Linked partition ideals, directed graphs and $q$-multi-summations

Finding an Andrews--Gordon type generating function identity for a linked partition ideal is difficult in most cases. In this paper, we will handle this problem in the setting of graph theory. With the generating function of directed graphs with an ``empty'' vertex, we then turn our attention to a $q$-difference system. This $q$-difference system eventually yields a factorization problem of a special type of column functional vectors involving $q$-multi-summations. Finally, using a recurrence relation satisfied by certain $q$-multi-summations, we are able to provide non-computer-assisted proofs of some Andrews--Gordon type generating function identities. These proofs also have an interesting connection with binary trees.

1. Introduction 1.1.Rogers-Ramanujan type identities.The two Rogers-Ramanujan identities [17,19], which state as follows, have attracted a great deal of research interest in the theory of partitions.
Theorem (Rogers-Ramanujan identities).(i).The number of partitions of a nonnegative integer n into parts congruent to ±1 modulo 5 is the same as the number of partitions of n such that each two consecutive parts have difference at least 2.
(ii).The number of partitions of a non-negative integer n into parts congruent to ±2 modulo 5 is the same as the number of partitions of n such that each two consecutive parts have difference at least 2 and such that the smallest part is at least 2.
There are many identities of the same flavor, including the Andrews-Gordon identity [1,10], the Göllnitz-Gordon identities [9,11], the Capparelli identities [7] and so forth.In 2014, Kanade and Russell [12] further proposed six challenging conjectures on Rogers-Ramanujan type identities, the latter two of which were proved in 2018 by Bringmann, Jennings-Shaffer and Mahlburg [6].
Among these Rogers-Ramanujan type identities, two types of partition sets are considered.One partition set is consist of partitions under certain congruence condition.For example, in the first Rogers-Ramanujan identity, we enumerate partitions into parts congruent to ±1 modulo 5.The other partition set contains partitions under certain difference-at-a-distance theme.Let us first adopt a definition in [12].Definition 1.1.We say that a partition λ = λ 1 +λ 2 +• • •+λ ℓ satisfies the difference at least d at distance k condition if, for all j, λ j − λ j+k ≥ d.
In this setting, we may paraphrase the corresponding partition set in the first Rogers-Ramanujan identity as the set of partitions with difference at least 2 at distance 1.
Although it is straightforward to find the generating function for partitions under given congruence condition, it is always difficult to obtain an analytic form of generating function for partitions under a difference-at-a-distance theme -this is why the six conjectures of Kanade and Russell remained mysterious for years.But this problem was recently settled by Kanade and Russell themselves [13] and independently by Kurşungöz [15,16] using combinatorial approaches, and later by Li and the author [8] using algebraic methods.For example, in the Kanade-Russell conjecture I 1 , we would like to count "partitions with difference at least 3 at distance 2 such that if two consecutive parts differ by at most 1, then their sum is divisible by 3." It was shown that its generating function is a double summation as follows: q n 2 1 +3n 2 2 +3n1n2 x n1+2n2 (q; q) n1 (q 3 ; q 3 ) n2 , where λ runs through all such partitions, ♯(λ) denotes the number of parts in λ and |λ| is the size of λ (that is, the sum of all parts in λ).
1.2.Span one linked partition ideals.In the 1970s, George Andrews [2][3][4] have already started a systematic study of Rogers-Ramanujan type identities and developed a general theory in which the concept of linked partition ideals was introduced.However, in this paper, we will not go into details of this concept due to its lengthy definition.The interested readers may refer to Chapter 8 of Andrews' book: The theory of partitions [5].
What we are interested in this paper is a special case of linked partition ideals -the span one linked partition ideals.In fact, this special case is enough to cover most partition sets under difference-at-a-distance themes.
Let us first fix some notations.Let P be the set of all partitions.We define a map φ : P → P by sending a partition λ to another partition which is obtained by adding 1 to each part of λ.For example, φ(5 . Also, for two partitions λ and π, their sum λ⊕π is constructed by counting the total appearances of each different part in λ and π.For example, if Let Π be a finite set of partitions containing the empty partition ∅.For each partition π ∈ Π, we define its linking set L(π) by a subset of Π containing the empty partition.Also, we require that the linking set of the empty partition, L(∅), equals Π.It is possible to construct finite chains . We may further extend such a finite chain to an infinite chain ending with a series of empty partitions Let S be a positive integer no smaller than the largest part among all partitions in Π.The above infinite chain C uniquely determines a partition by which is equivalent to Let us collect such partitions along with the empty partition λ = ∅ (which corresponds to the infinite chain ∅ → ∅ → • • • ) and obtain a partition set I := I ( Π, L , S).Then I is called a span one linked partition ideal.
Example 1.1.In the first Rogers-Ramanujan identity, we consider partitions with difference at least 2 at distance 1.It is not hard to verify that this partition set is a span one linked partition ideal I ( Π, L , S) where Π = {∅, 1, 2}, 1 the linking sets are and S = 2.
1.3.Generating function of span one linked partition ideals.Given a span one linked partition ideal I = I ( Π, L , S), one crucial problem is to determine its generating function Assume that Π = {π 1 , π 2 , . . ., π K } where π 1 = ∅, the empty partition.We define a (0, 1)-matrix A = A ( Π, L ) by and a diagonal matrix Let the S-tail of a partition λ be the collection of parts ≤ S in λ.
Theorem 1.1.For each 1 ≤ k ≤ K, we denote by I k the subset of partitions λ in I ( Π, L , S) whose S-tail is π k ∈ Π.We further write 1 Here 1 denotes a partition containing one part of size 1 and likewise 2 denotes a partition containing one part of size 2.
Let A and W (x) be defined as in (1.6) and (1.7), respectively.Then, for |q| < 1 It follows that all entries in the first row and column of A are 1.Further, the first entry in W (x) is also x 0 q 0 = 1.When |q| < 1 and |x| < |q| −1 , we have Remark 1.2.We have Hence, G (x) = G 1 (xq −S ). (1.10) In September 2018, George Andrews communicated to Zhitai Li and the author a conjecture on the generating function for linked partition ideals, which was recorded in [8].
By examining a number of examples in [8,13,15,16], it seems that in some cases the G k (x)'s in Theorem 1.1 are of a unified form of q-multi-summations.It motivates us to consider a matrix factorization problem involving column functional vectors of certain q-multi-summations.This, in turn, provides some crude ideas for the conjecture of Andrews.
Further, the algebraic method in [8] of proving generating function identities such as (1.1) relies heavily computer algebra (Mathematica packages qMultiSum [18] and qGeneratingFunctions [14]).Now we are able to present a new approach to get rid of such computer assistance.
1.4.Outline of this paper.This paper is organized as follows.
In §2, we first define the generating function for walks in a directed graph G.Then, by assigning an empty vertex to G, we obtain a modified directed graph G ! .The generating function of G ! can be defined naturally.Now we merely need to define the associated directed graph of a span one linked partition ideal I ( Π, L , S) and then deduce Theorem 1.1 from the generating function of this associated directed graph.
In §3, we will study a q-difference system arising from Theorem 1.1.Two examples will then be discussed: one example comes from the Rogers-Ramanujan identities and the other is about the Kanade-Russell conjectures I 1 -I 3 .Then, a matrix factorization problem will be identified from the two examples.
In §4, we turn to non-computer-assisted proofs of two identities obtained in §3.The two identities, in turn, can be used to prove Andrews-Gordon type generating function identities for span one linked partition ideals.Our approach relies on a key recurrence relation obtained in §4.1.Also, we are able to illustrate the proofs by binary trees.
Finally, we are going to raise some open problems in §5.

Directed graphs
Let G = (V, E) be a directed graph where V is the set of vertices and E is the set of directed edges.Throughout, we allow loops (that is, directed edges connecting vertices with themselves) in G but for any two vertices u and v, not necessarily distinct, we allow at most one directed edge connecting u with v.
Let A = A (G) be the adjacency matrix of G, that is, We say that w is a walk of step M in G if w is a chain of M + 1 vertices 2.1.Generating function for walks in a directed graph.To define the generating function for step M walks in a directed graph G = (V, E), we assign two weights to each vertex v: one is called length, denoted by ♯(v) ∈ N, and the other is called size, denoted by |v| ∈ N.
Let the shift S be a non-negative integer.
For any walk w ∈ W M , we define its generating function by Now we are able to define the generating function for step M walks from v i to v j for any 1 ≤ i, j ≤ K: Let us define a diagonal matrix W (x) = W (x, q) by Theorem 2.1.Let A be the adjacency matrix of G and let W (x) be as in (2.5).
) equals the number of walks of step M from vertex v i to vertex v j , Theorem 2.1 immediately leads to a well-known result in graph theory: Corollary 2.2.The number of walks of step M from vertex v i to vertex v j is the (i, j)-th entry of A M .
Proof of Theorem 2.1.We induct on M .When M = 0, that is, the chain w of vertices in (2.2) contains only one vertex ̟ 0 , it follows that which is identical to the (i, j)-th entry of W (x). Now let us assume that the theorem is true for some M ≥ 0. We also write for convenience On the other hand, , which is our desired result.

Assigning an empty vertex.
Let us assume that v 1 ∈ V is an empty vertex, that is, its length and size are both 0: We also assume that, for 2 ≤ k ≤ K, ♯(v k ) and |v k | are both positive integers.We require that, for each 1 ≤ k ≤ K, there is an edge from vertex v k to the empty vertex v 1 .Hence, the entries in the first column of the adjacency matrix A are all 1.
We call such modified directed graph For any finite walk in G ! , we may extend it to an infinite walk It follows from the assumptions ♯(v 1 ) = 0 and We are now in the position to define the generating function of G ! , by (2.10) ) denote the generating function for infinite walks in W ⋆ starting at v k .Let the shift S be a positive integer.Let A and W (x) be defined as in (2.1) and (2.5), respectively.Then, for |q| < 1 and |x| Proof.We simply observe that, for each 1 The desired result therefore follows.We first define the set of vertices.Since Π = {π 1 , π 2 , . . ., π K } is a finite set of partitions, we may treat each π k as a vertex.We also define the length of π k as the number of parts in π k and the size of π k as the sum of all parts in π k .In particular, since π 1 is an empty partition so that ♯(π 1 ) = 0 and |π 1 | = 0, we may treat π 1 as an empty vertex.
We next define the directed edges in a natural way.For 1 ≤ i, j ≤ K, if π j ∈ L(π i ), then we say that there is an edge from vertex π i to vertex π j .Since L(π 1 ) = L(∅) = Π, we know that, for each 1 ≤ k ≤ K, there is an edge from vertex π k to vertex π 1 .
We call this graph the associated directed graph of I , denoted by Recall from (1.4) that each partition λ in I can be uniquely decomposed as so that λ K = ∅ as long as λ = ∅.Hence, we have a natural bijection to infinite walks in G ! (I ) ending with Further, if λ is an empty partition, then the resulted infinite walk is simply Now let us define S to be the shift.Then Hence, The rest follows directly from Theorem 2.3.
3. q-Multi-summations 3.1.A q-difference system and the uniqueness of solutions.Recall that in Theorem 1.1 we have shown that Let us focus on .
If we further write F k (x) := F ⋆ k (xq −S ) for each k, then the column vector satisfies the q-difference system Recall that, we have defined in Theorem 1.1 that, for each 1 ≤ k ≤ K, I k denotes the subset of partitions in I ( Π, L , S) whose S-tail is π k .Further, G k (x) is the generating function of I k .Since A is a (0, 1)-matrix, it follows that More importantly, since the empty partition ∅ is contained in I 1 but not in I k for 2 ≤ k ≤ K, we have G 1 (0) = 1 and G k (0) = 0 for 2 ≤ k ≤ K. Since the entries in the first column of A are all 1, it follows that We next show the uniqueness of solutions of (3.3).
Proposition 3.1.In the q-difference system (3.3),we assume that, for each , then there exists a solution to (3.3).Further, the solution is uniquely determined by F(0).
Proof.For each 1 ≤ k ≤ K, let us write ] for n ≥ 0. We also write for notational convenience that Recall that ♯(π 1 ) = |π 1 | = 0 and A k,1 = 1 for all k.We have that, for n ≥ 0, Setting n = 0 gives the requirement

Two examples.
Recall that, for each 1 ≤ k ≤ K, I k denotes the subset of partitions in I ( Π, L , S) whose S-tail is π k .Further, 3.2.1.Example 1.In the first example, we consider "partitions with difference at least 2 at distance 1."This partition set obviously corresponds to the Rogers-Ramanujan identities.In Example 1.1, we have shown that it is a span one linked partition ideal I ( Π, L , S) where Π = {π 1 , π 2 , π 3 } with π 1 = ∅, π 2 = 1 and π 3 = 2, the linking sets are Notice that the generating function for partitions with difference at least 2 at distance 1 is and that the generating function for partitions with difference at least 2 at distance 1 with the smallest part ≥ 2 is We know from (3.4) that Hence, by (3.7) and (3.8), if we put and then we have the following relation from (3.3): Conversely, if we are able to prove (3.11) directly (notice that F 1 (0) = F 2 (0) = F 3 (0) = 1), then by Remark 3.1 and Proposition 3.1, we can compute that Also, (3.7) and (3.8) can be deduced with no difficulty.

Example 2.
In the second example, we consider "partitions with difference at least 3 at distance 2 such that if two consecutive parts differ by at most 1, then their sum is divisible by 3." This partition set corresponds to the Kanade-Russell conjectures I 1 -I 3 .It was shown in [8] that this partition set is a span one linked partition ideal I ( Π, L , S) where S = 3, and Π = {π 1 , π 2 , . . ., π 7 } along with the linking sets are given as follows.

3.3.
A matrix factorization problem.Motivated by (3.11) and (3.18), we turn our interest to a matrix factorization problem as follows.
Let R be a positive integer.Let α = (α i,j ) ∈ Mat R×R (N) be a fixed symmetric matrix.Let A = (A r ) ∈ N R >0 and γ = (γ r ) ∈ N R >0 be fixed.Let F be a set of q-multi-summations defined by where and the additional condition reads: for all (n 1 , . . .
Now we consider a column functional vector where H(β k ) ∈ F for all 1 ≤ k ≤ K.We expect F β (x) to satisfy the following factorization property.
Factorization Property.Let U be a (0, 1)-matrix such that all entries in the first row and column are 1.Let V be a diagonal matrix such that all (diagonal) entries are monic monomials in x and q with V 1,1 = 1.We say that F β (x) satisfies the Factorization Property if for some positive integer S.
Example 3.2.In the example in §3.2.2, we have .

Non-computer-assisted proofs
In [8], Li and the author provided an algebraic method to prove Andrews-Gordon type generating function identities such as (3.12), (3.13) and (3.14).However, one defect in that work is that the proofs rely heavily on computer assistance.Our aim here is to overcome this problem.
Recall that the Factorization Property says that Perhaps, if we expect to apply Theorem 4.1 to deduce Andrews-Gordon type generating function identities, we need to attach some additional conditions to the Factorization Property.
Additional Conditions.For all 1 ≤ s ≤ R: (3.11).We first prove (3.11), which is relatively easy. and We have shown in Example 3.1 that in this case S = 2, α = 2 , γ = (1), A = (1) and Further, it follows from (4.2) that and To prove (4.5), it suffices to show that and It follows from Theorem 4.1 that Also, Identities (4.8) and (4.9) are therefore proved.
x γr q βr Now the proofs of (3.11) and (3.18) can be illustrated by Figs. 3 and 4, respectively.x 2 q 3 In fact, it is relatively easy to deduce other much more complicated identities of the same flavor as (3.11) and (3.18).For example, the next result follows from the binary tree in Fig. 5.

Closing remarks
Our main concern is about the Factorization Property.Recall that U is a (0, 1)matrix such that all entries in the first row and column are 1, and V is a diagonal matrix such that all (diagonal) entries are monic monomials in x and q with V 1,1 = 1.The Factorization Property says that where S is a positive integer and q R r=1 αr,rnr (nr−1)/2 q 1≤i<j≤R αi,j ninj q R r=1 βrnr x R r=1 γrnr (q A1 ; q A1 ) n1 • • • (q AR ; q AR ) nR .
Probably we also require the Additional Conditions: for all 1 ≤ s ≤ R: (i).γ s S ∈ A s Z; (ii). for all 1 ≤ r ≤ R, α r,s ∈ A s Z.
Problem 5.1.For given U and V , is it possible to determine if there exist F β (x) and S such that (5.1) is true?
We have another problem from a different direction.
Problem 5.2.Are there any criteria of F β (x) that we are always able to find U , V and S such that (5.1) is true?
The last problem is perhaps simpler.x 2 q 6 x 3 q 10 xq 2 1 xq 3 1 xq 4 x 2 q 4 1 xq 4 x 3 q 7 x 2 q 6 x 3 q 10 x 2 q 2 1 xq 3 1 xq 4 x 3 q 4 If we are able to find such construction, then we may derive a family of span one linked partition ideals (or at least a family of modified directed graphs) with nice analytic generation functions.

2. 3 .
Proof of Theorem 1.1.To prove Theorem 1.1, let us define the associated directed graph of a span one linked partition ideal I = I ( Π, L , S).

Figure 1 .
Figure 1.The associated directed graph in Example 2.1