Polyhedral geometry, supercranks, and combinatorial witnesses of congruences for partitions into three parts
In this paper, we use a branch of polyhedral geometry, Ehrhart theory, to expand our combinatorial understanding of congruences for partition functions. Ehrhart theory allows us to give a new decomposition of partitions, which in turn allows us to define statistics called supercranks that combinatorially witness every instance of divisibility of by any prime , where is the number of partitions of into three parts. A rearrangement of lattice points allows us to demonstrate with explicit bijections how to divide these sets of partitions into equinumerous classes. The behavior for primes is also discussed.
Ramanujan  observed and proved the following congruences for the ordinary partition function:
In 1944, Freeman Dyson  called for direct proofs of Ramanujan’s congruences that would give concrete demonstrations of how the sets of partitions of , , and could be split into and equinumerous classes, respectively.
…it is unsatisfactory to receive no concrete idea of how the division is to be made. We require a proof which will not appeal to generating functions, but will demonstrate by cross-examination of the partitions themselves…
He conjectured that a very simple statistic on partitions, the largest part minus the smallest part, performs this division when considered modulo 5 and 7. He named this statistic the “rank” of a partition, and he further hypothesized the existence of a different statistic, called the “crank,” that would witness Ramanujan’s congruence modulo 11 in the same way. In , Atkin and Swinnerton-Dyer proved Dyson’s conjecture about the rank, and in 1988, Andrews and Garvan  found a crank that not only witnessed Ramanujan’s congruence modulo 11, but also witnessed Ramanujan’s congruences modulo 5 and 7 with a new division into 5 and 7 classes, respectively. However, in both cases, the proofs were analytic, and they did not employ a cross-examination of the partitions themselves as Dyson had hoped.
In , Garvan, Kim, and Stanton finally gave a combinatorial proof of Ramanujan’s congruences by finding explicit bijections among equinumerous classes. These bijections were realized by 5-, 7-, and 11-cycles, respectively, that exhaust the corresponding sets of partitions. Their cycles led to new crank statistics that were different from the rank and the Andrews-Garvan crank, but that witnessed the congruence modulo 11 as Dyson had requested, and also witnessed the congruences modulo 5 and 7 with another new division into 5 and 7 classes, respectively. Remarkably, more than 70 years after Dyson’s original request, there is still no known bijective proof that the rank witnesses Ramanujan’s congruences modulo 5 and 7.
In this paper, we explore the possibilities for giving bijective proofs of partition congruences by considering , the number of partitions of into three parts. We define a new crank-like statistic on partitions that has a remarkable property. Unlike the rank and the few known cranks for which witness congruences along certain arithmetic progressions, we discover cranks for that witnesses each and every instance of divisibility modulo a given prime. We call cranks with this remarkable property supercranks, and they were first treated by the second and third authors in . The new techniques we introduce here allow us to give the first infinite family of supercranks, and more generally establish an entirely new framework for treating certain types of partition congruences and crank statistics. Although we currently restrict our attention to , in  we address a much more general class of partition functions.
Our new method for discovering bijective proofs comes from an appeal to polyhedral geometry, specifically, Ehrhart Theory [3, 4, 10, 11, 12]. Partitions of an integer into three parts can be viewed as integer vectors lying inside a triangle in 3-dimensional space. We call this triangle the partition triangle, and the natural inequalities that determine partitions with three parts define a polyhedral cone that we call the partition cone. Following Ehrhart, the partition cone can be tiled by integer translates of a certain fundamental parallelepiped. This tiling of the partition cone then induces a tiling of the partition triangle with slices of the fundamental parallelepiped (Figures 2, 3 and 4). This construction allows us to decompose each such partition into a partition in the fundamental parallelepiped and a non-negative integer vector that determines which translate of the fundamental parallelepiped lies in. We call the box remainder and the box quotient, which together give the box decomposition of (Figure 5 and Definition 4.1). The box decomposition has a purely combinatorial description that we study in detail in . In this paper, we maintain a geometric vantage point whereby the box decomposition allows us to view the set of all partitions of into three parts as a union of six copies of triangular arrays of lattice points. In particular, as Ehrhart already pointed out, this provides a geometric interpretation of the coefficients of the quasipolynomial formula for in a binomial basis.
With this geometric insight in hand, when exhibits divisibility, it is often possible to reassemble the triangles into a rectangle of lattice points wherein the number of lattice points along the width or height of this rectangle is divisible by the modulus we are interested in (Figure 6). Cycling partitions along the rows of this rectangle provides a combinatorial witness for divisibility. Since this construction works for any arrangement of triangles into a suitable rectangle, this method provides a whole family of combinatorial witnesses for congruences of (Theorems 5.1 and 5.2). Cranks for every such witness are given by a composition of the piecewise linear functions that perform the original decomposition into triangles and the subsequent rearrangement into a rectangle.
In general, essentially the only way to write down a formula for these cranks is by this direct appeal to the mechanics of this rearrangement. However, for every prime , there is one special scheme for rearranging these triangles which causes all of the resulting cranks to simplify into a single simple formula: the largest part minus the smallest part modulo . This simple statistic, denoted , witnesses each and every divisibility of by any such . In other words, the polyhedral geometry approach provides us with a deep structural insight into the set of partitions into three parts, allowing us to construct a supercrank (Theorem 3.4).
The rest of the paper is arranged as follows. In Section 2, we give background on formulas for and fully characterize when is divisible by any prime . In Section 3, we define terms surrounding combinatorial proofs of partition congruences, and we state our main theorem. In Section 4, we give our key decomposition of partitions into three parts via Ehrhart theory. In Section 5, we use this decomposition to demonstrate a general method for constructing Ehrhart cranks that witness congruences for . In Section 6, we prove our main theorem based on the mathematics developed in the previous sections. Section 7 provides an alternate proof of the main theorem which, in contrast to Section 6, is confined to only the partition triangle. In Section 8, we use our techniques to consider the divisibility of by primes . In Section 9, we offer some concluding remarks and indicate some directions for future study.
2 The Arithmetic and Congruence Properties of
2.1 Historical Background for .
Historically, there have been several ways to compute the values of . One method is the expansion of the generating function for as rational function.
Unfortunately, the above expressions for have some shortcomings. The methods used to obtain them either do not easily generalize, or they require mathematics with a significant amount of depth. Moreover the expressions in (2), (3), and (4) elicit very little information about the partitions themselves. In the next section, we consider an alternative that eradicates all of these shortcomings.
2.2 Ehrhart’s method for computing quasipolynomials.
Half a century ago, the French geometer Eugène Ehrhart devised a very elegant method for computing formulas for functions such as , , and many others [3, 10, 11, 12]. While Ehrhart was interested in counting integer points in dilates of polytopes, or, in other words, the number of solutions of a linear Diophantine system as a function of the system’s right-hand side, his method boils down on the arithmetic level to straightforward manipulation of the relevant generating function. The key insight is that it is useful to bring the denominator into the form . For , we compute:
In this paper, our focus is on the structure underlying lines (8) through (13). We will show that these expressions contain fundamental information about arithmetic, geometric, and combinatorial properties of the partitions themselves. Notice that (8) through (13) are given in the binomial basis which, when simplified to the monomial basis, are equivalent to the expressions in (3).
In particular, the formulas (8) through (13) as well as (2), (3), and (4) show that is a quasipolynomial, as are all for fixed . A quasipolynomial is a polynomial in whose coefficients are periodic functions of . Equivalently, a quasipolynomial is a function such that there exist polynomials with the property for all and . The integer is called a period of . The minimal period of the partition function is as we have seen above. In general the minimal period of is , which is implicit in the work of Herschel , Cayley , Sylvester , Glaisher , and others.
2.3 Establishing Infinite Families of Divisibility for
With a quasipolynomial formula in hand, it is straightforward to determine various divisibility properties of . For any prime , we can determine when completely.
Let be prime. Then
We will often write these congruences as, for ,
Write for and appeal to (3).
Case 1: .
Then ; i.e., .
Case 2: or .
Then (i.e., ) or . Now , and since , the inverse of is , and so this is the same as . Multiplying both sides of this congruence by , adding , and observing that , we have that this is equivalent to
Thus, in Case 2, we have
i.e., exactly when .
Case 3: .
Observe that . Also observe that, modulo , the inverse of is , and so the inverse of is . Thus if and only if
Dividing by 3, this holds if and only if , which can only have solutions if is a square modulo . Since
we see that no matter what the parity of is, is not a square modulo . ∎
3 Crank, Supercrank, and Statement of Main Theorem
We saw in the last section that modulo any prime , the quasipolynomial formula for allows us to see exactly when . A natural question to ask is, “is there a combinatorial way we could have predicted this without appealing to the formula?” Also, “are there crank statistics that witness any of this divisibility?” The remainder of this paper is devoted to demonstrating affirmative answers to both of these questions. In this section, we make rigorous the notions of “combinatorial witness,” “rank,” and “crank,” we state our main theorem, and we outline the proof.
Let be a finite set of size , and let be a positive integer. A combinatorial witness for is an explicit partition
of into disjoint sets together with explicit bijections
which witness that any two of the are of the same size.
Dyson’s original request for a crank statistic has now been satisfied in a few different ways. Here, we will refer to any statistic that forms a combinatorial witness for divisibility as a crank.
A crank is a function defining the classes and cycles are given by a permutation such that is a bijection between and , where the index is understood modulo .
In this article, we introduce a new class of cranks, which we call Ehrhart cranks, in Sections 4 and 5 below. Ehrhart cranks make excellent combinatorial witnesses as they reveal a tremendous amount of structure in sets of partitions with a restricted number of parts.
One remarkable crank is the following. Let denote the set of partitions with three parts . Let a modulus be fixed. For every define
to be the largest part minus the smallest part modulo . As we will show in Section 6, defines a crank for all the congruences given in Proposition 2.1. This is a particularly strong property for which we coin a new name: Supercrank.
Let be a fixed integer. Let be a finite set for every nonnegative integer . A function defined on is a supercrank if for all such that , the function is a crank witnessing this divisibility.
A main result of this paper is that has this property for the set of partitions into three parts for every prime .
Let be prime. Then , largest part minus smallest part modulo , is a supercrank witnessing for each and every for which this divisibility holds, as characterized in Proposition 2.1.
We now give a rough outline of how Theorem 3.4 will be established, and we begin by describing a way in which one might geometrically demonstrate divisibility of by .
By treating a partition as an integer vector , we can see that the set of partitions of into three parts is
As can been seen in the example shown in Figure 1, these lattice points fit into an obvious triangular region , which is determined by the above equation and inequalities applied in . If we find a way to rearrange these points evenly into a rectangle such that the number of lattice points along the width or height of the rectangle is divisible by the modulus we are interested in, then we have a geometric demonstration that is divisible by (see Figure 6 for an illustration). Of course, we would like to do this not for one particular , but, if possible, for every single such that . By studying these lattice points in a way outlined by Ehrhart [3, 10, 11, 12], a great deal of structure is revealed in Section 4. In particular, for every , this collection of lattice points dissects nicely into six neatly arranged triangular collections of points (see Figure 5). As it turns out, for every prime , we are able to find a uniform method for arranging these triangles into rectangles with side lengths divisible by for every such that , and thus we have a geometric proof of the divisibility.
In fact, the rectangles proving divisibility offer an obvious way to divide the partitions into cycles of length . We can define a “crank” statistic on the lattice points (and equivalently, on the partitions) as simply “the distance from the appropriate edge of the rectangle,” and then our partitions divide into equal classes according to their “crank” modulo . As it turns out, whenever we have one arrangement of our triangles into a useful rectangle, we actually have many such arrangements. For each, we get a different crank statistic that witnesses the divisibility. We call cranks of this type “Ehrhart cranks”, which are defined in Section 5.
In Section 6, we find that among all of the possible arrangements of triangles into rectangles, there is one that is by far the most well poised. There is one way in particular of arranging the triangles such that a constant multiple of the distance modulo from one edge of the rectangle is identically the largest part minus the smallest part of the partition. In other words, we have Theorem 3.4, a statement as simple as Dyson’s original conjecture. What is quite striking is that, whereas Dyson’s rank witnesses the first two Ramanujan congruences, this new supercrank witnesses every congruence of the form for every prime .
4 The Box Decomposition of Restricted Partitions
In this section, we introduce the box decomposition of a restricted partition into a box remainder , which is a partition in the fundamental parallelepiped defined below, and a box quotient , which is a non-negative integer vector. This decomposition is motivated by polyhedral geometry, and is the result of applying a classic construction in Ehrhart theory [3, 10, 11, 12] in a partition theoretic context. While this decomposition can be defined purely in combinatorial terms , the geometric point of view will provide the key intuition for the rest of this paper, and so we introduce the relevant background from Ehrhart theory in this section. In particular, we take great care to visualize the construction in order to build geometric intution. To be clear, our use here of the word box is not motivated by the geometry at hand but by the Ferrers Diagram of . Since partitions into three parts are the topic of this paper, we will restrict our attention to the three-dimensional case. Note, however, that the constructions below generalize in a straightforward manner to partitions with any fixed number of parts; this is treated in detail in . For general introductions to Ehrhart theory and polyhedral geometry, we recommend the textbooks  and .
We now set the stage for our definition of the box decomposition in Definition 4.1 below. As illutrated in Figure 1, the starting point for the geometric approach is to view a partition as an integer vector . Experienced partition theorists may not be used to this convention, so just to be absolutely clear, throughout the rest of the paper, we will literally use the word “partition” to mean such a vector in . The height is the sum of coordinates of ; i.e., the number being partitioned. The set of all partitions into three parts is then the set of integer vectors or lattice points in the partition cone
The set of partitions of a fixed into three parts is then the set of lattice points in the partition polytope or partition triangle
which is the intersection of the partition cone with the plane at height . With this notation, the restricted partition function is simply .
One important property of the partition cone is that it has the dual description
The columns of are called the generators of the cone . Equation (18) states that is the set of all vectors that can be written as a linear combination of generators with non-negative real coefficients, where the coefficient of the last generator has to be strictly positive. Note that is simplicial; i.e., the generators are linearly independent. The generators are not uniquely determined; we could replace any generator by any positive multiple for . However, the generators in (18) have the crucial property that they all have integer components, and they are all at the same height. Notice is the lowest height at which this happens. The idea of choosing generators in this way goes back to Ehrhart [10, 11, 12].
Here, denotes the Minkowski sum; i.e., is obtained by translating by every vector , and then taking the union, which is disjoint. Again, we write . Note that the fundamental parallelepiped is not uniquely determined by the cone , but by our particular choice of generators.
For every , the tiling (19) of induces a tiling of the partition triangle as shown in Figure 2. To make this precise we define as the slice of at height , and let and denote the corresponding lattice point set and count, respectively. All slices through and the lattice points they contain are shown in Figure 3. The coefficients in (8) through (13) reflect the lattice point counts in the .
We define as the set of all non-negative integer vectors with coordinate sum . The elements of are necessarily arranged in a triangle pattern and are correspondingly counted by the triangular numbers . With this notation, the construction shown in Figure 2 then yields
for any and any . Here, we have made crucial use of the fact that our chosen generators are all at the same height and are integral. In particular, (21) shows that the coefficients count lattice points at a certain height in the fundamental parallelepiped, and that the binomial coefficients count translation vectors . This was Ehrhart’s key observation, whence the vector is called Ehrhart -vector or Ehrhart -vector.
Figure 4 shows examples of the tiling given in (20) for through . Passing from the continuous tiles to lattice point sets yields (21). Counting these lattice points we obtain (22), which is the general form of equations (8) through (13). This construction shows that the coefficients of (22), hence, (8) through (13) are precisely the lattice points counts in the as is evident in Figure 3.
Since all unions are disjoint, (21) allows us decompose any partition uniquely into a partition (the box remainder) in the fundamental parallelepiped and a partition of the form , where (the box quotient) is a non-negative integer vector. For brevity, we will simply say that we decompose into the pair . This decomposition is vital for our results, and so we summarize it in the following definition and lemma. See Figure 5 for an illustration.
For all , , and , write
We abbreviate . The pair is called the box decomposition of . The partition is the box remainder and the vector is the box quotient of .
As we have proven above, the crucial property of is the following.
For all and , the map
is a bijection.
Even though , we have since but . The underlying reason is that we are working with partitions with exactly three parts. In fact, is the magenta square in as shown in Figure 3.
The box decomposition has a very nice combinatorial interpretation in terms of a decomposition of the Ferrers diagram of into boxes, which is explored in . For the purposes of this paper our focus is on the geometric point of view.
5 Ehrhart cranks for witnessing divisibilities
In this section, we construct a whole family of combinatorial witnesses for each and every arithmetic progression of divisibility for modulo any prime . The basic strategy is the following: The decomposition from Definition 4.1 allows us to divide the set of partitions into copies of the triangle , copies of the triangle , and copies of the triangle . Any way in which these triangles can be reassembled into a rectangle of lattice points wherein the number of lattice points along the width or height of the rectangle is divisible by will provide us with cycles proving divisibility and a crank statistic that witnesses divisibility for that fixed . Here it is crucial that for a given there is an arrangement of triangles that works for all , so that we do indeed obtain a crank function that witnesses divisibility for the entire arithmetic progression with a fixed remainder.
We first give an informal statement of the theorem, and then in Theorem 5.2, we give the technical details.
For fixed and , and any , let be such that . Then, any way of reassembling triangles , triangles , and triangles into a rectangle with one side divisible by defines a crank function , as illustrated in Figure 6, that witnesses the congruence
where cycles are given by traversing either the rows or columns of .
To work our way up to a precise technical statement of this result, let us consider the case of remainder as an example, as shown in Figure 6. Our goal is to witness (24). To begin, is given by the decomposition from Section 4. To this end, we rewrite
where , , and
We now show that the six interlacing triangles in can be rearranged to make a rectangle having at least one side of length a multiple of . This is done by noting that , being the six triangles indicated by the binomial coefficients in , can be arranged into a rectangle of width and height . Formally, we let denote the continuous rectangle, and write for the set of lattice points therein. Since, in this example, is a multiple of for all , it follows that is divisible by , and this divisibility is witnessed by the crank
The cycles that go along with this crank are simply moving along a row of the rectangle. Our strategy will therefore be to reassemble the triangles in to form the rectangle . To this end, assume that for every we have an affine linear function such that for every , the restriction is injective, and
for all , where the unions are disjoint and is determined by . Then, the function
is a bijection. Affine linear functions with this property can be understood by an inspection of Figure 6. We will give explicit formulas for this example below, but first, let us reap the benefits of this construction: the composition ,
is a crank function witnessing the congruence (24) for . Cycles are defined by moving along a row of . One important aspect here is that the formulas for do not depend on . In this sense, is one function that witnesses divisibility for the entire congruence (24) for fixed .
The above construction generalizes, which proves the following theorem.
Let and be fixed. Let and such that for all . Let . Let and . Let . For each , let be an affine linear function such that for every , the restriction is injective, and