; 4; 2) and = ((6; 6; 4; 2; 2; 2); (0; 0; ; 2; 2; 2)) (iteration 3) We now dene the reverse bijetion. Let =( ;:::; l ) be a partition in D k and a omposition in E k.thenwe apply the following algorithm to reate 2 A k : For j from l downto do i d j maxft ijt =orl t, >l t g r j For t from d +toj do r d r t r t, + l d l d + Note that at the end of eah iteration, d is the largest index for whih l d is odd, l d >l d+ = :::= l i l i+, r d+ ::: r i >r d, and if l d = l i+ +thenr d r i+. Thus i is the smallest index greater than or equal to d suh that l i+ < l d + or l i+ = l d + and r d r i+. Therefore the mapping is reversible and we have dened a bijetion. The seond bijetion (between E k and P k ). For 2 E k, write as ((l ;:::;l k ); (r ;:::;r k )) where i = il i + r i and 0 r i i, for i k. (Then r =0.) We onstrut 2 P k by speifying the multipliity in, m (i), of eah i 2f2; 3;:::;k+g: m (k +) = r k + l k 2 k; m (i) = r i,, r i + l i,, l i (i, ); 2 i k: 2 It is easy to see that we an reonstrut from. 5
Note that m (i) isalways nonnegative asr i, <iand if r i, <r i then l i, >l i, that is, l i, >l i + (sine eah l i is even) and (l i,, l i )(i, )=2 i,. Now wemust show that jj = jj: jj = i im (i) = r k (k +)+ = = i(r i,, r i )+k(k +)l k =2+ (r i + l i ((i +)i, i(i, ))=2) (r i + il i )=jj: (i, )i(l i,, l i )=2 Finally, we show that the number of positive partsof, L(), is half the sum of the l i. L() = k+ = r k + = m (i) l i =2 (r i,, r i )+kl k =2+ =(i, )(l i,, l i )=2 Therefore the seond bijetion satises the required onditions and Theorem 2 is proved. 2 Example. Starting from = ((6; 6; 4; 2; 2; 2); (0; 0; ; 2; 2; 2))we apply the seond bijetion and get =(7; 7; 7; 7; 7; 7; 7; 7; 4; 4; 3)). 3 Conluding Remarks A renement of the Leture Hall Theorem (), dierent from (2) was proved in [3]: 2L k x jjo y jje = k, Y i=0, x i+ y i ; 6
where jj o = + 3 + 5 + :::,andjj e = 2 + 4 + 6 + :::. (This was further generalized in [4].) We have a onjeture as to the generating funtion for the orresponding renement of the Anti-leture Hall Theorem whih we hope to be able to prove in ongoing work. Referenes [] George E. Andrews. MaMahon's partition analysis. I. The leture hall partition theorem. In Mathematial essays in honor of Gian-Carlo Rota (Cambridge, MA, 996), pages {22. Birkhauser Boston, Boston, MA, 998. [2] George E. Andrews, Peter Paule, Axel Riese, and Volker Strehl. MaMahon's partition analysis. V. Bijetions, reursions, and magi squares. In Algebrai ombinatoris and appliations (Goweinstein, 999), pages {39. Springer, Berlin, 200. [3] Mireille Bousquet-Melou and Kimmo Eriksson. Leture hall partitions. Ramanujan J., ():0{, 997. [4] Mireille Bousquet-Melou and Kimmo Eriksson. Leture hall partitions II. Ramanujan J., (2):65{85, 997. [5] Mireille Bousquet-Melou and Kimmo Eriksson. A renement of the leture hall theorem. J. Combin. Theory. Ser. A, 86():63{84, 999. [6] Ae Ja Yee. On ombinatoris of leture hall partitions. Ramanujan Journal, 5:247{262, 200. [7] Ae Ja Yee. On the rened leture hall theorem. Disrete Math., 200. to appear. 7