A symmetric Bloch-Okounkov theorem

The algebra of so-called shifted symmetric functions on partitions has the property that for all elements a certain generating series, called the $q$-bracket, is a quasimodular form. More generally, if a graded algebra $A$ of functions on partitions has the property that the $q$-bracket of every element is a quasimodular form of the same weight, we call $A$ a quasimodular algebra. We introduce a new quasimodular algebra consisting of symmetric polynomials in the part sizes and multiplicities.


Introduction
Partitions of integers are related in interesting ways to modular forms, starting with the observation that the generating series of partitions is closely related to the Dedekind η-function, i.e. (1 − q n ) −1 = q 1/24 η(τ ) −1 (q = e 2πiτ ), where P denotes set of all partitions and |λ| denotes the integer λ is a partition of. Another example is the occurrence of modular forms in the proof of the partition congruences which go back to Ramanujan [AO01]. More recently, partitions were connected to (quasi)modular forms via the q-bracket. Given a function f : P → Q, the q-bracket of f is defined as the following power series (1) Before continuing, note that it is not surprising at all that for a well-chosen function f the qbracket f q is a quasimodular form, since it is easily seen that the map (1) from Q P to Q[[q]] is surjective. What is surprising is that one can find graded subalgebras A of Q P which (i) are 'interesting' in the sense that they have an interpretation in combinatorics, enumerative geometry or another field of mathematics and (ii) have the property that the q-bracket of a homogeneous function f ∈ A is quasimodular of the same weight as f . In this case we call A a quasimodular algebra. Note that the q-bracket is linear but not multiplicative, so in order to show that an algebra is quasimodular, it is not sufficient to show that the q-brackets of the generators of such an algebra are quasimodular. The aim of this paper is to introduce new quasimodular algebras. The Bloch-Okounkov theorem [BO00, Theorem 0.5] provided the first quasimodular algebra Λ * . Write a partition λ as a non-increasing sequence (λ 1 , λ 2 , . . .) of non-negative integers with |λ| = ∞ i=1 λ i finite. The Q-algebra Λ * is freely generated by the so-called shifted symmetric power sums where the c k are constants given by 1 x + k c k x k−1 (k−1)! = 1 2 sinh(x/2) . The function Q 3 naturally occurs in the simplest case of the Gromov-Witten theory of an elliptic curve, as discovered by Dijkgraaf [Dij95] and for which quasimodularity was proven rigorously in [KZ95]. Quasimodularity of Λ * is used in many recent works in enumerative geometry [CMZ18, BKY18, CMSZ20, AIL19, GM20]. There are many other functions in invariants of partitions which turn out to be elements of Λ * , for example symmetric polynomials in de modified Frobenius coordinates [Zag16,Eqn 19]; the hooklength moments [CMZ18,Thm. 13.5] (see § §7.1); central characters of the symmetric group [KO94,Prop. 3] and symmetric polynomials in the content vector of a partition [KO94,Proof of Thm. 4].
Previously, the Bloch-Okounkov algebra and some generalisations to higher levels (see e.g. [EO06,Eng17]), were the only known quasimodular algebras. However, there are many examples of functions on partitions admitting a quasimodular q-bracket (and in general not belonging to Λ * ) [Zag16, section 9], for example the Möller transformation of functions with quasimodular q-bracket (defined by [Zag16,Eqn 45] and recalled in Section 7), invariants A P for every even polynomial defined in terms of the arm-and leg-lengths of a partition and the moment functions that also occur in the study of so-called spin Hurwitz numbers in the algebra of supersymmetric polynomials [EOP08] (in that reference, these functions are only evaluated at strict partitionspartitions without repeated parts -and quasimodularity is shown for a correspondingly adapted qbracket).
In this paper we prove the stronger result that the algebra S generated by these moment functions S k is quasimodular. Moreover, besides the pointwise product of functions on partitions we define a second associative product , called the induced product as it is inherited from the product of power series. The vector space Sym (S) generated by the elements in S under the induced product is strictly bigger than S, is a quasimodular algebra for either of the two products, and has a particularly nice description in terms of functions T k,l depending not only on the parts of a partition, but also on their multiplicities. Here, the multiplicity r m (λ) of parts of size m in a partition λ is defined as the number of parts of λ of size m. More precisely, let F l be the Faulhaber polynomial of positive integer degree l, defined by F l (n) = n i=1 i l−1 for all n ∈ Z >0 . Then T k,l is given by T k,l (λ) = C k,l + ∞ m=1 m k F l (r m (λ)) (k ≥ 0, l ≥ 1, k + l even) with C k,l a constant equal to − B k+l 2(k+l) if k = 0 or l = 1 and 0 else. Let T be the algebra generated by all these T k,l under the pointwise product.
We show that Sym (S) and T are algebras for the pointwise product as well as for the induced product. In fact, the expression of elements of Sym (S) in terms of the T k,l implies that Sym (S) is a strict subalgebra of T (with respect to both products). Our main result is the following: Theorem 1.1. The algebras Sym (S) and T are quasimodular algebras, with respect to both the pointwise and the induced product.
A further main result of the paper is the following: Theorem 1.2. The q-bracket is an equivariant mapping T → M with respect to sl 2 -actions by derivations on both spaces.
A main theme throughout this paper is the principle to establish all identities in Q P or T before taking the q-bracket, instead of doing these computations in Q [[q]] or the space of quasimodular forms M . By doing so, we discover the algebraic structure of T . Without having the induced product at one's disposal, for example when studying the shifted symmetric algebra Λ * , this seems impossible. See the following table for an overview of situations where the principle is applied: formula for H p f q in [CMZ18, Eqn 152] 1 formula for T k,l f 6.2 Motivated by the fact that many functions in invariants of partitions are elements of Λ * , we describe many functions on partitions which are elements of T or are closely related. Among those are the border strip moments, generalizing the hook-length moments, which are defined in terms of the representation theory of the symmetric group. We show that the q-brackets of these functions are contained in the space of so-called combinatorial Eisenstein series, having the space of quasimodular forms as a subspace. By doing so, we hope that this work -besides advocating the notion of a 'quasimodular algebra' by giving a new example of such an algebra and studying its algebraic structure -may serve as a tool for enumerative geometers trying to show that generating series are quasimodular forms.
The contents of the paper are as follows. In Section 2 we recall notions (known to the experts) related to quasimodular forms, partitions and special families of polynomials. Next, in Section 3 we motivate all new notions in this work and prove quasimodularity of the algebra S. A study of the symmetric algebra T , including a proof of our main theorem can be found in Section 4. The sl 2 -action by differential operators, the proof of Theorem 1.2, and Rankin-Cohen brackets are the content of Section 5. In Section 6 further results that arise from comparing the two different products on T are given and finally in Section 7 we provide many examples of functions in or closely related to T .

Preliminaries
2.1. Quasimodular forms Let Hol 0 (H) be the ring of holomorphic functions ϕ of moderate growth on the complex upper half plane H, i.e. for all C > 0 one has ϕ(x + iy) = O(e Cy ) as y → ∞ and ϕ(x+iy) = O(e C/y ) as y → 0. A quasimodular form of weight k and depth at most p for SL 2 (Z) is a function ϕ ∈ Hol 0 (H) such that there exist ϕ 0 , . . . , ϕ p ∈ Hol 0 (H) so that for all τ ∈ H and all γ = a b c d ∈ SL 2 (Z) one has (5) Equation (5) is called the quasimodular transformation property. Note that if ϕ is a quasimodular form, the functions ϕ 0 , . . . , ϕ p are quasimodular forms uniquely determined by ϕ (the function ϕ r has weight k − 2r and depth ≤ p − r). For example, taking the identity I ∈ Γ yields ϕ 0 = ϕ.
for positive even integers k. For k > 2 the Eisenstein series are modular forms of weight k. The Eisenstein series G 2 is a quasimodular form of weight 2 and depth 1. Denote by M (≤p) k the vector space of quasimodular forms of weight k and depth at most p. Often we omit the depth and/or weight and simply write M k for the vector space of all quasimodular forms of weight k or M for the graded algebra of all quasimodular forms. Let M denote the graded algebra of modular forms. The quasimodular form G 2 generates the algebra of quasimodular forms as an algebra over the subalgebra of modular forms, that is, Often, when encountering an indexed collection of numbers or functions, we study its generating series. The generating series corresponding to the Eisenstein series, is called the propagator The propagator is closely related to the Weiserstrass ℘-function and Jacobi theta series 2.2. The action of sl 2 on quasimodular forms by derivations A way to produce examples of quasimodular forms is by taking derivatives of (quasi)modular forms under the differential opera- , given by In fact, every quasimodular form can uniquely be written as a linear combination of derivatives of modular forms and derivatives of G 2 . For more details, see [Zag08,. It may happen that a polynomial in the derivatives of two modular forms f ∈ M k and g ∈ M l is actually modular. This is the case for the Rankin-Cohen brackets of f and g, defined by That is, for all f ∈ M k , g ∈ M l and n ≥ 0, one has that [f, g] n is a modular form of weight k + l + 2n. Besides the differential operator D, an important differential operator on quasimodular forms is the operator d : M defined by ϕ → 2πiϕ 1 (with ϕ 1 defined in the quasimodular transformation property (5)). For example dG 2 = − 1 2 and in fact this property together with the fact that d annihilates modular forms defines d completely since d is a derivation and Let W be the weight operator, which multiplies a quasimodular form by its weight. Remark 2.2.2. By these commutation relations, for all n ≥ 1 one has which turns out to be useful later.
Following a suggestion of Zagier, we make the following definition: Definition 2.2.3. Given a Lie algebra g, a g-algebra is an algebra A together with a Lie homomorphism g → Der(A).
As D, d and W satisfy the Leibniz rule, the algebra M becomes an sl 2 -algebra.
Both P and Π(n) form a locally finite partially ordered set, i.e. a partially ordered set P for which for all x, z ∈ P there exists finitely many y ∈ P such that x ≤ y ≤ z. Namely, on P we define a partial order by κ ≤ λ if r m (κ) ≤ r m (λ) for all m ≥ 1. The ordering on Π(n) is given by α ≤ β if for all A ∈ α there exists a B ∈ β such that A ⊆ B. For instance, we have α ≤ 1 n for all α ∈ Π(n), where 1 n = {[n]}.
Recall that on a locally finite partially order set P the Möbius function µ : P 2 → Z is defined recursively by (see for example [Rot64]): µ(x, z) = − x≤y≤z µ(x, y) if x < z with initial conditions µ(x, x) = 1 and µ(x, z) = 0 else. For the above partial order on P the value of µ(κ, λ) depends on whether the difference of κ and λ considered as multisets, denoted by λ − κ, is a strict partition. That is, The Möbius function µ(α, β) of two elements α, β ∈ Π(n) is given by is the partition on B induced by α. A Möbius function satisfies the following two properties:  2.4. The connected q-bracket The q-bracket defined in the introduction (Eqn (1)) is a map Q P → Q [[q]]. In this section we define the connected q-bracket following [CMZ18, p. 55-57], which naturally arises in enumerative geometric when counting connected coverings. In our setting the connected q-bracket turns out to be easier to compute than the usual q-bracket.
Definition 2.4.1. Given an integer n ≥ 1, the connected q-bracket is defined as the multilinear map q : Q P ⊗ · · · ⊗ Q P n → Q extending the q-bracket such that for all f, f 1 , . . . , f n ∈ Q P any of the following equivalent conditions hold: (ii) f 1 ⊗ · · · ⊗ f n q = α∈Π(n) µ(α, 1 n ) A∈α a∈A f a ; (iii) f 1 ⊗ · · · ⊗ f n q is the coefficient of Note that by Definition 2.4.1(i) it follows that for n ≥ 2 the connected q-bracket of functions f 1 , . . . , f n vanishes if one of the f i is constant. Moreover, by invoking the Möbius inversion formula (Theorem 2.3.1(ii)) condition (ii) in Definition 2.4.1 implies that For example, We will use the third condition in Definition 2.4.1 in our proof that S is a quasimodular algebra.
2.5. The discrete convolution product and Faulhaber polynomials Let N denote the set of strictly positive integers. Given f, g : N → Q we denote by f · g or f g the pointwise product of f and g. We define the discrete convolution product of f and g by and denote the convolution product of functions f 1 , . . . , f n by Let the discrete derivative ∂ of f : N → Q be defined by ∂f (n) = f (n) − f (n − 1) for n ≥ 2 and ∂f (1) = f (1) and denote by id the identity function N → N ⊂ Q. Observe that id The Faulhaber polynomials F l for l ≥ 1 are defined as the unique polynomials with vanishing constant term satisfying ∂F l (n) = n l−1 for all n ∈ N, or equivalently by F l (n) = n i=1 i l−1 . The first four are given by Note that these polynomials are related to the Bernoulli polynomials B n (x), the unique family of polynomials satisfying which can also be deduced directly from the definition. The generating series F (n) of the Faulhaber polynomials equals 3. The moment functions, their q-bracket and a second product 3.1. Three proofs of the quasimodularity of the moment functions The q-bracket of the moment function S k defined in (3) equals the Eisenstein series G k . To motivative the rest of this work, we provide three different proofs-and three generalizations-of this statement using three different approaches. In the first approach we motivate the definition of the T k,l (Eqn (4)), the second approach gives an interpretation for these functions, and the last approach gives an example of our main principle of establishing all identities before taking the q-bracket.

First approach
The key observation in this first proof is that S k can be rewritten as More generally, for k > 0 and f : N → Q we set f (0) = 0 and we let In case when f is the identity, S k,f = S k+1 . Our first method of proof gives the following more general statement: Proposition 3.1.1. Let f be a polynomial of degree l without constant term and k a positive integer satisfying k ≡ l mod 2. Then, (ii) if S k,f q is a quasimodular form, then f is a multiple of the Faulhaber polynomial F l .
Substituting this result in the numerator of (18), we obtain Hence, Observe that applying x ∂ ∂x to the right-hand side of (19) has the same effect as applying 1 (16)). Hence, by taking l − 1 derivatives x ∂ ∂x = ∂ ∂z and setting z = 0, it follows that Part (ii) of the statement follows by writing f as a linear combination of Faulhaber polynomials.
Second approach The double moment functions T k,l (see (4)) are by definition equal to In this section we give a direct proof for the quasimodularity of the q-brackets of T k,l : Proposition 3.1.2. For all k ≥ 0, l ≥ 1 and k + l even one has The generating series of T 0 k,l is given by Let p(n) denote the number of partitions of n. The coefficient C a,b (n) equals the number of partitions of n with at least b parts of size a, i.e. C a,b (n) = p(n − ab). Hence, writing m = n − ab we obtain In other words, so that expanding this equation for X = e x and Y = e y yields Third approach In this last proof we start with the observation that one can rewrite the qbracket as In contrast to the previous two proofs, it is only in the last step of this proof that we take the qbracket: first we rewrite (21) considering u 1 , u 2 , . . . to be formal variables and in the last step we let u i = q i . We start with the denominator: Proposition 3.1.3. There exists a function µ : P → {−1, 0, 1} defined by any one of the following three equivalent definitions: (i) µ(λ) is given by the Möbius function µ(∅, λ) on the partial order on the set of partitions in (9); Proof. The first two definitions clearly coincide using (9). For the latter, it suffices to show that Let f (λ) = 1 and g(λ) = δ λ,∅ for λ ∈ P. Then f (α) = γ≤α g(γ) for all α ∈ P, so that by Möbius inversion and by using µ(γ, β) = µ(∅, β − γ) the last definition is equivalent.
The fact that S k q = G k follows directly from the following proposition: Proof. Fix m ≥ 1. By the previous proposition we have Denote by C(λ) the coefficient of u λ 1 u λ 2 · · · after expanding the right-hand side of above equation.
Observe that where α ∪ β denotes the union of α and β considered as multisets and it is understood that β is a strict partition. Suppose λ admits a part equal to m = m. Then, define an involution ω on all pairs (α, β) satisfying that α ∪ β = λ and β is strict by As ω changes the sign of (−1) (β) f (r m (α)), it follows that C(λ) = 0. Observe that C(∅) = 0 and that in case λ = (m, m, . . .) consists of a strictly positive number of parts all equal to m one has Therefore, the desired result follows.
3.2. The induced and connected product Motivated by the last of the three approaches in the previous section, we define the u-bracket of a function f ∈ Q P by Then, for all f ∈ Q P one has f q = f (q,q 2 ,q 3 ,...) . Observe that the u-bracket defines an isomorphism of vector spaces We now use the algebra structure of Q[[u 1 , u 2 , u 3 , . . .]] to define a product on Q P : Definition 3.2.1. Given f, g ∈ Q P we define their induced product f g by where the product of f u and g u is the usual product of power series.
Remark 3.2.2. Observe that Q P is a commutative algebra with the constant function 1 as the identity for both the pointwise and the induced product.
The following proposition gives an alternative definition for the induced product.
Proposition 3.2.3. For all λ ∈ P one has The result follows by expanding the products.
Analogous to the connected q-bracket, we define the connected product: . . , f n ∈ Q P , define the connected product f 1 | . . . |f n to be the following function P → Q: For example, for f, g, h ∈ Q P one has The induced and connected product allow us to establish many identities before taking the qbracket, as follows from the following result.
Proposition 3.2.5. For all f 1 , . . . , f n ∈ Q P one has • f 1 f 2 · · · f n q = f 1 q f 2 q · · · f n q ; Proof. Both statements follow directly from the definitions. For the first, note that for all f, g ∈ Q P one has so that the statement follows inductively. The second follows from the first, as Remark 3.2.6. Let R be the space of functions having a quasimodular form as q-bracket, i.e. R = · −1 q ( M ). Then, R is a graded algebra with multiplication given by the induced product. Namely, if f ∈ R and f q ∈ M k , we define the weight of f to be equal to k. Note that if f, g ∈ R and f q and g q are quasimodular forms of weight k and l respectively, then f g q = f q g q is a quasimodular form of weight k + l.
When establishing these identities, it turn out to be very useful to express the connected product of pointwise products of elements of Q P in terms of connected and induced products. This can be done recursively using the following result, where for a set S and f ∈ Q P we denote f S = s∈S f s .
Proof. Observe that both sides of the equation in the statement are a linear combination of terms of the form C∈γ f C over γ ∈ Π(n). We determine the coefficient of such a term on both sides of the equation. First of all, assume γ is such that {1, 2} ⊂ C for some C ∈ γ. Then on the right hand side such a term only occurs in f 1 | . . . | f n with coefficient µ(γ, 1). Moreover, letγ ∈ Π(n − 1) be given by γ ∩ {2, . . . , n} subject to replacing i by i − 1 for all i = 2, . . . , n. Note that the coefficient on the left-hand side equals µ(γ, 1). As (γ) = (γ), the coefficients on both sides agree.
Next, assume C 1 , C 2 ∈ γ with 1 ∈ C 1 and 2 ∈ C 2 . Then, the coefficient of C∈γ f C on right-hand side of (23) equals where the sum is over all I ⊂ {2, 3, . . . , (γ)} and A and B are given by A = C 1 ∪ i∈I C i and B = C 2 ∪ i∈I c C i . Letting i be the number of elements of I, we find that (24) equals Correspondingly, the coefficient of C∈γ f C on the left-hand side of (23) vanishes if there are C 1 , C 2 ∈ γ with 1 ∈ C 1 and 2 ∈ C 2 .
3.3. Quasimodularity of pointwise products of moment functions Not only do the moment functions S k admit quasimodular q-brackets, but also the homogeneous polynomials in the moment functions admit quasimodular q-brackets; here, each moment function S k has weight k in accordance with the fact that S k q has weight k. Given a tuple k = (k 1 , ..., k n ) of even integers, we write S k = S k 1 · · · S kn . Note that, as a vector space, S is spanned by these functions S k . We provide two approaches to proving the quasimodularity of the q-brackets of the S k . First, we give a direct proof of the statement in Theorem 3.3.1, after which, in accordance with our main principle of establishing all identities before taking the q-bracket, we prove a more general result which will be used frequently in the next section.
Theorem 3.3.1. The algebra S is a quasimodular algebra. More precisely, for k ∈ (2N) n one has Proof. Observe that it suffices to show that

The logarithm of this expression equals
Now, let n ≥ 2. In the expansion of (27) the coefficient of Hence, Note that, as n ≥ 2, both sides of the equation do not change if one replaces S 0 k by S k . In case n = 1 we have established (26) in Proposition 3.1.1 or in Proposition 3.1.2. Hence, (26) holds and (25) is then implied by Möbius inversion. Denoting and setting S 0 (λ) ≡ 1 one has the following expression for the generating series of the q-bracket of the generators of S: where z A = a∈A z a and is the totally even part of the propagator in (6).
3.4. Intermezzo: surjectivity of the q-bracket We deduce from Theorem 3.3.1 the surjectivity of the q-bracket: every quasimodular form is the q-bracket of some f ∈ S.
Note that this is not obvious since the q-bracket is not a homomorphism. Denote by ϑ k : M k → M k+2 the Serre derivative, given by ϑ k = D + 2kG 2 . Extend this notation by letting ϑ x : M → M for x ∈ Q be given by ϑ x = D + 2xG 2 .
Proof. Let f ∈ M k with f = 0. Observe that ϑ x f is modular precisely if k = x. By our assumption on x, this is not the case. Hence, ϑ x increases the depth strictly by one. The result follows by induction on p by the same argument as in [Zag08,Proposition 20]. Namely, if ϕ ∈ M ≤p k then the last coefficient ϕ p in the quasimodular transformation (5) is a modular form of weight k − 2p. Hence, ϕ is a linear combination of ϑ p x ϕ p and a quasimodular form of depth strictly smaller than p.
Proof of Theorem 3.4.1. First observe that (D + G 2 ) f q = S 2 f q . As D + G 2 is not a Serre derivative, by Proposition 3.4.2 it follows that it suffices to show that the q-bracket is surjective on modular forms. Every modular form can be written as a polynomial of degree at most 2 in Eisenstein series, see [Zag77, Section 5]. Hence, we show that the q-bracket is surjective on polynomials of degree at most 2 in all Eisenstein series, possibly involving the quasimodular Eisenstein series G 2 .
Eisenstein series are in the image of the q-bracket by Theorem 3.3.1. Note that DG k can be written a polynomial of degree 2 in Eisenstein series, explicitly: Also, we have an explicit formula for the q-bracket of S k S l : so that this q-bracket is expressible as a polynomial of degree at most 2 in the Eisenstein series. Now fix an integer m ≥ 4. We consider the equations (28) for all k + l = m. It suffices to show that we can invert these equations, i.e. write G k G l as a linear combination of q-brackets of products of at most two S i . A direct computation shows that the determinant of the matrix corresponding to the equations above equals Hence, the q-bracket is surjective.
Remark 3.4.3. Only the last step of above proof uses the explicit formula (28) for the derivative of Eisenstein series. The author expects one could conclude the proof by an abstract argument, but he is not aware of such an argument.
3.5. The connected product of moment functions In the second approach we compute the connected product S k 1 | . . . | S kn , which by Proposition 3.2.5 yields the left-hand side of (25) after taking the q-bracket. In order to do so, we start by introducing the following notation. For a partition λ and a subset A of N we write λ| A for the partition where a part of size m occurs r m (λ) times if m ∈ A and does not occur if m ∈ A. For example, (5, 4, 3, 3, 1, 1, 1)| {4,1} = (4, 1, 1, 1).
Definition 3.5.1. We say f : P → Q is supported on A if f (λ) = f (λ| A ) for all partitions λ.
The following lemma expresses the induced product of two functions F and G supported on disjoint sets as the pointwise product of these functions, and of two functions F and G supported on the same singleton set as a convolution product of functions.
Lemma 3.5.2. Suppose X and Y are subsets of N and F, F , G, G : P → Q are supported on X, X, Y and Y respectively. Then i) F F is supported on X; ii) If X and Y are disjoint, then where f and g are such that F (λ) = f (r m (λ)), G(λ) = g(r m (λ)).
Proof. By Proposition 3.2.3, we have where it is understood that γ is a strict partition. We have that Recall f 1 = f for all functions f , hence (F F )(λ) = (F F )(λ| X ), which is the first statement. Next, we have that where again it is understood that γ is a strict partition. Using the fact that F, F , G and G are supported on X, X, Y and Y respectively, we obtain where Z denotes the complement of X ∪ Y in N. We factor the right-hand side of (29) as By definition of the product , we conclude By taking F and G to be the constant function 1 (which is supported on every X and Y ), we see that F G = F G is implied by F G F G = (F F )(G G ).
Next, for (iii) we have Letting i = r m (α) and j = r m (β) we have The following result not only computes the connected product of the moments functions S k , but also is one of the main technical results needed to prove Theorem 1.1.
Theorem 3.5.3. Let k i , f i for i = 1, . . . , n be such that (17) defines S k i ,f i . Then (i) There exists a function g : N → Q such that S k 1 ,f 1 | . . . | S kn,fn = S |k|,g .
where f A = a∈A f a and * denotes the convolution product (10).
Remark 3.5.4. We extend g by g(0) = 0. Here and later in this work we usually omit the dependence of g on f 1 , . . . , f n in the notation.
Proof. For the first part, we let where r m i is considered as a function P → Q. In case n = 1 the result (i) is trivially true, so we assume n ≥ 2. By definition of the connected product and S k,f (see (22) and (17) respectively) we have Now, given m ∈ N n , Z = Z(m) and A ∈ α| Z , observe that the function ) is supported on N\{m 1 }. Hence, by Lemma 3.5.2(ii) we find that (31) equals Recall that α| Z = β| Z and α| Z c = β| Z c , so that the only dependence on α in the above equation is in µ(α, 1). We claim that if Z = [n] then for all β ∈ E(m) we have α∈[β] µ(α, 1) = 0. This will imply that we can restrict the first sum in (32) to m ∈ N n for which m i = m j for all i, j. To prove this claim, observe that α ∈ [β] precisely if for all A ∈ α we have (A ∩ Z = ∅ or A ∩ Z ∈ β| Z ) and similarly we have (A ∩ Z c = ∅ or A ∩ Z c ∈ β| Z c ). Hence, every A ∈ α is the union of some A 1 ∈ α| Z ∪ {∅} and A 2 ∈ α| Z c ∪ {∅} with not both A 1 = ∅ and A 2 = ∅. Write a = (β| Z ), b = (β| Z c ), and assume without loss of generality that a ≤ b. Write k for the number of A ∈ α for which both A 1 = ∅ and A 2 = ∅. Now, (α) = a + b − k. Moreover, given k, Z and β, there are In case A 1 = {1} (i.e. |A 1 | ≥ 2), one finds by (12) that As [β] contains one element for which (33) holds and |γ| elements for which (34) holds, one finds

By (11) and (13) this equals
Hence, summing over all conjugacy classes, we obtain The case when f 1 (x) = .
. . = f n (x) = x is the easiest example (for arbitrary n ∈ N) of the above result. In this case one generalises Theorem 3.3.1 by a result which, in accordance with our main principle of establishing identities before the q-bracket, yields this theorem after taking the q-bracket: Proof. Recall S k = S k−1,id and apply Theorem 3.5.3(ii) n − 1 times.
Later we will use Theorem 3.3.1 when the f i are Faulhaber polynomials. This is the situation in which we prove the main result of this paper, in which case the following lemma is useful.
Lemma 3.5.6. If f 1 , . . . , f n are Faulhaber polynomials of degrees d 1 , . . . , d n respectively and g : N → Q is as in Theorem 3.5.3, then there exists a polynomial p such that ∂g(m) = p(m) for all m ∈ N. Moreover, p is strictly of degree |d| − 1, is even or odd and p(0) = 0.
Proof. By Theorem 3.5.3(ii) we can assume w.l.o.g. that none of the degrees d i equals 1. Now, consider a monomial ∂ (α) * A∈α f A in ∂g. Note that both * and ∂ are operators on the space of polynomials, more precisely: Hence the degree of such a monomial is |d| − 1. Now observe that by the symmetry (15) one has Therefore, we see that ∂f A is even or odd and as the convolution product preserves this property, every monomial is even or odd. By the same arguments ∂f A (0) = 0 and hence the constant term of every monomial vanishes. Therefore, every monomial ∂ (α)−1 * A∈α f A in g satisfies the desired properties, so that is remains to show that the leading coefficient does not vanish. As F l = 1 l x l + O(x l−1 ), the leading coefficient of a monomial as above equals where for a set B we have set d B = b∈B d b . Hence, the leading coefficient of ∂g equals where α = {A 1 , . . . , A r }. Note that this number has the following combinatorial interpretation. Let n balls be given which are colored such that d 1 balls are colored in the first color, d 2 in the second color, etc. Suppose we use the same multiset of colors to additionally mark each ball with a dot (possibly of the same color), that is d 1 balls are marked with a dot of the first color, d 2 with a dot of the second color, etc. Given a subset C of the set of all colors, it may happen that if we consider all balls colored by the colors of C, all the dots on these balls are colored by the same set of colors C. We then say that the balls are well-colored with respect to C. For example, both the empty set of colors and the set of all possible colors give rise to a well-coloring of balls. If we independently at random color and mark the balls as above, the probability that the balls colored by a subset C are well-colored is |d| equals the probability that if we independently at random color and mark the balls as above, there does not exist a proper non-empty subset C of the colors such that the balls colored by C are well-colored. If we mark at least one ball of every color i with color i + 1 (modulo n), such a set C cannot exist. Hence, the number (35) is positive, so the polynomial p is strictly of degree |d|−1.

Three quasimodular algebras
4.1. Introduction Given integers k, l with k ≥ 0 and l ≥ 1 recall the definition of the double moment functions in (4) by Unless stated explicitly, we always assume that Moreover, it turns out to be useful to define T 0,0 ≡ T −1,1 ≡ −1 and T k,l ≡ 0 for other pairs (k, l) with k < 0 or l < 1.
Remark 4.1.1. The double moment functions specialize to the moments functions studied in the previous section whenever l = 1, i.e. T k,1 = S k+1 . Also, as F l (1) = 1, for a strict partition λ one has T k,l (λ) = S k (λ). Hence, our functions T k,l can be seen as an extension of the algebra of supersymmetric polynomials, mentioned in the introduction, to functions on all partitions (and not only on strict partitions).
Remark 4.1.2. In case k + l is odd, the q-bracket of T k,l does not vanish-in contrast to the shifted symmetric functions for which the q-bracket vanishes for all odd weights. However, the q-bracket of a polynomial involving the double moment functions in both even and odd weights also is a polynomial in the so-called combinatorial Eisenstein series, defined in Definition 7.2.4.
These double moment functions give rise to three different graded algebras, which turn out to be quasimodular (see page 2).
Definition 4.1.3. Define the Q-algebras S, Sym (S) and T by the condition that • S is generated by the moments functions S k under the pointwise product; • Sym (S) is generated by the elements of S under the induced product; • T is generated by the double moment functions under the pointwise product.
Our main result Theorem 1.1 is slightly refined by the following statement.
Theorem 4.1.4. Let X be any of the algebras S, Sym (S) and T . Then X is • quasimodular; • closed under the pointwise product; • closed under the induced product if X = S.
Moreover, the three algebras are related by S Sym (S) T .
Remark 4.1.5. Observe that being closed under the pointwise product is not implied by being quasimodular. For example, the algebra R = · −1 q ( M ) in Remark 3.2.6 is quasimodular, closed under the induced product and T ⊂ R, but R is not closed under the pointwise product [Zag16, Section 9].
In the next section we provide different bases for these algebras: in this way we obtain many examples of functions with a quasimodular q-bracket and moreover the study of these bases leads to a proof of Theorem 4.1.4.
Remark 4.1.6. The algebras T and Λ * are different algebras, as follows from the observation that f (λ) = (−1) k f (λ ) for all f ∈ Λ * k but not for all f ∈ T , where the first part of the observation follows by writing a shifted symmetric polynomial as a symmetric polynomial in the Frobenius coordinates and the latter can easily be check numerically. It is not true that f (λ) = f (λ ) for all f ∈ T , as Q 2 = T 1,1 with Q k defined by Equation (2). More precisely, one has: Namely, if f ∈ T ∩ Λ * , consider a partition λ consisting of different parts (i.e. r m (λ) ≤ 1 for all m). Then we have that f (λ) is symmetric polynomial in the parts λ 1 , λ 2 , . . .. On the other hand, as f ∈ Λ * , it follows that f (λ) is a shifted symmetric polynomial in the parts λ 1 , λ 2 , . . .. The only polynomials of degree d in the variables x i that are both symmetric and shifted symmetric are up to a constant given by

The basis given by double moment functions
In this section we show that T is closed under the induced product. Moreover, we show that S and Sym (S) are subalgebras of T . In the next section, we use these results to define a weight grading on T . Observe that as a vector space T is spanned by the functions T k,l , defined by T k,l = i T k i ,l i , for all k, l ∈ Z n satisfying the conditions (36) for all pairs (k, l) = (k i , l i ). Proof. Observe that T k,l T k ,l = T k,l T k ,l − T k,l | T k ,l .
Hence, it suffices to show that T k,l | T k ,l can be expressed in terms of elements of T . By Theorem 3.5.3 and Lemma 3.5.6 we have that an expression of the form Hence, by using this proposition recursively, we can replace the pointwise products in T k,l | T k ,l by a linear combination of connected products of double moments functions T k,l , showing that T k,l | T k ,l is an element of T . Now, we determine a basis for the three algebras. Let T mon be the set of all monomials for the pointwise product in T . Two elements of T mon are considered to be the same if one can reorder the products so that they agree, for example T 1,1 T 3,5 and T 3,5 T 1,1 are the same function. In other words: every elements of T mon can be written as T k,l in a unique way up to commutativity of the (pointwise) product. (37)

Moreover, a basis for
• T is given by T mon ; • Sym (S) is given by all T k,l ∈ T mon satisfying k i ≥ l i for all i; • S is given by all T k,l ∈ T mon satisfying l i = 1 for all i.
Proof. From the stated bases, statement (37) follows immediately. By definition the elements of T mon generate T as a vector space. Hence, it suffices to show that they are linearly independent, i.e. that if α∈I c α T α (λ) = 0 (38) for all λ ∈ P, where I is the set of all pairs (k, l) up to simultaneous reordering and c α ∈ Q, we have that c α = 0 for all α.
First of all, let λ = (N 1 , N 2 ) and consider (38) as N 1 → ∞. Note that T k,l (λ) grows as plus lower order terms, where k min is the smallest of the k i in k. Hence, |k| should be constant among all T α in (38). Moreover, we conclude that k min should be constant among all T α in (38).
Continuing by considering the lower order terms, we conclude that k is constant among all T α . Similarly, by instead considering partitions consisting of N 1 times the part 1 and N 2 times the part 2, we conclude that l is constant among all T α . Hence, there is at most one α with non-zero coefficient c α . We conclude that c α = 0 for all α ∈ I. For Sym (S) we show, first of all, that indeed T k,l ∈ Sym (S) if k i ≥ l i for all i. Let k ≥ l of the same parity be given. By Corollary 3.5.5 we find that Therefore, T k,l ∈ Sym (S) for all k ≥ l. Moreover, by applying Möbius inversion on Equation (22), which defines the connected product, we find As we already showed that T k,l ∈ Sym (S) if k ≥ l, we find T k,l ∈ Sym (S) if k i ≥ l i for all i. Next, we show that all elements in Sym (S) are a linear combination of the T k,l satisfying k i ≥ l i . As S clearly is contained in the space generated by the T k,l for which k i ≥ l i , it suffices to show that the latter space is closed under . For this we follow the proof of Theorem 4.2.1 observing that in each step k i ≥ l i , so that indeed the T k,l for which k i ≥ l i form a generating set for Sym (S).
As we already showed that the T k,l are linearly independent, we conclude that the T k,l ∈ T mon satisfying k i ≥ l i for all i form a basis for Sym (S).
The last part of the statement follows directly, as by definition all T k,l ∈ T mon satisfying l i = 1 for all i generate S, and by the above they are linearly independent.
4.3. The basis defining the weight grading By definition, the double moment functions generate T under the pointwise product. In this section we show that we can replace the pointwise product in the latter statement by the induced product. Again we will consider every reordening of the factors in T k 1 ,l 1 · · · T kn,ln due to commutativity of the products to be the same element. Then we have: Theorem 4.3.1. The elements T k 1 ,l 1 · · · T kn,ln form a basis for T . A basis for the subspace Sym (S) is given by the subset of elements for which k i ≥ l i for all i.
Proof. Consider the subspace of elements of weight w in T . The number of basis elements in the basis given by the pointwise product in the previous section equals the number of induced products of the T k,l . Hence, it suffice that the induced products of the T k,l generate T . For this we proceed by induction first on the weight and then on the degree. Here, by degree, we mean the unique grading under the pointwise product for which every T k,l has degree 1, usually called the total degree.
Trivially, every element of weight 0 or degree 0 is generated by (empty) induced products of the T k,l . Next, consider T k,l ∈ T and assume all elements of lower weight and of the same weight and lower degree are generated by induced product of the T k,l . Let T k,l ∈ T of weight w be given and write k , l for k, l after omitting the last (nth) entry. Then, T k,l = T k ,l T kn,ln − T k ,l | T kn,ln .
Note that T k ,l is of weight strictly less than w, hence is generated by induced products of the T k,l . Moreover, by 3.2.7 and 3.5.3 it follows that the weight of T k ,l | T kn,ln is at most n − 1. Hence, by our induction hypothesis it is generated by induced products of the T k,l . We conclude that T k,l is generated by induced products of the T k,l , which proves the first part of the theorem. The second part follows by the same proof, everywhere restricting to those T k,l for which k ≥ l.
By the above theorem, we can define a weight grading on T : Definition 4.3.2. Define a weight grading on T by assigning to T k,l weight k + l and extending under the induced product.
Note that both the grading on T and the grading on S correspond to the grading on quasimodular forms after taking the q-bracket. Hence, the grading on S is the restriction of the grading on T .
The weight grading defines a weight operator. In Section 5 we extend this weight operator to an sl 2 -triple acting on T , so that T becomes an sl 2 -algebra.
4.4. The n-point functions As induced products of the T k,l form a basis for T , knowing f q for all f ∈ T is equivalent to knowing the following generating function, called the n-point function for all n ≥ 0. Here the sum is over all k i , l i such that k i + l i is even and m! is consider to be 1 for m < 0. As the q-bracket is a homomorphism with respect to the induced product, we directly conclude that We also define the partition function by T k 1 ,l 1 · · · T kn,ln q t k 1 ,l 1 · · · t kn,ln .
The following result (together with (40)) expresses these functions in terms of the Jacobi theta series (see (7)): Theorem 4.4.1. For all n ≥ 0 one has where [x 0 y 0 ] denotes taking the constant coefficient.
Proof. We have that where in the sum it is understood that k + l is even, k ≥ 0, l ≥ 1. The expression for F 1 (u, v) in the statement now follows from [Zag91, § §3]. The expression for Φ follows immediately from this result.

Differential operators
5.1. The derivative of a function on partitions Note that for all f ∈ Q P one has Hence, by letting Df Moreover, D acts as a derivation: Proposition 5.1.1. The map D : Q P → Q P is an equivariant derivation, i.e. D is linear, satisfies the Leibniz rule and D f q = Df q .
In fact, for all k ≥ 1, the mapping f → S k | f is a derivation. Recall the defintion of the Möbius function µ defined in Proposition 3.1.3 and denote by S 0 k = S k − S k (∅).
since µ(λ ∪ (m)) = −µ(λ) for λ ∈ S m , so that for r ≥ 2 the coefficient of u r m u λ cancels. We conclude that S 0 k µ = −S 0 k µ. For the second part, note that (i) implies that Let f, g ∈ Q P be given. Then Therefore, i.e. the mapping f → S k | f is a derivation. The formula S m | T k,l = T k+m−1,l+1 follows directly from Theorem 3.5.3.
Proof of Proposition 5.1.1. As S 2 | f = S 2 f − S 2 f is derivation by the above lemma, the results follows directly from (41).

The equivariant q-bracket
In this section we extend the action by the sl 2 -triple (D, d, W ) on quasimodular forms to T . As the derivation d does not act on all power series in q, but only on quasimodular forms, we cannot hope to define d on all functions on partitions as we did with D.
On the algebra T , however, this is possible. We define an sl 2 -action on this space and we show that the q-bracket restricted to T is an equivariant map of sl 2 -algebras. Note that the following definition agrees with the definition of D in the previous section: One immediately checks that D, W and d satisfy the commutation relation of an sl 2 -triple on T . The corresponding acting of sl 2 on T makes the q-bracket equivariant, so that a refined version of Theorem 1.2 is: Theorem 5.2.2 (The sl 2 -equivariant symmetric Bloch-Okounkov theorem). The algebra T is an sl 2 -algebra with respect to the above action of sl 2 on T . Moreover, the q-bracket becomes an equivariant map of sl 2 -algebras, i.e. for f ∈ T one has Proof. We already observed that the first of the three equality holds, the second is the homogeneity statement. Hence, it suffices to prove the last statement. Using (8) we find that for and the last statement follows from the Leibniz rule.

Rankin-Cohen brackets
The sl 2 -action allows us to define Rankin-Cohen brackets on T .
Definition 5.3.1. For two elements f, g ∈ T the nth Rankin-Cohen bracket is given by Note that the formula (43) would have defined the Rankin-Cohen brackets on M if D acts by q ∂ ∂q and the induced product is replaced by the usual product, whereas in this line D acts on T as explained in the previous sections.
If f, g ∈ ker d, then f q and g q are modular forms. The Rankin-Cohen bracket of two modular forms is a modular form, analogously we have: Proof. Using (8), we find that where 1 (−1)! should taken to be 0. This is a telescoping sum, vanishing identically.
Remark 5.3.3. The algebra T becomes a Rankin-Cohen algebra by the above bracket, meaning the following. Let A * = ⊕ k≥0 A k be a graded K-vector space with A 0 = K and dim A k < ∞ (for us A = T The operator D is replaced by multiplication with S 2 . Lemma 5.4.1. The triple (S 2 , s, W − 1 2 ) forms an sl 2 -triple of operators acting on S.
As s and S 2 decrease respectively increase the weight by 2, the claim follows.
Theorem 5.4.2. The q-bracket · q : S → M is an equivariant mapping with respect to the sl 2triple (S 2 , s, W − 1 2 ) on S and the sl 2 -triple Proof. The first of the three equalities in (44) follows from the definition of the q-bracket, the second is the homogeneity statement of Theorem 5.2.2. Hence, it remains to prove the last equation d f q = sf q . Given k ∈ N n , let k i ∈ N n−1 be given by k i := (k 1 , . . . , k i−1 , k i+1 , . . . , k n ) omitting k i . Similarly, define k i,j ∈ N n−2 by omitting k i and k j . Then By Theorem 3.3.1 one finds For I ∈ β and l ∈ N I , let C(I, l) := i,j∈I,i =j (l i + l j − 2) = ( (I) − 1)(|l| − (I)). Therefore,

It follows that
which by the above reasoning is exactly equal to sS k q .
6. Relating the two products 6.1. A formula for the coefficients In Theorem 3.5.3 we deduced that In the particular case that f 1 = .
. . = f n is the identity function, we saw in Corollary 3.5.5 that g = F n . If f 1 , . . . , f n are Faulhaber polynomials, the function g is not necessarily equal Faulhaber polynomial on all m ∈ N, but, by Lemma 3.5.6, ∂g equals some polynomial. Also, using g is uniquely determined by ∂g, the function g equals some polynomial. We expand g as a linear combination of Faulhaber polynomals.
Definition 6.1.1. Given integers l 1 , . . . , l n , we define the structure constants C l i by Observe that C l i = 0 for odd i, as ∂g is even or odd. Corollary 3.5.5 is the statement More generally, by Theorem 3.5.3(ii) one has C 1,l i = C l i , so that w.l.o.g. we can assume l i > 1. In this section, we give an explicit, but involved, formula for these coefficients in terms of Bernoulli numbers and binomial coefficients. In order to do so, for l 1 , l 2 ≥ 1 and i ∈ Z, we introduce the following numbers The following polynomials can be expressed in terms of these coefficients: Lemma 6.1.2. For all l 1 , l 2 , . . . , l r ≥ 2 one has the following identities: Using these identities, one obtains These easy expressions for small n are misleading, as 6C l 1 ,l 2 ,l 3 i equals 1 4 δ i,2 + 3 up to full symmetrization, i.e. summing over all σ ∈ S 3 with l i replaced by l σ(i) . In general, given α ∈ Π(n), write α = {A 1 , . . . , A r } and denote A j = ∪ j i=1 A j . Also, for a vector k and a set B we let k B = b∈B k b . Then, the above observations allows us to write down the following formula, which is very amenable to computer calculation: Proposition 6.1.3. Let l 1 , . . . , l n > 1. Then Note that the latter formula is written in an asymmetric way, but (by associativity of the convolution product) is symmetric in the l i .
6.2. Derivatives on pointwise products Suppose an element of T is given, written in the basis with respect to the pointwise product. How do we determine its (possibly mixed) weight and its representation in terms of the basis with respect to the product? A first answer is given by applying Möbius inversion to Equation (22), as given by Equation (39), i.e.
However, as every factor T k A 1 ,l A 1 | T k A 2 ,l A 2 | . . . in the above equation is a linear combination of generators of different weights, it is useful to have a recursive version of this result. For this, we write ∂ ∂T k,l for the derivative of f ∈ T in the former basis (with respect to the pointwise product) and ∂ ∂T k,l for i ∂ ∂T k i ,l i . Proposition 6.2.1. Let k, l ≥ 1. There exist differential operators s i,j for all i, j ∈ Z such that s i,j = 0 if j < 0 and for all f ∈ T one has Explicitly, where the structure constants C l,b |b|−j are as in Proposition 6.1.3.
Proof. By linearity, it suffices to proof the statement for monomials T k,l . Hence, assume f = T k,l . Applying (45), extracting the factor containing T k,l and applying (45) again, yields f.
By Definition 6.1.1 this equals Replacing j by −j + |b| and writing i = |a| one obtains f, as desired.
Corollary 6.2.2. For all k, l ≥ 1 and f ∈ T one has where T a,b k,l = s a−l+1,a+b−k−1 + s a+b−l,a−k .
Proof. Distinguishing two cases in the previous result yields

Related functions on partitions
We apply our results to interesting functions on partitions.
7.1. Hook-length moments First of all, we focus on the hook-length moments H k [CMZ18, part III]. These functions form a bridge between the symmetric algebra studied in this note and the shifted symmetric functions: the H k themselves are shifted symmetric as and they are also equal to the Möller transform of the symmetric S k , i.e. H k = M(S k ), meaning the following. Denote z ν = n! |Cν | with |C ν | the size of the conjugacy class corresponding to ν. Recall that Given f ∈ Q P , the Möller transform of f at a partition λ ∈ P(n) is given by [Zag16,Eqn (45)] where the sum ν n is over all partitions of size n and χ λ (ρ) denotes the the character of the representation corresponding to the partition λ evaluated at the conjugacy class corresponding to ρ.
Then M(f ) q is a quasimodular form if and only if f q is a quasimodular form (which follows directly by the column orthogonality relations for the symmetric group). In the next section we study the Möller transform of elements of T , but first, we explain the Murnaghan-Nakayama rule, used in [CMZ18, part III] to show equality between M(S k ) and (46) and give two other expressions for the hook-length moments.
To start with the latter, the hook-length moments, as their name suggests, are defined as moments of the hook-lenghts, i.e.
where Y λ denotes the Young diagram of a partition λ and h(ξ) denotes the hook-length of a cell ξ ∈ Y λ . Next, the following constructions related to the Young diagram, give rise to the Murnaghan-Nakayama rule for the characters of the symmetric group. Given partitions λ, ν with ν i ≤ λ i for all i, we define the skew Young diagram λ/ν by removing the cells of Y ν from Y λ . Denote by |λ/ν| = |λ| − |ν| the number of cells of this diagram. We call λ/ν a border strip of λ if it is connected (through edges, not only through vertices) and contains no 2 × 2-block. If γ = λ/ν we write λ \ γ for ν. The height of a border strip γ is defined to be one less than the number of columns and denoted by ht(γ). Given m ∈ N s , we let a border strip tableau γ of type m be a sequence γ 1 , . . . , γ s such that γ i is a border strip of λ γ 1 · · · γ i−1 and |γ i | = m i . Write Y γ for the skew Young diagram consisting of all boxes of all the γ i and write ht(γ) = ht(γ 1 ) + . . . + ht(γ s ). Denote by BST(λ, m) and BST(λ/ν, m) the set of all border strip tableau of type m within λ and λ/ν respectively.  2, 1, 1), (2, 1, 2)).

Border strip moments
The hook-length moments are Möller transformations of the S k . In this section we study the Möller transformation of the algebra T , which contains the vector space spanned by all the S k . In order to do so, we express elements of T in terms of functions U k,l for which the induced product and Möller transformation are easy to compute. However, these function do not admit the property that the q-bracket is quasimodular if k i + l i is even for all i: each U k,l lies in the space generated by all the T k,l (possibly with k i + l i odd). Observe that this product converges since r a (m) = 0 for all but finitely many values of a. Generalise the hook-length moments in Definition 7.1.1(ii) by the following notion: Proof. By Proposition 7.2.3, f equals the Möller transform of some polynomial in the T k,l with respect to the product . Here, however, it may happen that k + l is odd. Generalizing either of the three approaches in § §3.1 yields that the q-bracket of T k,l lies in C k+l , which proves the result.
Proof. Observe that Proposition 7.2.3 implies that M(T k ) ⊂ X ≤k . Equation (48) follows from this proposition after noting that the Möller transformation of T k,l − (l − 1)!U k,l has degree strictly smaller than k + l.
See Appendix A for a table of elements in X with quasimodular q-bracket and of small degree. 7.3. Moments of other partition invariants Whereas the last two paragraphs provided many examples of functions on partitions in Λ * and T related to the representation theory of the symmetric group, in this paragraph we see that many purely combinatorial notions lead to different bases for S. We compare these bases to corresponding bases of Λ * . Most of these bases take the following form. Suppose for every λ ∈ P an index set I and a sequence {s i } ∞ i∈I of elements of Q P are given. Then we define the kth moment of s by (whenever this sum converges) For example, let the functions p, q for the index set N be given by Then, by definition, S k = S k (∅) + M k−1 (p), Q k = Q k (∅) + M k−1 (q) Note that by definition M k (s)(∅) = 0. As the functions below will not respect the weight grading anyway, we will not include a constant term. Define the sequences a, c, h, x of functions on partitions indexed by ξ = (i, j) ∈ Z 2 ≥0 by 0 if ξ ∈ Y λ and if ξ ∈ Y λ by a ξ (λ) : arm length of ξ h ξ (λ) : hook-length of ξ x ξ (λ) = i c ξ (λ) : content of ξ, i.e. i − j For h and c is it known that the corresponding moment functions are shifted symmetric, for the latter see [KO94,Theorem 4]. Observe that, considered as multisets, x and a are equal. Moreover, we have generate S, which corresponds to the first equality in the statement. By interchanging the sums one obtains Hence, the result is also true for s = x.
Remark 7.3.2. Note that for a given i the number of (i, j) ∈ Y λ equals λ i , where λ is the conjugate partition of λ. Hence, (49) can be written as ∞ i=1 i k−1 λ i and consequently these functions for k odd generate S. Note that these functions are different from the S k (λ ). In fact, the algebra generated by the S k (λ ) is distinct from the algebra S, in contrast to the algebra of shifted symmetric functions, for which Q k (λ ) = (−1) k Q k (λ).