Lecture Hall Partitions 2 1 Introduction

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partitions of an integer m such that the quotient between consecutive parts is greater than is ... For example, an obvious result is that the generating function for ... generating function for (1;2;3;:::;n)-lecture hall partitions was shown to have a similar nice form: .... Our goal is to compute the two factors of the expression above.
Lecture Hall Partitions 2 Mireille Bousquet-Melou  LaBRI, Universite Bordeaux 1 351 cours de la Liberation 33405 Talence Cedex, FRANCE [email protected]

Kimmo Eriksson y Dept. of Mathematics Stockholm University S-106 91 Stockholm, SWEDEN [email protected]

Abstract

For a non-decreasing integer sequence a = (a1; : : :; an) we de ne La to be the set of n-tuples of integers  = (1 ; : : :; n) satisfying 0  a11  a22  : : :  ann : This generalizes the so-called lecture hall partitions, corresponding to ai = i and previously studied by the authors and by G. Andrews. We nd sequences a such that the weight generating function for these a-lecture hall partitions has the remarkable form X qjj = (1 ? qe1 )(1 ? q1e2 )    (1 ? qen ) : 2La In the limit when n tends to in nity, we obtain a family of identities of the kind "the number of partitions of an integer m such that the quotient between consecutive parts is greater than  is equal to the number of partitions of m into parts belonging to the set P ," for certain real numbers  and integer sets P . We then underline the connection between lecture hall partitions and Ehrhart theory and discuss some reciprocity results.

1 Introduction Suppose a lecture hall is to be built with n rows of seats placed at distances a1 ; a2; : : :; an from the speaker, at heights 1 ; 2; : : :; n. If the people in the audience are of negligible height, then the condition on the architecture for every seat to give a clear view of the speaker is that the slopes are weakly increasing, that is, aii  aii for 1  i < n. This justi es the following de nition (Figure 1). +1

+1

De nition 1.1 Given a non-decreasing sequence of n positive integers a = (a1; a2; : : :; an), an a-lecture hall partition is an n-tuple of integers  = (1; : : :; n) satisfying

0  1  2  : : :  n :

a1

a2

We let La denote the set of a-lecture hall partitions.

an

Let jj = 1 +    + n bePthe weight of . Let Sa (q ) denote the weight generating function for a-lecture hall partitions, i.e. Sa (q ) = 2La q jj . For example, an obvious result is that the generating function for (1; 1; : : :; 1)-lecture hall partitions is S(1;1;:::;1) (q) = (1 ? q)(1 ? q12 )    (1 ? q n ) :  Partially y Partially

supported by EC grant CHRX-CT93-0400 and GDR AMI. supported by EC grant CHRX-CT93-0400.

1

c c

c

c

c

1 2 4 6

Figure 1: The design of a lecture hall of four rows corresponding to the (1; 2; 3; 4)-lecture hall partition (1; 2; 4; 6). The study of lecture hall partitions was initiated in a previous paper [3] by the same authors, where the generating function for (1; 2; 3; : : :; n)-lecture hall partitions was shown to have a similar nice form: S(1;2;:::;n) (q) = (1 ? q)(1 ? q 3)1   (1 ? q 2n?1 ) : (1) For general a, the generating function Sa(q ) is always a rational function, but usually with a nontrivial numerator. We call the sequences a for which the series Sa(q ) is the multiplicative inverse of a polynomial polynomic sequences. The result above, though simple to state, seems far from trivial. Two di erent proofs were presented in [3], one using Bott's formula for the ane Coxeter group Cen , and one more combinatorial. Recently, Andrews [2] gave a third proof, starting from MacMahon's method of partition analysis (see [5], Vol. 2, Section VIII). In the present paper, we shall generalize the combinatorial approach of our previous paper to exhibit an in nite family of nontrivial polynomic sequences. This family includes the example (1) above and also for instance the sequence of every other Fibonacci number: S(1;3;8;21;:::;F n )(q) = (1 ? q 1)(1 ? q 1+3)(1 ? q 3+8)(11? q 8+21)    (1 ? q F n? +F n ) : Our results will allow us to generalize the following classical theorem, rst published in 1748 by Euler [4]: the generating function for partitions into distinct parts is Y X q jj = 1 ? 1q 2i+1 : :i >i i0 2 +1

2

1

2 +1

+1

In other words, the number of partitions of an integer m into distinct parts is equal to the number of partitions of m into odd parts. The standard way to perceive the condition i+1 > i is that the di erence between consecutive parts must be at least one. The famous Rogers-Ramanujan identities and related formulas (see Andrews's book [1]) are results of the same type, counting partitions with conditions on the di erence between parts. However, the condition in Euler's theorem can equivalently be perceived as demanding that the quotient between consecutive parts be greater than one. In this way one can see that we derive Euler's theorem by letting n tend to in nity in (1). More generally, we will derive from our main result many identities of the same type: we will describe an in nite family of real numbers  and integer sets P for which the number of partitions of m such that the quotient between any two parts is greater than  is equal to the number of partitions of m into parts belonging to the set P . Furthermore, we will discuss strict lecture hall partitions, obtained by taking strict inequalities in De nition 1.1 (which is in fact quite reasonable from the architectural point of view). We shall nd that they are connected to lecture hall partitions through a reciprocity theorem of the kind discussed by Stanley in [6], and give a simple proof of this theorem. The outline of the paper is as follows. After discussing polynomic sequences in Section 2 we go on with an approach to enumerating a-lecture hall partitions for general sequences a in Section 3. We will nd that in order to obtain explicit results further assumptions on a are needed. This leads in Section 4 2

to the de nition of a family of sequences a for which we compute nice generating functions. Taking the limit, we obtain the Euler-like results mentioned earlier. Finally, Section 5 deals with strict lecture hall partitions, establishes the reciprocity theorem and relates self-reciprocal sequences to continued fractions.

2 Polynomic sequences As we mentioned in the introduction, we will exhibit several sequences a = (a1 ; a2; : : :; an ) such that the generating function for a-lecture hall partitions has the remarkable form Sa(q) = (1 ? q e )(1 ? q1e )    (1 ? q en ) for some positive integers e1; : : :; en , all di erent. In combinatorial terms, this can equivalently be stated as the a-lecture hall partitions of weight m being equinumerous with the partitions of m into parts belonging to the set fe1 ; : : :; eng. We say that such a sequence a is a polynomic sequence since the series Sa is the inverse of a polynomial. In this section we collect what we know to be true about polynomic sequences and make some conjectures on their nature, based on computer search. First, there is a trivial kind of polynomic sequence, namely (a1 ; : : :; an ) such that ai+1 is divisible by ai for all i. This property will be denoted ai jai+1. Then, integer multiples of polynomic sequences are polynomic sequences, for L(ka ;:::;kan ) = L(a ;:::;an ) . Our rst conjecture says that all polynomic sequences are integer multiples of polynomic sequences starting with a one. Conjecture 2.1 If (a1; : : :; an) is a polynomic sequence, then all ai are divisible by a1. Polynomic sequences can be appended to trivial sequences to produce more polynomic sequences, according to the following result. Proposition 2.2 Let (a1; : : :; an) be a positive non-decreasing integer sequence, and suppose that there exists m  n such that (i) a1 ja2j : : : jam , (ii) am?1 j(am+1 ; : : :; an ), and (iii) (am ; : : :; an ) is polynomic. Then (a1; : : :; an) is also polynomic. Note that condition (ii) would follow from the two other conditions if Conjecture 2.1 was proved to be true. Proposition 2.2 is a straightforward consequence of the following lemma. Lemma 2.3 Let (a1; : : :; an) be a positive non-decreasing sequence satisfying the rst two conditions of Proposition 2.2. Then there exist m ? 1 integers e1 ; : : :; em?1 such that 1

1

2

1

S(a ;:::;an) (q) = S(am;:::;an) (q) 1

mY ?1 i=1

1 1 ? q ei :

Proof. For any partition  = (1; : : :; n), augmented by 0 = 0, we de ne the integers 1; : : :; n by i = aai i?1 + i for i?1

and

i = a ai m?1 + i for m?1 3

1  i  m;

m < i  n:

(2)

Note that (2) holds for i = m as well. Then  is an (a1; : : :; an )-lecture hall partition if and only if i  0 for 1  i < m and  = (m ; : : :; n ) is an (am ; : : :; an )-lecture hall partition. Moreover, there exist m ? 1 integers e1 ; : : :; em?1 such that mX ?1 jj = jj + eii : i=1

This leads to the announced result.

Our second conjecture is basically the reciprocal of Proposition 2.2. Conjecture 2.4 Let (a1; : : :; am) be a polynomic sequence, and let m be the least integer such that am+2 is not divisible by am+1 . Then (am ; : : :; an ) is a polynomic sequence. If this conjecture is true, a complete characterization of polynomic sequences would follow from the characterization of polynomic sequences (a1 ; : : :; an ) such that a3 is not divisible by a2 . A consequence of our main theorem in Section 4 is that we obtain such sequences by taking a1 to be any positive integer, a2 = `a1 , and letting a3; : : :; an be determined by (

a2i = `a2i?1 ? a2i?2 ; a2i+1 = ka2i ? a2i?1

for any two integers `  2 and k  2. We call such a sequence a (k; `)-sequence. For instance, a = (1; 2; 3; : : :; n) is the (k; `)-sequence obtained with a1 = 1 and k = ` = 2, while a = (1; 3; 8; : : :; F2n+1 ) is the (k; `)-sequence obtained with a1 = 1 and k = ` = 3. Note that the (k; `)-sequences satisfy Conjecture 2.1. Our third conjecture is that we have obtained all polynomic sequences such that a2 6 ja3. Conjecture 2.5 Let a = (a1; : : :; an) be a polynomic sequence such that a3 is not divisible by a2. Then a is a (k; `)-sequence. Finally, we have a fourth conjecture that seems very natural and would follow from the combination of Conjectures 2.4 and 2.5: that the class of polynomic sequences is hereditary. Conjecture 2.6 If (a1; : : :; an) is a polynomic sequence, then (a1; : : :; an?1) is a polynomic sequence. This last conjecture is related to material in the last section. There we characterize self-reciprocal sequences, show that they generalize the (k; `)-sequences and form an hereditary class.

3 An approach to enumerating lecture hall partitions In this section, we x a non-decreasing sequence a = (a1; : : :; an ) of positive length n. We let La be the set of a-lecture hall partitions. We will often omit the reference to a.

3.1 Reduced and standard partitions

We will re ne the weight generating function to consider the even and odd weights of  = (1; : : :; n), de ned respectively as

jje = n + n?2 + n?4 + : : : and jjo = n?1 + n?3 + n?5 + : : : Of course, jj = jje + jjo. Let Sa(x; y ) be the two-variable generating function for lecture hall partitions: X Sa (x; y) = xjje y jjo : 2La

4

All further work will be done in the two-variable setting. Taking q = x = y gives the generating functions discussed in previous sections. P We will describe a way of splitting any lecture hall partition  as  =  + ni=1 ki (i) (with partitions considered as members of an n-dimensional vector space), where  belongs to a nite set R of reduced lecture hall partitions, the standard lecture hall partitions (i) constitute a natural basis of La , and the ki are nonnegative integers. This implies that the generating function for lecture hall partitions factors as n Y X 1 j  j j  j o e Sa(x; y) = x y i je j i jo : j  y i=1 1 ? x 2R ( )

( )

Our goal is to compute the two factors of the expression above. The second one is easy, for we will give the (i) explicitly. The rst one with the reduced lecture hall partitions is harder and we will succeed only for some particular sequences a, which generalize slightly the (k; `)-sequences mentioned in the previous section. We shall now de ne the reduced and standard lecture hall partitions. To begin with, note that an alternative way of expressing the lecture hall condition is that i  dai i?1 =ai?1 e. Given an n-tuple  = (1; : : :; n) 2 IR n , we de ne its D-sequence to be the n-tuple D() = (d1; : : :; dn) where 

 a  i i ? 1 d1 = 1 and di = i ? a for 2  i  n: i?1 Clearly the sequence D() completely de nes , and  2 INn is a lecture hall partition if and only if di  0 for all i. For 1  i  n, de ne the standard lecture hall partitions (i) by (i) = (0; : : :; 0; ai; ai+1; : : :; an): They are obviously lecture hall partitions, and D((i)) = (0; : : :; 0; ai; 0; : : :; 0). More generally, if  belongs to La , then the sum  + (i) also belongs to La , and its D-sequence is (d1; : : :; di?1 ; ai + di ; di+1; : : :; dn) where (d1; : : :; dn ) = D(). The following lemma describes for which  also the di erence  ? (i) lies in La . Lemma 3.1 Let  2 La , and let (d1; : : :; dn) be its D-sequence. Let 1  i  n. Then  ? (i) belongs to La if and only if di  ai. Proof. Let (d01; : : :; d0n) = D( ? (i)). Then d0i = di ? ai and d0j = dj if j =6 i. Thus d0j  0 for all j if and only if di  ai .

Now we de ne a lecture hall partition to be reduced if every subtraction of a standard lecture hall partition destroys the lecture hall property. By the lemma above this can be stated as follows. De nition 3.2 A lecture hall partition is reduced if its D-sequence (d1; : : :; dn) satis es 0  di < ai for 1  i  n. Since the D-sequence completely de nes the partition, there are exactly a1 a2    an reduced lecture hall partitions. The set of reduced partitions of La will be denoted R. Iterating Lemma 3.1 leads to the following reduction result. Proposition 3.3 The map  7! (; k1; : : :; kn) given by ki = bdi=aic, with (d1; : : :; dn) = D(), and

= + de nes a bijection from La to R  INn .

n X i=1

5

ki (i)

Consequently, the generating function for lecture hall partitions is Sa(x; y) = Qn P (x;j yi )je j i jo y ) i=1 (1 ? x where the polynomial P (x; y ) enumerates reduced lecture hall partitions: X P (x; y) = xjje y jjo : ( )

( )

(3)

2R

Computing this polynomial is thus the central problem one has to solve in order to enumerate lecture hall partitions using our approach. In Section 4, we compute it by induction on the length of a for some particular sequences a, using the involution on reduced lecture hall partitions described next.

3.2 An involution on reduced lecture hall partitions

For  2 INn , let  = (1 ; : : :; n ) be the unique n-tuple such that (  n?2k = j n?2k for n ? 2k  1 k a n ? k ?  dn?2k?1 = an? k n?2k ? n?2k?1 for n ? 2k ? 1  1; 2

1

(4)

2

where (d1; : : :; dn) is the D-sequence associated with  . The second equation is of course a short way to de ne n?2k?1 . Equivalently,

    n?2k?1 ? aan?2k?1 n?2k?2 = aan?2k?1 n?2k ? n?2k?1 : (5) n?2k?2 n?2k In this expression, and always in this section, we have set i = i = 0 if i  0. Example. Let n = 6 and take (a1; : : :; a6) = (1; 2; 3; 5; 6; 9). Then  = (0; 1; 3; 9; 13; 21) is a reduced lecture hall partition, since D() = (0; 1; 1; 4; 2; 1). We obtain  = (0; 1; 4; 9; 12; 21) and D( ) = (0; 1; 2; 2; 1; 3). Thus  is also a reduced lecture hall partition. Moreover, computing ( ) shows that ( ) = . We shall see that this holds in general. Proposition 3.4 The correspondence  7!  de nes an involution on the set R.

In the proof of this proposition, we will use the following simple lemma. Lemma 3.5 For integers `; m  0 and i; j  1, 

i ` ? m 2 [0; i ? 1] () ` ?  j m 2 [0; j ? 1]: j i

Proof of Proposition 3.4. Let  2 R. We rst want to prove that  2 R, that is, di 2 [0; ai ? 1] for 1  i  n. According to (4) and Lemma 3.5, this is true when n ? i is odd. Now, since n?2k = n?2k for all k, we can rewrite (5) as 

  an?2k?1   ?  a n ? 2k?1 (6) n?2k?2 = dn?2k?1 : n?2k?1 = n?2k?1 ? a an?2k n?2k n?2k?2 But dn?2k?1 2 [0; an?2k?1 ? 1] since  is a reduced lecture hall partition; hence, according to Lemma 3.5,   dn?2k = n?2k ? aan?2k n?2k?1 2 [0; an?2k ? 1]: n?2k?1 Thus  is indeed a reduced lecture hall partition. Finally, equation (6) says precisely that the involutive property ( ) =  holds.

6

Let us now indicate how we are going to use this involution to enumerate reduced lecture hall partitions. Let an+1  an , and let R0 denote the set of reduced (a1; : : :; an ; an+1 )-lecture hall partitions. We extend the involution  7!  into a bijection  from R  [0; an+1 ? 1] onto R0, by de ning 



an+1   + i : an n For convenience, let us denote the extended partition (; i) by  . We want to compute j je and j jo. As n?2k = n?2k , it is clear that jjo = jje . Moreover,   X a n +1  n?2k?1 jje = i + a n + n 12[1;n?1]    n?2k?X     a a a n ? 2k?1 n ? 2k?1 n +1 = i + a n + (7) an?2k?2 n?2k?2 + an?2k n?2k ? n?2k?1 n n ? 2 k ? 1 2 [1 ;n ? 1]  X an?2k+1   + X  an?2k?1   ? jj : = i+ n?2k n?2k o n?2k2[1;n] an?2k n?2k2[2;n] an?2k (; i) = 1 ; : : :; n ;

We cannot go further in this calculation without any additional assumptions on the sequence a. In the next section we study some particular sequences for which the weight j je can be expressed in a very simple way in terms of jje and jjo . This will allow us to enumerate reduced lecture hall partitions by induction on the length of a.

4 Generating functions for (

-sequences

k; `)

In this section, we x two integers k; `  2 and two initial values a1; a2 satisfying one of the following conditions: a2 = a1` or a2 = a1` ? 1 or a1 = 1: (8) We de ne an in nite sequence (ai )i1 by (

a2i = `a2i?1 ? a2i?2 for i  2 a2i+1 = ka2i ? a2i?1 for i  1:

(9)

This generalizes the (k; `)-sequences de ned in Section 2. For n  1, we will be able to enumerate reduced (a1 ; : : :; an )-lecture hall partitions. We will then compute the weights of the standard lecture hall partitions (i), which will nally provide the generating function for lecture hall partitions thanks to (3).

4.1 Reduced lecture hall partitions for (

-sequences

k; `)

For n  1, we denote the set of reduced (a1; : : :; an)-lecture hall partitions by Rn , and their generating function by Pn (x; y ). Let us return to the last lines of the previous section: the correspondence  is a bijection from Rn  [0; an+1 ? 1] to Rn+1 . We combine the last equation of (7) with the following simple lemma. Lemma 4.1 Let i  2 and m  0. Then 

ai+1 m +  ai?1 m = ai ai 7

(

km if i is even; `m if i is odd:

We obtain

8 >
:

which gives, using (8),

i + kjje ? jjo i + aa2 1 + ` 1

X

n?2k2[3;n]

if n is even, n?2k ? jjo if i is odd;

(

jje = ii ++ k`jjjjee ?? jjjjoo ifif nn isis even, odd;

(10)

(remember that 1 2 [0; a1 ? 1]). Thus the bijection  : Rn  [0; an+1 ? 1] ! Rn+1 is such that, if  = (; i), then j jo = jje , and j je is given by (10). This implies that the polynomials Pn (x; y ) can be computed inductively via the following recurrence relations: an an P2n+1 (x; y) = 1 ?1 x? x P2n (xk y; x?1) and P2n(x; y) = 1 1??xx P2n?1 (x` y; x?1 ); with the initial condition P0 (x; y ) = 1. Proposition 4.2 Given an in nite sequence (a1; a2; : : :) satisfying (8) and (9), the generating functions for reduced (a1; : : :; a2n)-lecture hall partitions and reduced (a1 ; : : :; a2n?1 )-lecture hall partitions are respectively given by:   a n?i 2n 1 ? xbi y bi Y P2n(x; y) = 1 ? xbi y bi i=1 and  a n?i 2Y n?1 1 ? xbi y bi? P2n?1 (x; y) =  1 ? xbi y bi? i=1 where the sequences of exponents b and b are de ned by b0 = 0, b1 = 1, b1 = 0, b2 = 1, and the same recurrence relations as for a. Proof. One readily veri es that the sequences b and b satisfy the relations b2i = b2i?1 and `b2i+1 = kb2i: (11) This implies further that bi+1 = kbi?1 ? bi?1 and bi = `bi ? bi?2 ; which are exactly the relations forced upon us by the recurrence for Pn . 2

2 +1

2

+1

2

1

+1

+1

1

Example. Let k = 2, ` = 3, a1 = 1 and a2 = `a1 = 3. We have (a1; a2; a3) = (1; 3; 5), (b0; b1; b2) = (0; 1; 3)

and (b2; b3; b4) = (1; 2; 5). The reduced (1; 3; 5)-lecture hall partitions are listed below: (0; 0; 0); (0; 0; 1); (0; 0; 2); (0; 0; 3); (0; 0; 4); (0; 1; 2); (0; 1; 3); (0; 1; 4); (0; 1; 5); (0; 1; 6); (0; 2; 4); (0; 2; 5); (0; 2; 6); (0; 2; 7); (0; 2; 8): Thus: P3(x; y) = 1 + x + x2 + x3 + x4 + x2y + x3 y + x4 y + x5 y + x6 y + x4y 2 + x5 y 2 + x6 y 2 + x7 y 2 + x8 y 2 = (1 + x + x2 + x3 + x4 )(1 + x2 y + x4y 2 ) 2 3 5 3 1 5 = 1 1??(xx) 1 1??(xx2yy) 1 1??(xx5yy 3) b y b )a 1 ? (xb y b )a 1 ? (xb y b )a 1 ? ( x = : 1 ? x b y b 1 ? xb y b 1 ? xb y b 0

2

2

3

0

1

3

3

2

1

2

4

4

8

1

2

4.2 Standard lecture hall partitions for (

-sequences

k; `)

Recall that the standard (a1; : : :; an )-lecture hall partitions are the (i) = (0; : : :; 0; ai; ai+1; : : :; an ), for 1  i  n. Our computation of their even and odd weights will depend on the following lemma. Lemma 4.3 For 1  i  n, let us de ne the sums E (n; i) and O(n; i) by E (n; i) = a2n + a2n?2 + : : : + a2n?2i+2 and O(n; i) = a2n?1 + a2n?3 + : : : + a2n?2i+1 : Then we have, for the sequences a, b and b of Proposition 4.2, E (n; i) = bia2n?i+1 and O(n; i) = bi+1a2n?i : Proof. We shall prove each of these equalities by induction over i. We have E (n; 1) = a2n = b1a2n and E (n; 2) = a2n + a2n?2 = `a2n?1 = b2a2n?1 as desired. De ne i to be ` if i is even and k if i is odd. Then induction gives, for i  3, E (n; i) = E (n; i ? 1) + E (n ? 1; i ? 1) ? E (n ? 1; i ? 2) = bi?1 a2n?i+2 + bi?1 a2n?i ? bi?2 a2n?i+1 = i bi?1 a2n?i+1 ? bi?2 a2n?i+1 = bi a2n?i+1 ; just as we wanted. The argument for O(n; i) is similar. Thus the weights of standard (a1; : : :; a2n )-lecture hall partitions are given by ( j(2n?2i)je = a2n +    + a2n?2i = E (n; i + 1) = bi+1a2n?i for 0  i  n ? 1; j(2n?2i)jo = a2n?1 +    + a2n?2i+1 = O(n; i) = bi+1a2n?i and ( j(2n?2i+1)je = a2n +    + a2n?2i+2 = E (n; i) = bia2n?i+1 for 1  i  n: j(2n?2i+1)jo = a2n?1 +    + a2n?2i+1 = O(n; i) = bi+1a2n?i The weights of standard (a1; : : :; a2n?1 )-lecture hall partitions are given by ( j(2n?2i+1)je = a2n?1 +    + a2n?2i+1 = O(n; i) = bi+1 a2n?i for 1  i  n; j(2n?2i+1)jo = a2n?2 +    + a2n?2i+2 = E (n ? 1; i ? 1) = bi?1a2n?i and ( j(2n?2i)je = a2n?1 +    + a2n?2i+1 = O(n; i) = bi+1a2n?i for 1  i  n ? 1: j(2n?2i)jo = a2n?2 +    + a2n?2i = E (n ? 1; i) = bia2n?i?1 Combining these results with Proposition 4.2 and equation (3), we nally obtain the generating function for lecture hall partitions. Proposition 4.4 Given an in nite sequence (a1; a2; : : :) satisfying (8) and (9), the generating functions for (a1 ; : : :; a2n )-lecture hall partitions and (a1 ; : : :; a2n?1)-lecture hall partitions are respectively given by:  a 2n 2n 1 ? xbi y bi n?i Y Y 1 S(a ;:::;a n ) (x; y) = 1 ? xbi ybi  (12) b n?i ai y bn?i ai? i=n+1 1 ? x i=1 and a n?i   2Y n ? 1 1 ? xbi y bi? 2Y n?1 1 (13) S(a ;:::;a n? )(x; y) = i bi? b b ai b n?i ai? i=n+1 1 ? x n?i y i=1 1 ? x y where the sequences b and b are de ned by b0 = 0, b1 = 1, b1 = 0, b2 = 1, and the same recurrence relations as for a. 2

1

2

2

+1

2

+1

1

2

1

+1

1

9

2

+1

+1

1

+2

1

2

2

1

Examples 1. If k = `, then bi+1 = bi for i  0. The proposition above implies that the generating function for (a1; : : :; an )-lecture hall partitions is

S(a ;:::;an )(x; y) = 1

nY ?1 i=0

nY ?1

1

1 ? xbi y bi i=b n +1

+1 2



a

1 ? xbi y bi n?i ai ai bn?i : c 1 ? (x y ) +1

+1

For example, if we take k = ` = 2, a1 = 1 and a2 = 3, then property (8) is satis ed and we have ai = 2i ? 1 and bi = i for all i. We obtain that the generating function for (1; 3; : : :; 2n ? 1)-lecture hall partitions is

S(1;3;:::;2n?1) (x; y) = In particular,

nY ?1 i=0

nY ?1

1

1 ? xi+1 y i i=b n

+1 2

1 ? xi+1 y i 2n?2i?1 : 2i+1 2i?1 n?i c 1 ? (x y ) ?



3 4 5 7 8 S(1;3;5;7)(q) = 1 ?(1q?+qq)2(1? ?q q+12q)(1??qq 16+)q :

Observe the nontrivial numerator of this series. 2. The (k; `)-sequences de ned in Section 2 satisfy a2 = a1 `. As noted in that section, it is sucient to study the case a1 = 1, a2 = `, or, equivalently, a0 = 0, a1 = 1. Then we have ai = bi for i  0. Moreover, properties (11) imply that if the integers i and j are equal modulo two, then bi+1bj = bibj+1: This implies that all the terms occurring in the last product of equations (12) and (13) are equal to 1, and leads to the following result, which states that the (k; `)-sequences are indeed polynomic sequences.

Theorem 4.5 De ne an in nite sequence by a0 = 0, a1 = 1 and for n  1, ( a2n = `a2n?1 ? a2n?2 ; a2n+1 = ka2n ? a2n?1 where ` and k are two integers  2. The generating functions for (a1 ; : : :; a2n )-lecture hall partitions and

(a1; : : :; a2n?1 )-lecture hall partitions are respectively given by

S(a ;:::;a n )(x; y) = 1

and

2

S(a ;:::;a n? )(x; y) = 1

2

2n Y

i=1 1 ? x

2Y n?1

1

1

ai y ai 

1

1 ? xai+1 y ai?1

i=1  where the numbers an are de ned by the initial conditions a1 = 0, a2 = 1 and the same recurrence relations

as for a.

Note (17=06=96). The method we used to enumerate reduced a-lecture hall partitions for sequences a

satisfying the recurrence relations (9) and the initial condition (8) can be applied to general a-lecture hall partitions for sequences a satisfying (9) and the more restrictive initial condition a2 = `a1. This gives a much shorter, direct proof of Theorem 4.5, which goes as follows: the transformation  7?!  given by 10

(4) also induces an involution on general lecture hall partitions (this is actually true for any sequence a). We de ne a bijection  from L(a ;:::;an )  IN onto L(a ;:::;an ;an ) , by 1

1



+1



an+1   + i : an n Then, denoting  = (; i), we have j jo = jje , and j je is given by (10). This implies that the series S(a ;:::;an)(x; y) can be computed inductively via the following recurrence relations: (; i) = 1 ; : : :; n ;

1

S(a ;:::;a n ) (x; y) = 1 ?1 x S(a ;:::;a n ) (xk y; x?1) and S(a ;:::;a n ) (x; y) = 1 ?1 x S(a ;:::;a n? )(x`y; x?1 ); 1

1

2 +1

2

1

2

1

2

1

with the initial condition S(a ) (x; y ) = 1=(1 ? x), and we thus obtain the theorem above. 1

4.3 Limit theorems

We give two limit theorems in the style of Euler's theorem, as discussed in the introduction. The sequence (a0; a1; a2; : : :) is de ned as in Theorem 4.5. First, note that removing the empty parts of lecture hall partitions puts the set L(a ;:::;an ) in one-to-one correspondence with the following set: 1



Dn = (1; : : :; m) : m  n and 0 < a

1

a

n?m+1

2

n?m+2

The conditions on the partitions of Dn can be restated as

  m  :::  a : n

i+1  an?m+i+1 : i an?m+i The following lemma implies that Dn  Dn+2 for all n. Lemma 4.6 For i  1, let us denote qi = ai+1=ai. Then the two sequences (q2i)i1 and (q2i?1)i1 are both decreasing, and converge respectively towards e and o where p p k` ( k` ? 4) k`(k` ? 4) : k` + k` + e = and o = 2` 2k Consequently, the sequence D2n converges, when n tends to in nity, to the set De of partitions (1 ; : : :; m ) such that 1 > 0 and, for i  1, ( i+1 > e if m + i is even; o if m + i is odd: i Similarly, the sequence D2n+1 converges, when n tends to in nity, to the set Do of partitions (1 ; : : :; m ) such that 1 > 0 and, for i  1, ( i+1 > o if m + i is even; e if m + i is odd: i Taking the limit n ! 1 in Theorem 4.5 leads to the following result. Proposition 4.7 The generating function for the elements of De is X

2De

xjje y jjo =

The generating function for the elements of Do is X

2Do

xjje yjjo =

Y

1

i1 1 ? x

Y

i1

11

ai y ai  :

1

: 1 ? xai+1 y ai?1

p When k = `, both e and o are equal to  = k+ 2k ?4 . The proposition above implies the following generalization of Euler's result: 2

p

the partitions of m such that the quotient of consecutive parts is greater than k+ 2k ?4 are equinumerous with the partitions of m into parts belonging to the set fe1; e2; : : :g, with e1 = 1, e2 = k + 1 and ei+1 = kei ? ei?1 for i  2. 2

p

For instance, the partitions of m such that the quotient of consecutive parts is greater than (3 + 5)=2 are equinumerous with the partitions of m into parts belonging to the set f1; 4; 11; : : :; F2i?1 + F2i+1 ; : : :g where Fi is the ith Fibonacci number.

5 Strict lecture hall partitions and reciprocity The enumeration of lecture hall partitions ts in the framework of Ehrhart theory and, more generally, of the theory of linear homogeneous diophantine systems of inequalities [6]. It is thus not surprising that a reciprocity theorem, relating lecture hall partitions and strict lecture hall partitions, holds. We rst show that this theorem can be simply derived from the reduction procedure described in Section 3, without using the general theory. Then we study the sequences a such that the generating function for a-lecture hall partitions is self-reciprocal. We give a partial characterization of these sequences in terms of continued fractions. Let us rst state more precisely the connection between lecture hall partitions and Ehrhart's theory on the enumeration of integer points lying in the integer multiples of a rational convex polytope of IR n . Let a = (a1 ; : : :; an) be a non-decreasing sequence of positive integers, and let P be the following convex polytope of IRn :  x x n 1 P = (x1; : : :; xn x1 +    + xn = 1 and 0  a  : : :  a : 1 n Then an a-lecture hall partition of weight m is exactly an integer point of mP . 

) 2 IRn :

5.1 A reciprocity theorem

Let La denote, as usually, the set of a-lecture hall partitions, and let La denote the set of strict lecture hall partitions:   La = (1; : : :; n) 2 INn : 0 < 1 < : : : < n :

a1

We de ne the complete generating functions for La and La by X G(X ) = X1 : : :Xnn and G (X ) = 1

2La

an

X

2La

X1 : : :Xnn 1

where G(X ) means G(X1; : : :; Xn). Note that G(X ) and G (X ) completely describe the sets La and La . Proposition 5.1 For any non-decreasing sequence a = (a1; : : :; an), the complete generating functions G(X ) and G(X ) for a-lecture hall partitions and strict a-lecture hall partitions are rational functions in X1; : : :; Xn. Moreover, they are related by the following identity: G(X ) = (?1)nG(1=X ); where G(1=X ) = G(1=X1; : : :; 1=Xn).

12

Example. The generating function for partitions (1; 2) satisfying 0  1=3  2=4 is 2 2 3 1X2 + X1 X2 G(X1; X2) = (11 +? X X )(1 ? X 3X 4) 2

1

2

whereas the generating function for partitions (1; 2) satisfying 0 < 1=3 < 2=4 is + X1X2 + X12X23 = G(1=X ; 1=X ): G(X1; X2) = X1 X22 (11 ? 1 2 X )(1 ? X 3X 4) 2

1

2

Proof. This result can be derived from Proposition 7.1 in [6], but we are going to give a direct proof.

The decomposition of lecture hall partitions into standard and reduced lecture hall partitions described in Section 3 gives (X ) G(X ) = Qn (1 ?RX (14) ai    X an ) ; i=1

n

i

where R(X ) = 2R X1 : : :Xnn is the complete generating function for reduced lecture hall partitions. Recall that the reduced lecture hall partitions are the partitions  such that 0  1 < a1 and for i  2 P

1

0  i ? i?1 < 1:

ai

ai?1

(15)

As there is a nite number of reduced lecture hall partitions, the series G(X ) is a rational function. A similar reduction, performed on strict lecture hall partitions, leads to  (X ) (16) G(X ) = Qn (1 ?RX ai an ; i=1 i    Xn ) where R (X ) is the complete generating function for reduced strict lecture hall partitions, that is, partitions (1 ; : : :; n ) satisfying 0 < 1  a1 and for i  2 0 < ai ? ai?1  1: i?1

i

(17)

As the set R of reduced strict lecture hall partitions is nite, G (X ) is also a rational function. Combining (14) and (16) proves that the identity of Proposition 5.1 is equivalent to

R(X ) = R(1=X )

n Y i=1

Xiiai :

But this is clear, as the correspondence (1 ; : : :; n ) 7?! (a1 ? 1 ; : : :; iai ? i ; : : :; nan ? n ) de nes a bijection from R to R (this can be easily checked thanks to (15) and (17)). Q Remark. The proof above shows that G(X ) and G (X ) are rational functions with denominator ni=1 (1 ? a

Xi i : : :Xnan ). We can actually be a bit more precise. The de nition of reduced partitions implies that the polynomials R(X ) and R (X ) will haveQa factor (1+ Xn +    + Xnan?1 ). Thus G(X ) and G (X ) are rational ?1 (1 ? X ai : : :X an ). functions with denominator (1 ? Xn ) ni=1 n i Proposition 5.1 provides the following result for strict lecture hall partitions associated with the sequences of Theorem 4.5. 13

Corollary 5.2 Let (an)n0 be the sequence de ned by a0 = 0, a1 = 1 and for n  1, ( a2n = `a2n?1 ? a2n?2 ; a2n+1 = ka2n ? a2n?1 where ` and k are two integers  2. The generating functions for strict (a1; : : :; a2n)-lecture hall partitions and strict (a1 ; : : :; a2n?1)-lecture hall partitions are respectively given by X

2L(a1 ;:::;a2n )

and

xjje y jjo =

xai y ai ai ai  i=1 1 ? x y 2n Y

xai yai? a a i=1 1 ? x i y i? 2L a ;:::;a n? where the sequence (an )n1 is de ned by the initial conditions a1 = 0, a2 = 1 and the same recurrence relations as for a. X

( 1

2Y n?1

xjje yjjo =

2

1

+1

+1

1

1)

Of course this result follows from Proposition 5.1. However, it can be proved directly, that is, without involving reduced lecture hall partitions: the correspondence  = (1; : : :; n) 7?!  = (1 + 1; : : :; n + n), where 2i = a2i?1 + a2i and 2i?1 = a2i?1 + a2i ; (equivalently, i = ai + ai ) de nes a bijection from L(a ;:::;an ) to L(a ;:::;an ). To check this point, one uses the fact that 1 = 1 and for i  2, ai?1 i ? ai i?1 = 1: But this argument actually proves something more precise than Corollary 5.2; it implies that the complete generating function for strict lecture hall partitions is 1

G (X ) = G(X )

According

1

n Y

Xi i :

i=1 to Proposition 5.1, we also have G (X ) = (?1)n G(1=X ), and

G(1=X ) = (?1)n G(X )

n Y

i=1

thus G(X ) is self-reciprocal:

Xi i :

The following paragraph is devoted to the study of sequences a such that the series G(X ) is self-reciprocal.

5.2 Self-reciprocal sequences

De nition 5.3 A non-decreasing sequence a = (a1; a2; : : :; an) of positive integers is said to be self-

reciprocal if the complete generating function G(X ) for a-lecture hall partitions is self-reciprocal: G(1=X ) = X G(X ) for some monomial X = X1 : : :Xn n . According to the reciprocity result 5.1, the sequence a is self-reciprocal if and only if there exists a (strict) lecture hall partition = ( 1; : : :; n) such that the correspondence  7?!  + de nes a bijection from La to La. In this case, G(1=X ) = (?1)nX G(X ). If such a bijection exists, any strict lecture hall partition (1; : : :; n) is such that i  i for all i, and thus has to be the (unique) minimal strict lecture hall partition, de ned by 1 = 1 and i+1 = 1 + bai+1 i =aic for i  1. 1

14

Proposition 5.4 The sequence a is self-reciprocal if and only if the minimal strict lecture hall partition ( 1; : : :; n), de ned by 1 = 1 and i+1 = 1 + bai+1 i =ai c, satis es ai i+1 ? ai+1 i = gcd(ai; ai+1) for i  1. In particular, if (a1; : : :; an ) is self-reciprocal, then (a1; : : :; an?1 ) is also self-reciprocal. Proof. As stated above, the sequence a is self-reciprocal if the correspondence  7?!  + is a bijection from La to La . Clearly,  + belongs to La when  belongs to La . Conversely, given  = (1; : : :; n ) 2 La , does  ? belong to La ? We have 1 ? 1 = 1 ? 1  0. Moreover, i+1 ? i+1  i ? i () a  ? a   a ? a : ai+1

ai

i i+1

i+1 i

i i+1

i+1 i

This inequality must hold for all strict lecture hall partitions , and thus the condition for a to be selfreciprocal is that for i  1, ai i+1 ? ai+1 i = min faii+1 ? ai+1 ig : 2La

Let gi denote the right-hand term of this identity. Clearly, gi is a positive multiple of gcd(ai ; ai+1). A standard consequence of Euclid's algorithm (sometimes called Bezout's theorem) is that there exist two integers n0 and m0 such that ai n0 ? ai+1 m0 = gcd(ai ; ai+1). Moreover, for any integer k, the pair (nk ; mk ) = (n0 + kai+1 ; m0 + kai ) also satis es ai nk ? ai+1 mk = gcd(ai; ai+1 ). For a large k, it is clearly possible to construct a strict lecture hall partition  such that i = mk and i+1 = nk . This proves that gi = gcd(ai; ai+1 ) and concludes the proof. This proposition implies that the trivial polynomic sequences, i.e., the sequences (a1; : : :; an ) such that ai jai+1 , are self-reciprocal. More interestingly, we give below a complete characterization of selfreciprocal sequences such that ai and ai+1 are relatively prime. We show that these sequences are natural generalizations of the sequences studied in Section 4. Without proving any of the conjectures given in Section 2, our result partially explains why the (k; `)-sequences are particular. Proposition 5.5 Let a = (a1; : : :; an) be a non-decreasing sequence of positive integers. The following conditions are equivalent: (i) a is self-reciprocal and gcd(ai ; ai+1) = 1 for 1  i < n, (ii) there exist n integers u1 ; : : :; un such that u1  1, ui  2 for 2  i  n, and for i  0 ai+1 = ui+1 ai ? ai?1 with the initial values a?1 = 0 and a0 = 1. If these conditions are satis ed, the minimal strict lecture hall partition ( 1; : : :; n) is de ned by 1 = 1 and i+1 = ui+1 i ? i?1 for i  1, with the initial condition 0 = 0. Proof. Assume that a is self-reciprocal and gcd(ai; ai+1) = 1. According to the criterion of Proposition 5.4, the minimal strict lecture hall partition satis es ai i+1 ? ai+1 i = 1 (18) for i  0 (with 0 = 0 and a0 = 1). Taking the di erence of two consecutive such identities gives, for i  1, ai ( i+1 + i?1 ) = i(ai+1 + ai?1 ): (19) But according to (18), ai and i are relatively prime. Thus (19) implies that there exists an integer ui+1 such that ai+1 + ai?1 = ui+1 ai and i+1 + i?1 = ui+1 i . Taking u1 = a1 completes the construction of the sequence (u1; : : :; un ). 15

Conversely, in order to prove that the existence of this sequence implies the rst point of Proposition 5.5, we rst check that the minimal strict lecture hall partition has the announced form, and then prove, by induction on i, that ai i+1 ? ai+1 i = 1. This proves that gcd(ai ; ai+1 ) = 1 and, via the criterion of Proposition 5.4, that a is self-reciprocal.

Example. Let us take n = 3 and (u1; u2; u3) = (2; 3; 5). We obtain (a1; a2; a3) = (2; 5; 23). The generating

function G(X ) for a-lecture hall partitions is 1 + X2 X35 + X22X310 + X23X314 + X24X319 + X1 X23X314 + X1X24X319 + X1 X25X323 + X1X26X328 + X1 X27X333 : (1 ? X12X25X323)(1 ? X25X323)(1 ? X3) It satis es G(1=X ) = ?X1 X23X314G(X ). The minimal strict lecture hall partition = (1; 3; 14) satis es the conditions of Proposition 5.4: 2  3 ? 5  1 = 5  14 ? 23  3 = 1.

Remark. Another way of describing the sequences a of the proposition above is the following: for 1  i  n, the quotient i =ai is the ith convergent of the continued fraction u1 ? That is,

i = ai u1 ?

1

1

u2 ? :::1 1

u2 ?

1

:

1

:

::: ? u1i

For example, the continued fractions associated with the sequences of Theorem 4.5 are 1 : 1 1? 1 `+1? 1 k? 1 `? k? 11 `?

:::

Since every positive rational number has a unique continued fraction expansion of the form above, this establishes a bijection between self-reciprocal sequences such that gcd(ai ; ai+1 ) = 1 and positive rational numbers.

Acknowledgements. We are grateful to George Andrews for his great encouragement during this work.

We would like also to thank Philippe Flajolet, Sergei Kerov and Svante Linusson for helpful comments and discussions.

References [1] G. E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, AddisonWesley, 1976. 16

[2] G. E. Andrews, MacMahon's partition analysis: I. The lecture hall partition theorem, preprint, 1996. [3] M. Bousquet-Melou and K. Eriksson, Lecture hall partitions, Report LaBRI 1107-96, Universite Bordeaux 1, 1996. To appear in The Ramanujan Journal. [4] L. Euler, Introductio in analysin in nitorum, Chapter 16. Marcum-Michaelem Bousquet, Lausannae, 1748. French translation: Introduction a l'analyse in nitesimale, ACL editions, Paris, 1987. [5] P. A. MacMahon, Combinatory analysis, 2 vols., Cambridge University Press, Cambridge, 1915{1916 (Reprinted: Chelsea, New-York, 1960). [6] R. P. Stanley, Combinatorial reciprocity theorems, Adv. Math. 14 (1974) 194{253.

17