An Arithmetic Property of Moments of the β-Hermite Ensemble and Certain Map Enumerators

Moments of the β-Hermite ensemble are known to be related to the enumerative theory of topological maps. When β ∈ {1, 2}, asymptotic information about these moments has been used to deduce asymptotics on the number of maps of given genus, and arithmetic information about these moments can sometimes be explained by underlying group actions on the set of maps. In this paper we establish a new arithmetic property about the 2q-th moment of the β-Hermite ensemble, for any prime q > 3 and real number β > 0, that has a combinatorial interpretation in terms of maps but no known combinatorial explanation. In the process, we derive several additional results that might be of independent interest, including a general integrality statement and an efficient algorithm for evaluating expectations of multipart elementary symmetric polynomials of bounded length.


Introduction
Since the work of Tutte [28,29] over fifty years ago, the study of map enumeration has been a popular area of research among combinatorialists.One basic question in this field asks for the number of maps with V vertices and F faces that exist on a given surface of genus g, for fixed integers V , F , and g.In most cases, closed form expressions for these numbers are unavailable, so researchers have instead asked for properties that these numbers satisfy.
For instance, questions asking for asymptotic properties of these numbers date back at least to 1986 with the work of Bender and Canfield [3].Since then, there has been significant effort in trying to find more asymptotic results in map enumeration; see, for instance, [4,7,14,15,27,31].Earlier work in this area, such as [14], established asymptotics in entirely combinatorial ways.However, many more recent works, such as [7,15,27,31], take advantage of a connection between random matrices and map enumeration; by studying the asymptotics of random matrix integrals, one can sometimes obtain asymptotics on map enumerators.
Instead of trying to find asymptotic properties of matrix integrals, our goal in this paper is find their arithmetic properties; these can then be interpreted as arithmetic properties of map enumerators.As we will see later (in Section 1.2), results of this type can sometimes be combinatorially understood through an underlying group action on the set of maps on a given surface.
Before discussing this further, let us begin by introducing some of the notions of random matrix theory.

Matrix Integrals
We first need some notation about partitions and symmetric functions; we will follow the conventions used by Macdonald in [24].A partition λ = (λ 1 , λ 2 , . . ., λ r ) is a finite, nondecreasing sequence of positive integers.The size of λ is equal to the sum r i=1 λ i and will be denoted by |λ|.The length of λ is r and will be denoted by (λ).For any positive integer i, let m i (λ) denote the number of indices j for which λ j = i; observe that (λ) = |λ| i=1 m i (λ).For any positive integer i r, let λ \ λ i denote the partition (λ 1 , λ 2 , . . ., λ i−1 , λ i+1 , . . ., λ r ) obtained from removing one occurence of λ i from λ.For any positive integer j, let (j, λ) denote the partition λ ∪ {j} obtained from adjoining j to λ.
Now, let N be a positive integer and X = {x 1 , x 2 , . . ., x N } be a set of N real variables.For each nonnegative integer k, let p k (X) = N i=1 x k i denote the kth power sum.Moreover, let e k (X) = S⊆X |S|=k s∈S s, denote the kth elementary symmetric polynomial, where the sum ranges over all unordered k-element subsets S ⊆ X.For any partition λ, let p λ (X) = (λ) i=1 p λ i (X) denote the multipart power sum and let e λ (X) = (λ) i=1 e λ i (X) denote the multi-part elementary symmetric polynomial.
In this paper we will be interested in expectations of multi-part power sums and multipart elementary symmetric polynomials against the β-Hermite ensemble (also called the Gaussian β Ensemble or β-Ensemble), which is a probability distribution that generalizes the joint eigenvalue distributions of particular Gaussian ensembles of random matrices.Let us define these random matrices, following the notation used by Mehta in [25].
the electronic journal of combinatorics 25(1) (2018), #P1.29 A random matrix is a matrix whose entries are random variables.Examples of such matrices are given by the Gaussian Orthogonal Ensemble (GOE), which is a collection of symmetric random matrices GOE n = {g i,j } 1 i,j n whose upper triangular entries {g i,j } 1 i<j n are independently distributed, normalized, real Gaussian random variables; whose lower triangular entries {g i,j } 1 j<i n are symmetrically determined by the upper triangular entries through the equality g i,j = g j,i ; and whose diagonal entries are independently distributed real Gaussian variables with variance 2. The Gaussian Unitary Ensemble (GUE) and Gaussian Symplectic Ensemble (GSE) are complex and quaternionic analogs of the GOE, respectively.
The joint eigenvalue distributions of the GOE, GUE, and GSE are part of a oneparameter family of probability measures P β = P β,N , which are defined by for β = 1, 2, 4, respectively, where is a normalizing function in β.
Although P β is defined above for β ∈ {1, 2, 4}, it is a probability distribution for all β > 0. This distribution is called the β-Hermite ensemble.Dense matrix models (such as the GOE, GUE, and GSE) are not known to exist for the β-Hermite ensemble if β / ∈ {1, 2, 4}.However, Dumitriu and Edelman [9] showed that P β has a tridiagonal matrix model for all positive β; we will discuss this further in Section 2.2.
For any function f (X) of N variables X = {x 1 , x 2 , . . ., x N }, let denote the β-expectation of f .For instance, if f (X) = p k (X) for some positive integer k, , where E denotes expectation.By symmetry, this is equal to 0 if k is odd.
An intriguing result of Harer and Zagier [20] states that p λ (X) 2 is a nonnegative integer polynomial in N , for all partitions λ.When λ = (2s) consists of one part, Harer and Zagier expressed p 2s (X) 2 explicitly through the identity which holds for all integers s 1 (see also [19] and [23] for combinatorial proofs of this fact).This identity implies the following arithmetic property about the integer polynomial p 2q (X) 2 when q 3 is a prime.Proposition 1.1.We have that p 2q (X) 2 ≡ 2N q+1 − 2N 2 mod q for any prime q 3. Proof.The equality (1.2) may be rewritten as (1.3) Now, 2q q /(q + 1) is the q-th Catalan number and thus is integral.Furthermore, q divides q j for all nonnegative integers j < q, unless j = 0, in which case q divides (q +1)!/(j +1)!.Therefore, the right side of (1.3) is divisible by q.Since q = 2 and p 2q (X) 2 is an integer polynomial in N , we have that The proposition now follows from the facts that 2), we find that p 4 (X) 2 = 2N 3 + N .Thus, Proposition 1.1 does not hold for q = 2.
Similar results for p λ (X) 1 were found by Goulden and Jackson in [18].They showed that p λ (X) 1 is a nonnegative integer polynomial in N , for all partitions λ.When λ = (2s) consists of one part, Goulden and Jackson established the identity for all positive integers s.The following result is analogous to Proposition 1.1 and can be verified similarly; we omit its proof.
Proposition 1.1 and Proposition 1.3 might initially seem arbitrary in this matrixtheoretic context, but in the next section we will discuss their combinatorial significance; we will show how these two results can be deduced in a different way, using combinatorial interpretations of the coefficients of the polynomials p 2q (X) 2 and p 2q (X) 1 as enumerators of topological maps.

Maps
In this section, we will describe how to establish Proposition 1.1 and Proposition 1.3 in a more combinatorial way, using the enumerative theory of topological maps.Let us begin by introducing some of the terminology in this field, following the notation given in Chapter 2.2 of [22].
Let G be a topological connected graph (possibly with multiple edges and loops), and let M be a surface.A map m : G → M is a two-cell embedding of G into M.The edges and vertices of m are the edges and vertices of G, respectively; the faces of m are the connected components of M \ m(G).The map m is orientable if the surface M is orientable and is otherwise non-orientable; following Goulden and Jackson [18], we call all maps locally orientable.We call m planar if M has genus 0.
A map m : G → M is equivalent to m if there is an orientation-preserving homeomorphism from M to M that induces a graph isomorphism from G to G .A map is rooted by distinguishing an edge e ∈ G and a flag f , which is a side-end position of e; each edge has two sides and two ends, so four flags.In what follows, all maps will be rooted.
In his original paper [28], Tutte was interested in the number n(F, E) of planar rooted maps, up to equivalence, with a specified number of faces F and edges E. The number of vertices V is determined by the equality V + F = E + 2 because the map is planar.More generally, we would have that where g is the Eulerian genus of M, which we will also refer to as the genus of the map m.In [29], Tutte combinatorially deduced recursive relations for n(F, E) and used them to solve for a generating function of the form ∞ F =0 ∞ E=0 n(F, E)x F y E .Remark 1.4.Euler's equality describing the standard genus g of an orientable map states that V + F = E + 2 − 2g .We use the Eulerian genus, which is twice the standard genus, in order to discuss the genera of orientable and non-orientable surfaces simultaneously.
For the remainder of this section, we will restrict ourselves to maps with one face (F = 1), although analogs of the following results are known for maps with multiple faces.For any nonnegative integers E and g, let k g (E) be the number of genus-g orientable rooted maps with one face and E edges, and let k v (E) denote the number of orientable rooted maps with one face, v vertices, and E edges.Observe that n(1, E) = k 0 (E); furthermore, (1.6) implies that k E+1−g (E) = k g (E).Similarly, let K g (E) denote the number of genusg locally orientable rooted maps with one face and E edges, and let K v (E) denote the number locally orientable rooted maps with one face, v vertices, and E edges.
Extending on work of 't Hooft [21] and Brézin-Itzykson-Parisi-Zuber [6], Harer and Zagier showed that a generating function for the orientable map enumerators k v (E) could be found from moments of P 2 [20].Specifically, they established that the electronic journal of combinatorics 25(1) (2018), #P1.29 for any positive integer E; we refer to [20] or Section 5 of the survey [31] for a proof.Now, Proposition 1.1 may be restated as follows.
In the previous section, we deduced this fact from (1.2), which follows from matrix theoretic methods.However, as stated, Proposition 1.5 also has a direct combinatorial proof.
Proof of Proposition 1.5.For each nonnegative integer g, k g (2q) counts the number of genus-g rooted maps with one face of degree 2q.This is equal to the number of gluings (that is, the number of ways to identify pairs of edges; see Section 5.1 of [31]) of a labelled regular 2q-gon P to form a surface of Eulerian genus g.Label the edges of P with the integers 1, 2, . . ., 2q in clockwise order; any gluing may be represented by a collection of q pairs {(a i , b i )} of integers between 1 and 2q, specifying which edges to identify.For instance, when q = 3, the collection {(1, 4), (2,5), (3,6)} glues opposite sides of the hexagon P to form a surface of Eulerian genus 2.
The cyclic group C q of order q acts on the set of genus-g gluings; a generator of this group acts on a gluing by rotating P clockwise by an angle of 2π/q around its center.Specifically, this generator sends any gluing identifying the pairs {(a i , b i )} i∈ [1,q] to the gluing identifying the pairs {a i + 2, b i + 2} i∈ [1,q] (here, the a i + 2 and b i + 2 are taken modulo 2q).Any gluing not fixed by the action of this group has an orbit of size q; therefore, the orbits of such gluings do not contribute to the residue of k g (2q) modulo q.
Remark 1.6.The proof of Proposition 1.5 uses the fact that the cyclic group C q acts on the set of genus-g gluings by rotations.Rotating a genus-g gluing is equivalent to re-rooting its corresponding genus-g map.Therefore, the above proof implicitly uses the fact that re-rooting a map does not change its genus or orientability.As we will mention in Section 1.3, this is not true in the general β setting.
The analog of (1.7) for locally orientable maps was established by Goulden and Jackson in [18].Specifically, they showed that for any positive integer E. Now, Proposition 1.3 can be rewritten as follows.
Similar to Proposition 1.5, Proposition 1.7 might be shown using the action of C q on the set of oriented gluings; we omit this proof.
Thus, while both Proposition 1.1 and Proposition 1.3 may initially appear as statements about matrix integrals, they can also be interpreted as arithmetic properties of the k v (2q) and K v (2q) that can be combinatorially explained through an underlying group action on the set of orientable or locally orientable genus-g topological maps; the nonzero residues in these statements result from the fixed points of this group action.In Section 1.3, we will present our main result, which is a generalization of both Proposition 1.1 and Proposition 1.3 that has also has a combinatorial interpretation but no known combinatorial explanation.

Results
The extension of (1.7) and (1.8) to the general β-Hermite ensemble was established by La Croix [22].He showed as Lemma 4.13 of Chapter 4 of [22] that moments of the β-Hermite ensemble satisfy the recursive relations for all integers j −1 (when j = −1, only the second term on the right side of (1.9) should be viewed as nonzero), positive reals β > 0, and partitions λ.In (1.9), b = 2/β − 1 denotes the shifted Jack parameter.Later, we will also use the notation α = 2/β = b + 1 to denote the Jack parameter.
Since p 0 (X) β = N , (1.9) and induction on |λ| imply that p λ (X) β is a polynomial in N and b with nonnegative integer coefficients, for any partition λ.Restricting to the case when λ = (2E) consists of one even part (again, analogs exist for (λ) > 1, but we will not discuss them here), this implies that there exist nonnegative integers K v,η (E) such that (1.10) Inserting b = 0 into (1.10) and applying (1.7) yields k v (E) = K v,0 (E); in this case, only orientable maps are counted in (1.10).Similarly, inserting b = 1 into (1.10) and applying (1.8) yields K v (E) = E η=0 K v,η (E); in this case, all maps are counted in (1.10).Therefore, it might be reasonable to guess that K v,η (E) enumerates the number of rooted maps with v vertices, one face, E edges, and some "degree of non-orientability" that increases with the integer η.This was conjectured implicitly by Harer, Goulden, and Jackson as Conjecture 3.5 of [17], and later in its above form in Section 5 of [16].
Through a suitable combinatorial interpretation of the recurrence in (1.9), LaCroix showed this conjecture to be true; the parameter η is known as a b-invariant.One may refer to Definition 4.1 of [22] for a precise definition of the b-invariant η.
Since the K v,η (λ) have a combinatorial interpretation as "partially orientable" map enumerators, one might expect the K v,η (2q) to have arithmetic properties similar to k v (2q) and K v (2q) for all odd primes q.The following result, which appears to be a new, simultaneous generalization of Proposition 1.5 (b = 0) and Proposition 1.7 (b = 1), shows this to be true.We will establish this theorem (in fact its equivalent Theorem 1.9) in Section 2.5.
Similar to Proposition 1.5 and Proposition 1.7, Theorem 1.8 suggests the existence of an underlying group action on the set of maps with one face of degree 2q, a fixed number of vertices, and a fixed b-invariant.As before, the nonzero residues appearing in Theorem 1.8 might then correspond to fixed points of this group action.However, Goulden and Jackson [17] (see also Theorem 3.30 of [22]) showed that the b-invariant η of a map depends on the map's rooting.Therefore, due to Remark 1.6, the cyclic group C q does not necessarily act on the set of maps with a fixed b-invariant.
This leads us to ask whether there is a combinatorial proof of Theorem 1.8, perhaps through a different group action or through a different combinatorial interpretation of the integers K v,η (λ).We remark that a different arithmetic property with no known combinatorial explanation (unless b ∈ {0, 1}, in which case it follows from re-rooting) was given by La Croix as Corollary 4.17 of [22].
Our proof of Theorem 1.8 is matrix theoretic.Due to (1.10), we can rewrite Theorem 1.8 as follows.
Theorem 1.9.For each prime number q 3, we have that Unlike the case β ∈ {1, 2}, there is no known analog of (1.2) and (1.5) that explicitly evaluates p 2s (X) β for general β > 0. Instead, we know of five ways to find exact expressions for p 2s (X) β .The first is by evaluating traces of powers of tridiagonal matrix models, given in Chapter 5.4 and Chapter 6.2 of [8] by Dumitriu; the second is through a change of basis from the Jack polynomials to power sums, given in Section 4 of [10] by Dumitriu, Edelman, and Shuman; the third is through the recursive equation (1.9), given in Chapter 4.4 of [22] by La Croix; and the fourth uses loop equations for the resolvent, given in Sections 2 and 3 of [12] by Forrester and Witte (see also the references therein).We do not know of a way to deduce arithmetic properties about p 2q (X) β through any of these four existing methods.
Therefore, we introduce the fifth method by expressing p 2q (X) β in terms of expectations e λ β of multi-part elementary symmetric polynomials; through an integrality result about these expectations (see Proposition 2.5), we will be able to establish Theorem 1.9 in Section 2.
In the process, we will derive several results about expectations of symmetric polynomials that might be of independent interest.The first is an integrality result (Corollary 2.15) about f (X) β that holds for arbitrary symmetric polynomials f with integer coefficients; as we will discuss in Remark 2.17, special cases of this result have appeared implicitly before, but not in full generality until now.The second is a recursion (Theorem 3.1) that explicitly evaluates expectations of multi-part elementary symmetric functions.This gives rise to an algorithm (Corollary 3.10) that finds e λ (X) β efficiently for partitions λ of small length; as we will discuss in Section 3, this complements the algorithm given by Dumitriu, Edelman, and Shuman in [10].

Proof of the Arithmetic Property
In this section, we will prove Theorem 1.9.Let us begin by outlining the proof.

Outline of Proof of Theorem 1.9
Instead of attempting to evaluate the power-sum expectation p 2q (X) β directly, we will first express the 2q-th power sum as a rational linear combination of multi-part elementary symmetric functions through the Newton-Girard identity.This yields where the c λ are explicit rational numbers, and we simplify notation by using f (X) in place of f (X) β for any function f of N variables.The prefactor of 2q in (2.1) might initially appear useful for reduction modulo q, but there seem to be two issues.First, the coefficients c λ are rational, and their denominators might be divisible by q.Thus, 2qc λ might not be divisible by q, so there might be many summands on the right side of (2.1) that are nonzero modulo q.For instance, if λ = 2 q = (2, 2, . . ., 2) (with q twos), then c λ = −q −1 .
Second, as seen from the following proposition (originally due Aotomo [2]), e λ (X) is not always an integer polynomial in N and b.
Remark 2.2.Proposition 2.1 implies that e k (X) is independent of β.It can be viewed as a restatement of the fact that the expectation of the characteristic polynomial of a random matrix from the Gaussian β Ensemble is the Hermite polynomial, for each β ∈ R >0 .
Remark 2.3.In fact, Aomoto established a more general result than Proposition 2.1; he found the expectations of single-part elementary symmetric polynomials against the β-Jacobi ensemble, which exhibits a limit degeneration to the β-Hermite ensemble.
The first issue can be quickly resolved through the following lemma, which we will establish in Section 2. 5.In what follows, 1 2q denotes the partition (1, 1, . . ., 1) (with 2q ones).
Lemma 2.4.For any partition λ / ∈ {1 2q , 2 q } of size 2q, there exists an integer k λ , not divisible by q, such that k λ c λ ∈ Z; equivalently, the denominator of c λ is not divisible by Lemma 2.4 shows that all but two of the coefficients c λ are well-behaved with respect to q; it also evaluates the other two coefficients explicitly.Using Lemma 2.4, we may rewrite (2.1) as where the c λ are rational numbers whose numerators are multiples of q.The polynomials e 1 (X) 2q and e 2 (X) q will be addressed separately in Section 2.4; the coefficients of all other polynomials on the right side of (2.2) are divisible by q.
Now, let us turn to the second issue.Although e λ (X) is not always an integer polynomial, Proposition 2.1 indicates that e k (X) is in the Z[b]-span of the binomal coefficients N 0 , N 1 , . ... The following lemma shows that this is also true for β-expectations of arbitrary multi-part elementary symmetric polynomials.Lemma 2.5.For each partition λ, there exist integer polynomials a λ,0 , a λ,1 , . . ., a λ,|λ| ∈ 6. Lemma 2.5 does not seem to be immediate from integrating e λ (X) against (1.1).In Section 2.3, we will present a combinatorial proof of this result using a tridiagonal matrix model.In view of Lemma 2.5, we can rewrite (2.2) as Since the index i in (2.3) runs from 0 to 2q, the issue remains that q appears in the denominators of many summands on the right side of (2.3).Moreover, the fact that i can equal 2q in (2.3) seems to suggest that p 2q (X) is of degree 2q in N .However, we will see in Section 2.5 that it is in fact a polynomial of degree q + 1.Thus, there is cancellation among the high-degree terms in (2.3).Specifically, we may rewrite (2.3) as where now the index i does not exceed q + 1.Furthermore, we will see from a result of Forrester and Witte (which will be more precisely stated in Section 2.5; see Proposition 2.24) that the degree-(q + 1) and degree-q terms on the right side of (2.4) are explicit.Therefore, (2.4) may be rewritten as for explicit constants s q and t q .Now, the first two terms on the right side of (2.5) are integer polynomials that can be evaluated modulo q directly, since s q and t q are explicit.The third term (which contributes the most summands to (2.5)) is zero modulo q since q divides c λ and does not divide the denominator of N i for any integer i < q.The remaining two sums in (2.5) will be evaluated modulo q through an examination of the polynomials e 1 (X) 2q and e 2 (X) q in Section 2.4.Thus, the right side of (2.5) can be evaluated modulo q termwise, from which we will be able to deduce Theorem 1.9.
The organization for the remainder of this section is as follows.In Section 2.2, we will introduce the tridiagonal matrix model for the β-Hermite ensemble.In Section 2.3, we will use this matrix model to establish Lemma 2.5.In Section 2.4, we will understand the modulo q properties of e 1 (X) 2q and e 2 (X) q .In Section 2.5, we will prove Lemma 2.4 and Theorem 1.9.

The Tridiagonal Matrix Model
Our proof of Lemma 2.5 will use the tridiagonal matrix model for the β-Hermite ensemble, introduced by Dumitriu and Edelman in [9].In this section, we will describe this matrix model.
The tridiagonal matrix models T 3 and T 4 are depicted above.
For any real number β > 0 and positive integer N , let T N = T N,β denote the N × N tridiagonal symmetric random matrix whose (i, j) entry t i,j is distributed as follows.If |i − j| 2, then t i,j = 0; if i = j, then t i,j = G i 2/β, where G i denotes a normalized, real Gaussian random variable; if j = i + 1, then t i,j = X iβ / √ β, where X iβ denotes a χ iβ distributed random variable; and if i = j + 1, then t i,j is determined through the symmetric equality t i,j = t j,i .Here, the upper diagonal entries are mutually independent.Examples of T N for N ∈ {3, 4} are shown in Figure 2.
The following result of Dumitriu and Edelman appears as Theorem 2.12 in [9].
Proposition 2.7 ([9, Theorem 2.12]).For any positive integer N , the joint eigenvalue distribution of T N is P β,N .
Remark 2.8.The tridiagonal model introduced above is slightly different from the tridiagonal model given by Dumitriu and Edelman.Specifically, they use the tridiagonal model T N β/2 (meaning that all entries in T N are multiplied by β/2).This is due to a difference in normalization; they define a β-Hermite ensemble P β through where c (β) = (2/β) n(βn−β+2)/4 c(β).Their normalization is useful for understanding how the β-Hermite distribution behaves as β tends to ∞; however, our normalization is useful for understanding the connection between the β-Hermite ensemble and map enumeration (see Section 1.2 and Section 1.3).Observe that our normalization (1.1) is related to the normalization (2.6) through the identity P β (X) = P β (X 2/β), where Xc = {cx 1 , cx 2 , . . ., cx N } for any c ∈ R.
For any nonnegative integer k and N × N matrix M, let P k (M) denote the sum of the N k principal minors of M. The following result shows how to obtain β-expectations of multi-part elementary symmetric polynomials from T N .Corollary 2.9.Let N be a positive integer, and let λ = (λ 1 , λ 2 , . . ., λ r ) be a partition.Then, e λ (X) = E r i=1 P λ i (T N ) .
When M is a tridiagonal matrix, we can evaluate P k (M) explicitly in terms of the entries of M. We begin with the following recursion.
Proposition 2.10.Let u 1 , u 2 , . . ., u N −1 ; v 1 , v 2 , . . ., v N ; and w 1 , w 2 , . . ., w N −1 be real numbers.Suppose that M is N × N tridiagonal matrix whose (i, j) entry is and is 0 otherwise.Let M denote the (N − 1) × (N − 1) matrix obtained from removing the bottommost row and rightmost colum of M, and let M denote the (N − 2) × (N − 2) matrix obtained from removing the bottommost row and rightmost column of M .For each integer k, we have that Proof.Any principal k × k minor of M either contains the entry v N (in which case we call it a type 1 minor) or does not contain the entry v N .The sum of all type 1 minors is v N P k−1 (M ).Now, any k × k principal minor that is not type 1 is either contained in M (in which case we say that it is a type 2 minor) or contains the entry u N −1 (in which case we say it is a type 3 minor).The sum of all type 2 minors is P k (M ).Any nonzero type 3 minor also contains the entry w N −1 , so the sum of all type 3 minors is Summing over all types of minors yields the proposition.
The following corollary now expresses P k (M) using the entries of M. Corollary 2.11.Suppose that the conditions of Proposition 2.10 hold.For any nonnegative integer k, we have that where the sum is over all subsets and such that the following holds.For any i ∈ S 2 , we have that i, i + 1 / ∈ S 1 and, for any i ∈ S 1 , we have that i − 1, i / ∈ S 2 .
Proof.One can verify the cases k ∈ {0, 1} and N ∈ {1, 2} directly.Now, let the right side of (2.7) be P k (M).Then, the P k satisfy the recursion The corollary now follows by induction on k and Proposition 2.10.

Proof of Lemma 2.5
In this section, we will use Corolary 2.9 and Corollary 2.11, to prove Lemma 2.5.However, it might be useful to first give an alternate proof of Proposition 2.1, in order to indicate the types of methods that we will use.As mentioned in Remark 2.3, our proof of Proposition 2.1 will complement the one given by Aomoto in [2].Instead of being analytic, it will be combinatorial, along the lines of the one given by Ullah [30] in the case β = 1.

Proposition 2.1 ([2]
).If k is a nonnegative even integer, then Proof.For any positive integer N and nonnegative integer k, let E P k (T N ) = P k (N ); observe that P k (N ) = e k (X) by Corollary 2.9.Due to Proposition 2.10, we have that where G N is a normalized, real Gaussian random variable that is independent from all entries in T N −1 , and X (N −1)β is a χ (N −1)β random variable that is independent from all entries in T N −2 .Taking expectations, we obtain that Proof.By Corollary 2.11, we have that where the sum is over all subsets S 1,i ⊂ [1, N ] and k and such that the following holds.For any j ∈ S 2,i , we have that j, j + 1 / ∈ S 1,i and, for any j ∈ S 1,i , we have that j − 1, j / ∈ S 2,i .Here, the G j are normalized, real Gaussian random variables, and the X jβ are χ jβ distributed random variables.
Taking expectations, applying Corollary 2.9, and recalling that α = 2/β, we deduce that where the sums are as above.Expanding the product yields that e λ (X) is equal to where the S 1,i and S 2,i are summed as above; n 1,j denotes the number of integers i ∈ [1, r] for which j ∈ S 1,i ; and n 2,j denotes the number of integers i ∈ [1, r] for which j ∈ S 2,i .
If |λ| is odd, then e λ (X) = 0 due to symmetry of (1.1).If |λ| is even, then Therefore, the exponent of α in (2.8) is integral.Furthermore, for all nonnegative integers n 1,j and n 2,j , we have that Thus, (2.8) is the sum of polynomials in α with integral coefficients, so e λ (X) Using Proposition 2.12, we may now establish Lemma 2.5.
Lemma 2.5.For each partition λ, there exist integer polynomials a λ,0 , a λ,1 , . . ., a λ,|λ| ∈ Z[b] such that e λ (X) = |λ| i=0 a λ,i N i .Proof.By (1.9) and induction on |λ|, we find that p λ (X) ∈ Q[N, b] is a rational polynomial whose degree in N is at most equal to |λ|.Since the multi-part elementary symmetric polynomials are in the Q-span of the multi-part power sums, we deduce that e λ (X) ∈ Q[N, b] is a rational polynomial whose degree in N at most equal to |λ|.Thus, there are rational polynomials From Proposition 2.12, R λ,i (N ) ∈ Z for each fixed positive integer N , partition λ, and integer i ∈ [0, d λ ].Therefore, the R λ,i are integer linear combinations of the binomials N 0 , N 1 , . ... Applying this to the equality e λ (X) = d λ i=0 b i R λ,i (N ) and using the fact that e λ (X) is of degree at most |λ| in N , we deduce the lemma.
the electronic journal of combinatorics 25(1) (2018), #P1.29 Remark 2.13.In the proof of Lemma 2.5, we used the fact that p λ (X) ∈ Q[N, b], which was shown by LaCroix through analytic methods (in fact, he shows the stronger statement that p λ (X) ∈ Z[N, b]).It is also possible to give a combinatorial proof of this fact using the tridiagonal matrix model reviewed in Section 2.2, but we will not pursue this here.
Remark 2.14.Lemma 2.5 does not explicitly evaluate e λ (X) .We will investigate the question of how to explicitly evaluate this expectation in Section 3.
The following corollary of Lemma 2.5 will not be used but may be of independent interest.Corollary 2.15.For any symmetric polynomial f with integer coefficients, there are integers d f and h Proof.This follows from Lemma 2.5 and the fact that f is in the Z-span of the multi-part elementary symmetric polynomials e λ .
Remark 2.16.One can show that d f deg f /2 using the fact that p λ (X) is of degree at most deg f /2 in b (which follows by (1.9) and induction on |λ|).
Remark 2.17.Corollary 2.15 may be of independent interest, since it has implicitly appeared previously in multiple articles in random matrix theory but has not been stated or proven in full generality until now.For instance, (1.10) (and thus its special cases (1.7) and (1.8)) exhibit the phenomenon stated in Corollary 2.15.Furthermore, in response to Conjecture 3.4 of Goulden and Jackson in [18], Okounkov [26] (and later Dumitriu in Theorem 8.5.1 in [8]) explicitly evaluated the β-expectation of the Jack polynomial J (2/β) (X); one may also verify that this expectation satisfies the statement of Corollary 2.15.
Remark 2.18.The proofs of the facts mentioned in Remark 2.17 are analytic [8,22,26].However, the proof of Corollary 2.15 is mainly elementary and combinatorial, as it used the tridiagonal matrix model for the β-Hermite ensemble instead of explicit integration against the distribution (1.1).

2.4
The Polynomials e 1 (X) 2q and e 2 (X) q In this section, we will discuss the modulo q properties of the polynomials e 1 (X) 2q and e 2 (X) q .Let us begin with the following result that evaluates p 1 (X) 2s and p 2 (X) s explicitly.
Remark 2.20.An alternative proof of (2.9) can be obtained by changing variables from x i to x i + t in (1.1), integrating over all x i , and differentiating with respect to t.Similarly, an alternative proof of (2.10) can be deduced from changing variables from x i to tx i in (1.1), and then integrating and differentiating.The author is grateful to Peter Forrester for mentioning this.
Next, we transfer from power sums to elementary symmetric polynomials to understand the polynomials e 1 (X) 2q and e 2 (X) q modulo q.
Remark 2.22.In the proof of Corollary 2.21, we explicitly evaluated e 1 (X) 2q , but did not explicitly evaluate e 2 (X) q .Instead, we found the residue of this term modulo q through the expectation p 2 (X) q of a multi-part power sum.This leads to the question of whether we can find e 2 (X) q explicitly; Proposition 3.5 in Section 3 will lead to a partial answer.
We can now establish the following lemma.
Lemma 2.23.For any prime number q 3, we have that where the a λ,i are the polynomials defined by Lemma 2.5.
Proof.From Corollary 2.21, we obtain Therefore, we deduce that Now the lemma follows from the fact that the electronic journal of combinatorics 25(1) (2018), #P1.29 2.5 Proof of Theorem 1.9 In this section, we will establish Theorem 1.9.However, we first require a proof of Lemma 2.4.Throughout, we will follow the notation used in Section 2.1.
Next, we will need the following proposition of Forrester and Witte [12], which states that p 2q (X) is a polynomial of degree q + 1 whose high-degree coefficients are explicit.Proposition 2.24 ([12, Theorem 3]).For each integer s 1, we have that where r is a polynomial of degree s − 1 in N .
Remark 2.25.Forrester and Witte prove Proposition 2.24 as Theorem 3 in [12] by finding loop equations for the resolvent of the β-Hermite ensemble; in fact, they find the first six leading order terms of p 2s (X) .Though interesting, their methods seem unrelated to ours; therefore, we refer the reader to the original paper [12] for the proof of this proposition.Now, we may establish Theorem 1.9.
Theorem 1.9.For each prime number q 3, we have that Proof.Due to the fact that is a polynomial of degree at most s − 1 in N , Propostion 2.24 states that there exists a polynomial r(N, b) of degree q − 1 in N such that Thus, (2.3) may be rewritten as Now let us clear denominators in (2.14).Let K = (q − 1)! λ k λ , where λ ranges over all partitions, not equal to 1 2q or 2 q , of size 2q, and the k λ are defined by Lemma 2.4.Let K = 2 q−1 K; then, q does not divide K. Multiplying (2.14) by K, we obtain that The first and second terms in (2.15) can be evaluated directly in terms of K. Indeed, from (1.4), we deduce that (2.16) Moreover, since we have that (2.17) The third term on the right side of (2.15) can be evaluated in terms of K using Lemma 2.23 since 2 q−1 ≡ 1 mod q, and the fourth term is zero modulo q due to Lemma 2.4.Thus, we obtain that Since 2 q−1 ≡ 1 mod q, we have that K ≡ K mod q.Thus, the theorem follows after multiplying (2.18) by K −1 mod q.

Expectations of Elementary Symmetric Polynomials
Several times in the proof of Theorem 1.9, we asked the question of how to explicitly evaluate the β-expectation e λ (X) of a multi-part elementary symmetric polynomial (see, for instance, Proposition 2.1, Remark 2.14, and Remark 2.22).In this section, we will show one way of doing this through a recurrence that parallels the power-sum recursion (1.9).

Historical Context
In general, the question of how to explicitly evaluate the expectations of symmetric polynomials against the β-Hermite ensemble has been one of both probabilistc and combinatorial interest since the mid 1980s.Let us review some of the known results in this direction.
In 1986, Harer and Zagier [20] found an explicit form for the 2-expectation p 2k (X) 2 of a single-part power sum and gave a combinatorial interpretation to the 2-expectation p λ (X) 2 of arbitrary multi-part power sums.The analogs of these results for β = 1 the electronic journal of combinatorics 25(1) (2018), #P1.29 were found by Goulden and Jackson [18] in 1997.In connection with map enumeration, Goulden and Jackson conjectured an explicit form for the β-expectation of the Jack polynomial J (2/β) (X) in 1997 [18].This conjecture was first proven in [26] by Okounkov and then in a different way in Chapter 8.5 of [8] by Dumitriu.The extension of Harer and Zagier's interpretation to the general β-Hermite ensemble was established by La Croix [22] in 2009.Moments of the β-Hermite ensemble have also been understood through loop equations for the resolvent, studied recently by Forrester and Witte in [12] (see also references therein and [13] for related work).
Also in 1986, Ullah [30] found a closed form for the 1-expectation e k (X) 1 of a singlepart elementary symmetric polynomial (this now follows as a special case of Proposition 2.1).Aomoto later found a different, more analytic, proof of Proposition 2.1 that generalizes to yield expectations of one-part elementary symmetric polyomials against the β-Jacobi ensemble [2].In Chapter 17.8 of his book [25], Mehta uses Aomoto's method to explicitly find the β-expectations m λ (X) β of monomial symmetric polynomials m λ (X), in which all parts of the partition λ are less than or equal to 3.These statements may be extended to more general partitions λ but with more complex results as the parts of λ increase.
Relatedly, Andrews, Goulden, and Jackson explicitly evaluated the expectations of moments of determinants for the GOE and GUE in 2003 [1].Although an explicit form for the expectation e N (X) k β of the k-th moment of the determinant of the general β-Hermite ensemble is not currently known, Dumitriu found the second moment explicitly and found recursive equations for the third and fourth moments (see Chapter 8.5 of [8]).Recently, Edelman and La Croix [11] used a singular value decomposition to find the law of the absolute value of the determinant of the GUE; the analog for the GOE was later found by Bornemann and La Croix in [5].
Thus, there has been a significant effort over the past 30 years in explicitly evaluating f (X) β , for various symmetric functions f .We will be interested in the case when f = e λ is a multi-part elementary symmetric polynomial.In Section 3.2, we will present a recurrence relation that evaluates e λ (X) β explicitly; we will also show how it can be applied in several cases.As a related application, we will show in Section 3.3 how this recursion can be used to give an algorithm that finds e λ (X) β efficiently for all partitions λ of small length.

The Recursion
The recursion for expectations of elementary symmetric polynomials may be stated as follows.
We will give the proof of Theorem 3.1 in Section 3.4, but let us first show how it can be used to explicitly evaluate e λ (X) β for several classes of partitions λ; some of our results are summarized in the table in Figure 3.For instance, we can obtain another proof of Proposition 2.1.

Proposition 2.1 ([2]
).If k is a nonnegative even integer, then Proof.Observe that e 0 (X) β = 1 since P β is a probability measure.Applying Theorem Proof.If k = 0, then the result follows from Proposition2.1, so suppose that k 2. Inserting the single-part partition λ = (2) into Theorem 3.1 yields The proposition now follows by induction on k.
Remark 3.6.The first statement of Proposition 3.5 coincides with (2.9) since α = b + 1.The second statement of Proposition 3.5 allows one to evaluate e 2 (X) s β through an efficient recurrence, thereby partially answering the question asked in Remark 2.22.We do not know of a compact, explicit expression for e 2 (X) s β , but Proposition 3.5 might be sufficient to provide an alternative derivation of the arithmetic statement Corollary 2.21.Proposition 3.7.For any partition λ, e λ (X) Proof.We have that e 0 (X) = 1 and e 1 (X) = 0; therefore, the statement holds when |λ| ∈ {0, 1}.Now the proposition follows from Theorem 3.1 and induction on |λ|.Remark 3.8.Observe that Proposition 3.7 also follows from Lemma 2.5.We do not know how to establish the latter result directly from Theorem 3.1.

Algorithm
Due to the interest and applications of the β-Hermte ensemble, Dumitriu, Edelman, and Shuman introduced an algorithm that evaluates f (X) β explicitly in terms of N and α = 2/β, for any symmetric polynomial f of N variables [10].Though effective, their algorithm involves a change-of-basis to the Jack symmetric polynomials, which is a highcomplexity process.For instance, in order to evaluate e 2k (X) β , this algorithm requires Expectations of Multi-part Elementary Symmetric Polynomials F (X) Figure 2: Some applications of Theorem 3.1 are listed in the above table.
One may observe that LaCroix's recursive relation (1.9) also yields an algorithm that finds β-expectations of multi-part power sums p λ (X) β explicitly in terms of N and b, for any partition λ.A change-of-basis (which, depending on f , might be a high-complexity process) would then let one evaluate f (X) β , for any symmetric polynomial f , as above.
However, one may verify that this algorithm also requires complexity exp Θ( √ k) in order to evaluate e 2k (X) β .
In general, we do not know if there is an algorithm that evaluates β-expectations of general multi-part elementary symmetric polynomials, power sums, or monomials efficiently.More specifically, it is unknown whether there exists an integer d and an algorithm that finds e λ (X) β (or p λ (X) β or m λ (X) β ) in O |λ| d space and time, for any partition λ.However, in this section, we will provide a partial result in this direction by presenting a algorithm A λ that finds e λ (X) β , as a polynomial in N and α, with complexity O |λ| d (d 2 + |λ|) for all partitions λ satisfying (λ) d − 1.
the electronic journal of combinatorics 25(1) (2018), #P1.29 3. Let i be the minimum integer such that e µ (i) has not been defined.Let µ where k is the largest part of µ (i) .
4. Define e k,µ through the equality 5. Repeat Steps 3 and 4 until e λ is defined; then output e λ , and stop.
By Remark 3.2, the recursion given by Theorem 3.1 expresses e λ (X) β as a linear combination of |λ| + 1 terms of the form e µ (X) β for µ ∈ S λ .Furthermore, since S µ ⊆ S λ for any partition µ ∈ S λ , we deduce that each e ν on the right side of (3.2) has been previously defined by A λ (due to the minimality of |(k, µ)|).Therefore, Theorem 3.1 ensures that A λ outputs e λ (X) β , since the e λ and the e λ (X) β satisfy the same recurrence relation with the same initial conditions.
It remains to analyze the complexity of A. Proof.This follows from Proposition 3.9 and the fact that |S λ | |λ| (λ) .Remark 3.11.Generally, (λ) is not bounded, which means that A does not provide a polynomial-time and polynomial-space algorithm that evaluates e λ (X) β .However, (λ) is bounded for some classes of partitions λ; in such cases, A is efficient.For instance, A is efficient when evaluating small moments of determinants of the β-Hermite ensemble.

Proof of Theorem 3.1
Our proof of Theorem 3.1 will be similar to Aomoto's proof [2] of Proposition 2.1.In particular, we will use his "de-symmetrizing" idea, which involves transferring between expressions of the form e k (X) and expressions of the form e j (X \ {x 1 }) .As in Section 2, we write f (X) in place of f (X) β for any symmetric polynomial f .
The following lemma, whose proof will be deferred Section 3.5, will faciliate this transferal.
Lemma 3.12.Let a b 0 be integers, λ be a partition, and X = (x 1 , x 2 , . . ., x N ) be a set of real variables.Then, Using this lemma, we can prove Theorem 3.1.Theorem 3.1.For any partition λ = (λ 1 , . . ., λ r ) and integer k max i∈ [1,r] λ i , we have that denote the density function of the β-Hermite ensemble (1.1).Observe that where the second equality holds because both limits are equal to 0 (due to the exponential term in Ω β (X)).The derivative on the left side of (3.3) is equal to the electronic journal of combinatorics 25(1) (2018), #P1.29 Integrating first over x 1 , then integrating over X \ {x 1 }, and applying (3.3) yields   Inserting this into (3.5)yields the theorem.

Proof of Lemma 3.12
Here, we will give a proof of Lemma 3.12.

Further Directions
In this paper we established an arithmetic property of moments of the β-Hermite ensemble, and interpreted this result combinatorially.However, instead of combinatorial, our methods were matrix theoretic, which led to us to come across several intermediate results that might be of independent interest to combinatorialists and random matrix theorists alike.
Still, our study was not exhaustive.We would like to conclude this discussion by listing several related directions that might be interesting to pursue.
1. Is there a combinatorial proof of Theorem 1.8? 2. Our methods seem specific to evaluating p 2q (X) β modulo q.What modulo q properties do p λ (X) β satisfy for other partitions λ?For instance, what happens when λ = (4q)?
3. The result (1.10) shows that the coefficients h (f ) i,j from Corollary 2.15 are nonnegative and have combinatorial intepretations when f is a power sum.For what other symmetric polynomials f are these coefficients nonnegative, and when do they have a combinatorial interpretation?

Proposition 3 . 9 .
The algorithm A λ runs in time O |λ||S λ | and space O (λ) 2 |S λ | .Proof.The fact that A λ requires space O (λ) 2 |S λ | is due to the fact that the data stored by A λ consists of the variable set {e µ } µ∈S λ , and each variable is defined by a polynomial in N and α of degree at most (λ) in each variable.Now let us turn to the run time of A. Since |µ| ∈ 0, |λ| , for each µ ∈ S λ , the run time of Step 1 is O |λ||S λ | .The run time of Step 2 is O |S λ | .The run time of Step 3 is O(1), and the run time of Step 4 is O |λ| , since there are |λ| + 1 terms on the right side of (3.2).Step 5 repeats Steps 3 and 4 O |S λ | times, so the run time of Step 5 is A is O |λ||S λ | .Summing over the steps of A, we deduce that the run time of A is O |λ||S λ | .Corollary 3.10.Suppose that λ is a partition satisfying (λ) d.Then, A evaluates e λ (X) β in space O d 2 |λ| d and time O |λ| d+1 .