Five Guidelines for Partition Analysis with Applications to Lecture Hall-type Theorems

Five simple guidelines are proposed to compute the generating function for the nonnegative integer solutions of a system of linear inequalities. In contrast to other approaches, the emphasis is on deriving recurrences. We show how to use the guidelines strategically to solve some nontrivial enumeration problems in the theory of partitions and compositions. This includes a strikingly different approach to lecture hall-type theorems, with new $q$-series identities arising in the process. For completeness, we prove that the guidelines suffice to find the generating function for any system of homogeneous linear inequalities with integer coefficients. The guidelines can be viewed as a simplification of MacMahon's partition analysis with ideas from matrix techiniques, Elliott reduction, and ``adding a slice''.


Introduction
This continues our work in [18,19] studying nonnegative integer solutions to linear inequalities as they relate to the enumeration of integer partitions and compositions.Define the weight of a sequence λ = (λ 1 , λ 2 , . . ., λ n ) of integers to be |λ| = λ 1 + • • • + λ n .If sequence λ of weight N has all parts nonnegative, we call it a composition of N ; if, in addition, λ is a nonincreasing sequence, we call it a partition of N .
Given an r × n integer matrix C = [c i,j ], we consider the set S C of nonnegative integer sequences λ = (λ 1 , λ 2 , . . ., λ n ) satisfying the constraints c i,0 + c i,1 λ 1 + c i,2 λ 2 + . . .+ c i,n λ n ≥ 0, 1 ≤ i ≤ r. (1) We seek the full generating function which can be viewed as an encapsulation of the solution set S C : the coefficient of q N in F C (qx 1 , qx 2 , . . ., qx n ) is a listing (as the terms of a polynomial) of all nonnegative integer solutions to (1) of weight N and the number of such solutions is the coefficient of q N in F C (q, q, . . ., q).
Variations of this problem arise in other areas of mathematics, e.g., solving systems of linear equations, finding volume of polytopes, as well as in enumeration.In the papers [18,19] we demonstrated that in the area of partition and composition enumeration many familiar sets of linear constraints can be easily handled a matrix inversion: for homogeneous systems, if the constraint matrix C is an n × n invertible matrix, and if all entries of C −1 = B = [b i,j ] are nonnegative integers then by Theorem 1 in [19]: . This theorem (in its full generality) suffices to handle the enumeration of such families as Hickerson partitions [22], Santos' interpretation of Euler's family [28], Sellers' generalization of Santos [29,30], partitions with nonnegative second differences [3], super-concave partitions [31], partitions with r-th differences nonnegative [3,14,33], partitions with mixed difference conditions [3], and examples (0-5) of Pak in [27].The theorem provides bijections as well as generating functions.
However, it is easy to find simple examples where the "C matrix" technique fails.In Section 2, we propose five simple guidelines for computing the generating function F C of a system C of linear diophantine inequalities.The guidelines can be viewed as a simplification of MacMahon's partition analysis [25], with ideas from matrix methods, Elliott reduction [20], and "adding a slice" (e.g.[23]).
Our focus is on the use of the guidelines to derive a recurrence for the generating function F Cn of an infinite family {C n |n ≥ 1} of constraint systems.This is in contrast to the focus of the Omega package [6], a software implementation of partition analysis, well-designed to compute the generating function of a given fixed, finite system of linear constraints.The advantage of a recurrence for F Cn is a program which computes F Cn for any given n.But more significantly, if the recurrence can be solved, it provides a closed form for the generating function for the infinite family.
In Sections 3-6, we show how to use the guidelines of Section 2 strategically to solve some nontrivial enumeration problems in the theory of partitions and compositions.Sections 3 and 4 address well-studied problems, included as "warm-up" exercises to illustrate the approach and the handling of the recurrences that result.Sections 5 and 6 apply the method to the problem of enumerating anti-lecture hall compositions [16] and truncated lecture hall partitions [17], giving a simpler approach than in [16,17].For completeness, in Section 7 we prove that the guidelines suffice to find the generating function for the nonnegative integer solutions of any homogeneous system of linear inequalities with integer coefficients.This work was inspired by the the work of Andrews, Paule, and Riese in the sequence of papers [2,3,6,4,12,7,8,9,5,10,11], which illustrate many applications of partition analysis.The Omega Package software [6] was an invaluable tool in our early investigations.As illustrated in papers such as [2,3,12,9,10], recurrences can certainly be derived using partition analysis.However, we found that the task became easier with a simpler set of tools which appear to be no less powerful.In Section 8 we discuss MacMahon's partition analysis and show how the proposed guidelines can be be viewed as essential ideas underlying his theory.

The Five Guidelines
Let C be a set of linear constraints in n variables, λ 1 , . . ., λ n , each constraint c ∈ C of the form for integer values a 0 , a 1 , . . ., a n .Let S C be the set of nonnegative integer sequences λ = (λ 1 , . . ., λ n ) satisfying all constraints in C. Since we are only interested here in nonnegative integer solutions, we will always assume that C contains the constraints [λ i ≥ 0] for 1 ≤ i ≤ n.Define the full generating function of C to be: Let C λi←λi+aλj denote the set of constraints which results from replacing λ i by Lemma 1 Let C be a set of linear constraints on variables λ 1 , . . ., λ n which contains the constraints [λ k ≥ 0], 1 ≤ k ≤ n.Let a be any integer (possibly negative).Suppose [λ i − aλ j ≥ 0] is implied by C and let C ′ = C λi←λi+aλj .Then Proof.By the remarks preceding the lemma, the constraints C and C ′ guarantee that S C and S C ′ contain only nonnegative integer solutions.So, it suffices to show that β satisfies a constraint in C iff β ′ satisfies the corresponding constraint in Finally, to simplify notation, we will let X n refer to the parameter list x 1 , . . ., x n , so that F (X n ) denotes F (x 1 , . . ., x n ).Let F (X n ; x i ← x i x a j ) denote the function F (X n ) with all occurrences of x i replaced by x i x a j .
Theorem 1 (The Five Guidelines) 2. If C 1 is a set of constraints on variables λ 1 , . . ., λ j and C 2 is a set of constraints on variables λ j+1 , . . ., λ n , then 3. Let C be a set of linear constraints on variables λ 1 , . . ., λ n and assume C contains the constraints 4. Let c be any constraint with the same variables as the set C. Then Proof.

Let
So, 4. S C can be partitioned into those λ that satisfy c and those that do not.
Let C n be the set of constraints , where c has been removed from the next-to-last system, making it C n−1 ∪ [λ n > 0], and c has been replaced by ¬c in the last system.By guidelines 1 and 2, , so by guideline 3, the last system has generating function Putting this together with guideline 5 and the initial condition Let C n (q, s) = C n (q, q, . . ., q, s).Then the above recurrence gives C 1 (q, s) = s/(1 − s) and for n ≥ 2, C n (q, s) and use the recurrence for C n (q, s) to get Iterating the recurrence for C(q, s) gives .

Two-Rowed Plane Partitions
This example illustrates the advantage of guideline 3 of Theorem 1 when a < 0. The two-rowed plane partitions are those integer sequences (a 1 , b 1 , . . ., a n , b n ) satisfying the constraints It is well-known that the generating function for P n is [24] P n (q) = 1 (q; q) n (q 2 ; q) n .
In [3], Andrews shows how MacMahon's partition analysis can be used to compute P n (q) by considering an intermediate family G n .We will use this approach, but with a slight twist, to show how the generating function for P n , can be computed via G n from the guidelines of Theorem 1.
We will use the convention that when a constraint system is represented by a calligraphic letter, its generating function is represented by the corresponding roman letter.Also, to keep notation simple, when the meaning is clear from context, we will use the same letter to refer to multivariable and single variable forms of the generating function.
Define G n to be the set of constraints below: Denote the full generating functions for P n and G n by Note that P n can be transformed into G n by the sequence of substitutions: We focus on G n .Since for 1 ≤ i ≤ n − 1, a i − a i+1 ≥ 0 and b i − b i+1 ≥ 0 in P, by guideline 3 of Theorem 1, P n is obtained from G n by the sequence of substitutions: In particular, the generating function (3) for two-rowed plane partitions is obtained by setting x i = y i = q in P n for i = 1, . . ., n: P n (q, q, q, . . ., q) = G n (q, q, q 2 , q 2 , . . ., q n , q n ).( 4) Since a n − b n ≥ 0 in G n , by guideline 3, we can do the substitution a n ← a n + b n in G n to get F n and recover G n from F n as shown below.
Since a n−1 + a n ≥ 0 in F n , by guideline 3, we can substitute a n−1 ← a n−1 − a n in F n to get H n and recover F n from H n as shown.
Summarizing to this point, we have Now apply guideline 5 to and Now observe that so by guidelines 1 and 2, Returning to L n , since a n − a n−1 ≥ 0 in L n , we can do the substitution so by guidelines 1, 2, and 3, Combining ( 6), (7), and (8), we have Finally, substituting this expression for H n into (5) gives a recurrence for G n : with initial condition Let G * n (q, s) = G n (q, q, q 2 , q 2 , . . ., s, q n ).Then from the recursion (9), It is straightforward to show by induction that G * n (q, s) satisfies Substituting s = q n gives P n (q) = G n (q, q, q 2 , q 2 , . . ., q n , q n ) = G * n (q, q n ) = 1 (q; q) n (q 2 ; q) n , the desired generating function for 2 × n plane partitions.

Anti-Lecture Hall Compositions
In [16], we considered the set of sequences λ = (λ 1 , . . ., λ n ) satisfying the constraints We referred to these as anti-lecture hall compositions and showed that the generating function is Here we show how to apply the guidelines of Theorem 1 to get a recurrence for the full generating function A n (x 1 , x 2 , . . .x n ) and use it to give an "easy" proof of (10).The idea is easily extended to the truncated anti-lecture hall compositions studied in [17].We start with B n , a slight variation of A n .

Lemma 2
The full generating function for the integer sequences defined by the constraints satisfies Proof.The following sequence of substitutions transforms B n into A n−1 ∪ {[λ n ≥ 0]}, as illustrated in Figure 1: Note that the constraint λ i−1 ≥ (i − 1)λ n is implied at each stage, so by guidelines 1,2, and 3, B n is recovered from A n by performing the sequence of substitutions on A n−1 (x 1 , . . ., x n−1 )/(1 − x n ): ✷ Proposition 1 The full generating function for anti-lecture hall compositions satisfies: where Note that C n = A n−1 ∪ {[λ n ≥ 0]}, so by guideline 2, C n has generating function Since λ n ≥ λ n−1 is implied by D n in (13), by guideline 3, substituting and where X n represents the argument list x 1 , . . ., x n .Using guideline 5 again, with c where C n is (12) and where B n is (11).Putting this all together, we have Substituting from ( 14) and Lemma 2 gives the result.✷ In order to make use of the recurrence of Proposition 1 to prove the generating function (10) for anti-lecture hall compositions, let A n (q, s) A n (q, q, q, . . ., q, s).Then the recurrence of Proposition 1 becomes with initial condition A 0 (q, s) = 1.If we were to proceed as with two-rowed plane partitions, we would (i) "guess" the form of A n (q, s), (ii) prove by induction that it satisfies ( 16), and then (iii) show that setting s = q gives (10).This would be the easiest proof and it would give a refinement of the anti-lecture hall generating function, enumerating solutions according to both the weight and the size of the last part: λ∈SA n q |λ| s λn = A n (q, qs).
Since we have not succeeded in guessing A n (q, s), we follow a different approach.Iterating the recurrence of ( 16) gives: Now, setting s = q gives a recurrence independent of s: We show by induction that the solution to (18) is We will prove that Therefore, we need to prove the following identities for C n : A few q-series manipulations show that the two previous equations are equivalent to: Recalling that n j q = (q −n ) j (−1) j q nj−j(j−1)/2 (q) j , we see that the identity follows from the case Now we need The same q-series manipulations show that the two previous equations are equivalent to: This follows in a similar way from the "other" q-Chu Vandermonde summation (1.5.3 in [21]), under the substitutions a = −1, c = 0. ✷

Lecture Hall Partitions
In [13], Bousquet-Mélou and Eriksson studied the set of integer sequences λ = (λ 1 , . . ., λ n ) satisfying the constraints They referred to these as lecture hall partitions and showed that the generating function is In [2], Andrews showed how to use partition analysis to derive a recurrence for the full generating function of L n .However, substantial new ideas, outside of partition analysis, were required to move from this to the solution (27).
In this section, we show that by strategic application of Theorem 1, we can derive a recurrence for the full generating function of a generalization of L n that will reduce the proof of ( 27) to a q-series calculation (albeit nontrivial).Our derivation here via the five guidelines is both simpler and more elementary than the approach in [17] (at the expense of a more challenging q-series calculation).
In [17], we defined truncated lecture hall partitions to be the integer sequences satisfying: that is, all parts must be positive, the generating function is It can be checked that setting k = n and dividing by q ( n+1 2 ) gives (27).

Proposition 2
The generating function for truncated lecture hall partitions (28) satisfies The first system on the right, D, implies the constraint λ k−1 > 0, so it can be added.Now apply guideline 5 to The first system on the right of ( 31) is just Ln,k−1 ∪ {[λ k > 0]}.The second system on the right becomes Ln,k−1 ∪ {[λ k ≥ 0]} after the substitution λ k ← λ k + λ k−1 .So, by Theorem 1 and summarizing so far, we have where E(x 1 , . . ., x k ) is the generating function for the last constraint system, E, in (30).Apply .
By guideline 3, Finally, starting from F , the last set of constraints in (33), perform the following sequence of substitutions as illustrated below: The resulting system of constraints, G, in (35) can be viewed as L n,k−1 , where λ k−1 has been replaced by λ k , together with the constraint [λ k−1 > 0].Thus, By guideline 3, the generating function for F is obtained from G by the sequence of substitutions Returning to E in (34) and using (36) and (37), where c 1 , c 2 , . . ., c r is an ordered list of the non-basic constraints in C. If r = 0, all constraints are basic and the generating function follows from guidelines 1 and 2 (and F [λi=0] (x i ) = 1).
Otherwise, define: M : the largest positive coefficient of c 1 (0, if none); If m = 0, all coefficients of c 1 are nonnegative, so c 1 is redundant and can be deleted.Otherwise, if M = 0, all coefficients of c 1 are nonpositive and so we get an equivalient system replacing λ j by 0 in c 1 , . . ., c r and setting b j = [λ j = 0].In so doing we have decreased |m| or e min .
Otherwise, m < 0 and M > 0; we do a version of Elliott reduction [20].Let i and j be such that m is the coefficient of λ i in c 1 and M is the coefficient of λ j .We would like to use guideline 3 and reduce to a system with smaller M or e max or |m| or e min .First use guideline 4 with c = [λ i ≥ λ j ]: For the first term, F C∪[λi≥λj ] (X n ), do the substitution λ i ← λ i +λ j into constraints c 1 , c 2 , . . ., c r in C.This decreases the coefficient of λ j , thereby decreasing M or e max .By guideline 3, the substitution x j ← x j x i in the generating function of the resulting constraint system gives F C∪[λi≥λj ] (X n ).

✷
(Note that we have optimized the proof for simplicity at the expense of algorithmic efficiency.)It follows from Theorem 2 and its proof that the full generating function of (40) can be built up from the functions 1/(1−x i ) by a finite number of additions, subtractions, and substitutions.We get then as a corollary the following well-known result: The full generating function for the nonnegative integer solutions to any system of linear inequalities in n variables with integer coefficients has the form p(x 1 , . . ., x n ) , where t ≥ 0, p is a polynomial in x 1 , . . ., x n and each α i is a monomial in x 1 , . . ., x n .
• • • c ar,j r ) λj The core of partition analysis is a system of Omega-rules designed to be applied strategically to transform P (x 1 , x 2 , . . ., x n , c 1 , c 2 , . . ., c n ) step-by-step into F C (x 1 , . . ., x n ).This view converts the combinatorial problem into an algebraic one, opening the possibility, for example, of a partial fraction decomposition of (41) to assist in the transformation from P to F .A list of basic Omega-rules appears in [25](pp.103-106) and [2].
This approach has proven both powerful and systematic in the computer solution of systems of inequalities.However, for deriving recurrences for infinite families, we found that a return to some of the basic underlying ideas simplified the process.We note the roots of guidelines 3-5 of Theorem 1 in the work of MacMahon [25].
Our use of guideline 3 (which performs limited column operations on the constraint matrix) is used to much the same effect as the following Omega-rule: .
The utility of guideline 4 was recognized by MacMahon.He writes in [25], p. 103, "A very useful principle is that of adding an inequality which is a fortiori true."It is also used in a decomposition shown at the beginning of [25], Section 379, p. 131.Guideline 5 is one of MacMahon's Omega-rules, found in [25]

Concluding Remarks
The "five guidelines" of Theorem 1 provide a unified setting for computing the full generating function for many challenging families of constraints.However, even though they are guaranteed to be sufficient to find the generating function for any homogeneous linear system, we are not necessarily guaranteed to be able to use them to devise a recurrence for a parametrized family of constraint sets.
In continuing work we consider the case when all constraints have the form λ i ≥ λ j or λ i > λ j , forming a directed graph.We show how to get a recurrence by strategically manipulating the diagrams.Many examples are presented, including two-and three-rowed plane partitions, plane partitions with diagonals, plane partition diamonds, and hexagonal plane partitions.
Finally, we note that in [32], Xin offers a speed-up to the Omega package for implementing MacMahon's partition analysis.Xin's method uses the theory of iterated Laurent series and partial fraction decompositions.

*
Research supported in part by NSF grant DMS-0300034 † Research supported in part by NSF grants DMS-0300034 and INT-0230800

e
max : the number of occurrences of M among the coefficients of c 1 ; m: the smallest negative coefficient of c 1 (0, if none); e min : the number of occurrences of m among the coefficients of c 1 .When r > 0 we show that we can use the guidelines to reduce the computation of the generating function of C to the computation of the generating function of one or more systems C ′ in which at least one of the statistics {r, M, e max , |m|, e min } has been reduced.