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 T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}$$\end{document} 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 ηfunction, i.e., λ∈P q |λ| = n>0 (1 − q n ) −1 = q 1/24 η(τ ) −1 (q = e 2π iτ ), 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 [1]. 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 Before continuing, note that it is not surprising at all that for a well-chosen function f the q-bracket 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 [3, Theorem 0.5] provided the first quasimodular algebra * . 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 * 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 [7] and for which quasimodularity was proven rigorously in [14]. Quasimodularity of * is used in many recent works in enumerative geometry [4][5][6]12,13]. There are many other functions in invariants of partitions which turn out to be elements of * , for example symmetric polynomials in de modified Frobenius coordinates [23,Eq. 19]; the hook-length moments [5,Theorem 13.5] (see Sect. 7.1); central characters of the symmetric group [15,Proposition 3] and symmetric polynomials in the content vector of a partition [15, Proof of Theorem 4].
Previously, the Bloch-Okounkov algebra * and some generalizations to higher levels (see, e.g., [8,9]), 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 * ) [23,Sect. 9], for example the Möller transformation of functions with quasimodular q-bracket (defined by [23,Eq. 45] and recalled in Sect. 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 [10] (in that reference, these functions are only evaluated at strict partitions-partitions without repeated parts-and quasimodularity is shown for a correspondingly adapted q-bracket).
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) (4) 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 the induced product.
With respect to the pointwise product, these algebras are not quasimodular because of the following subtlety: The q-bracket of a homogeneous function f in T (with respect to the pointwise product) often is of mixed weight (i.e., a linear combination of quasimodular forms of weights bounded by the weight of f ). By making use of the induced product, one can explain these lower weight quasimodular forms, as we do in Sect. 6. For example, where G 2 and G 4 are the Eisenstein series defined by (6). The right-hand side is a quasimodular form of mixed weight, which is explained by the fact that is a linear combination of elements of T of different weights with respect to the induced product.
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 * , this seems impossible. See the following Motivated by the fact that many functions in invariants of partitions are elements of * , in Sect. 7 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. The corresponding space X of border strip moments is the image of a space U under the aforementioned Möller transform M, where U is generated by the double moment functions T k,l ∈ T as well as the odd double moments functions (those for which k + l is odd). The q-brackets of these functions are contained in the space C of so-called combinatorial Eisenstein series, having the space of quasimodular forms as a subspace. Moreover, the space of hook-length moments H is contained in both * and X -this contrasts the situation for T , which by Remark 4.1.6 has a trivial intersection with * . See the commutative diagram below for an overview of the spaces related to T with their corresponding mappings.
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 Sect. 2 we recall notions (known to the experts) related to quasimodular forms, partitions and special families of polynomials. Next, in Sect. 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 Sect. 4. The sl 2 -action by differential operators, the proof of Theorem 1.2 and Rankin-Cohen brackets are the content of Sect. 5. In Sect. 6 further results that arise from comparing the two different products on T are given, and finally, in Sect. 7 we provide many examples of functions in or closely related to T .

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 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 ∈ yields ϕ 0 = ϕ. Quasimodular forms of depth 0 are called modular forms.
Besides the constant functions, the simplest examples are the Eisenstein series 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.
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 or the Kronecker-Eisenstein series of weight 2 and given by The propagator is closely related to the Weierstrass ℘-function and Jacobi theta series

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 operator D : M , 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 [22, p. 58-60]. 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 M = M[G 2 ].
Let W be the weight operator, which multiplies a quasimodular form by its weight. The triple (D, d, W ) forms an sl 2 -triple with respect to the commutator bracket [A, B] = AB − BA: 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: 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 (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 (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 ∈ (n), where 1 n = {[n]}.
Recall that on a locally finite partially ordered set P the Möbius function μ : P 2 → Z is defined recursively by (see for example [16]): 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 ∈ (n) is given by is the partition on B induced by α. A Möbius function satisfies the following two properties: 3.1 Let f, g be functions on a partially ordered set P. Then

The connected q-bracket
The q-bracket defined in the introduction (Eq. 1) is a map Q P → Q [[q]]. In this section we define the connected q-bracket following [5, 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 two equivalent conditions hold: (ii) f 1 ⊗ · · · ⊗ f n q is the coefficient of By invoking the Möbius inversion formula (Theorem 2.3.1(ii)) condition (i) in Definition 2.4.1 implies that For example, and fg q = f ⊗ g q + f q g q , We often make use of the fact that the connected q-bracket of functions f 1 , . . . , f n vanishes if one of the f i is constant.
Observe that A∈α f A q takes the same value for all α ∈ (n + 1) which agree on [n] (but differ in the subset A of α containing n + 1). Then, summing μ(α, 1 n ) over all such α yields a · (−1) a−1 (a − 1)! + (−1) a a! = 0 as there are a choices for α for which {n+1} is not a subset of α, where a is the length of such an α, and there is only one choice for α for which {n + 1} is a subset. By Definition 2.4.1(i) the result follows.
We will use the second condition in Definition 2.4.1 in our proof that S is a quasimodular algebra.

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 fg 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 x+1 x B n (u) du = x n , by the formula lF l (x) = B l (x+1)−B l . Hence, the Faulhaber polynomials admit the symmetry 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

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 motivate the results in 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 (see (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: (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 (19), we obtain Hence, Observe that applying x ∂ ∂x to the right-hand side of (20) has the same effect as applying 1 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 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 Given 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 In other words, X a Y b q ab so that expanding this equation for X = e x and Y = e y yields T 0 k,l q = a,b≥0 a k b l−1 q ab .
Third approach In this last proof we start with the observation that one can rewrite the q-bracket as In contrast to the previous two proofs, it is only in the last step of this proof that we take the q-bracket: First we rewrite (22) 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, where we encounter the Möbius function on partitions also defined in [17]. (i) μ(λ) is given by the Möbius function μ(∅, λ) on the partial order on the set of partitions in (10); Proof The first two definitions clearly coincide using (10). 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.

The induced and connected product
Motivated by the last of the three approaches in the previous section, we define the ubracket 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. This observation should be compared with the q-bracket arithmetic in [17].
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. For a set S and functions f s ∈ Q P for all s ∈ S, we denote f S = s∈S f s .

Definition 3.2.4
For f 1 , . . . , 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 q-bracket, as follows from the following result.
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 identities on the level of functions on partitions (before taking the q-bracket), it turns 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.

. enumerate the elements of A (and similarly for B).
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 ∈ (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 ∈ (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 (24) 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 (25) equals Correspondingly, the coefficient of C∈γ f C on the left-hand side of (24) vanishes if there are C 1 , C 2 ∈ γ with 1 ∈ C 1 and 2 ∈ C 2 .

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 k n . 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.
The algebra S is a quasimodular algebra. More precisely, for k ∈ (2N) n one has Proof Observe that it suffices to show that for all positive even k. Euler's formula for the generating series of partitions follows from writing |λ| = m≥1 mr m (λ) and summing over all possible values of r 1 (λ), r 2 (λ), etc. By the same idea, we find The logarithm of this expression equals Now, assume all parts of k are distinct. In the expansion of (29) the coefficient of Hence, Denoting and setting S 0 (λ) ≡ 1, one has the following expression for the generating series of the qbracket of the generators of S: where z A = a∈A z a and is the totally even part of the propagator in (7).

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 an algebra 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 .
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 [22,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 [19,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 Eqs. (30) 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 (30) 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.

The connected product of moment functions
In the second approach we compute the connected product S k 1 | . . . | S k n , which by Proposition 3.2.5 yields the left-hand side of (26) after taking the q-bracket. The result is formulated in Theorem 3.5.4 and depends on two technical lemma's which we state first. 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).
The first 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.
Proof By Proposition 3.2.3, we have where it is understood that γ is a strict partition. We have that , 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 (31) 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 ), Next, for iii we have Letting i = r m (α) and j = r m (β), we have The second lemma is concerned with the vanishing of certain sums of the Möbius functions of set partitions. Given α ∈ (n) and a subset Z of [n], we let where (Z) denotes the set of all partitions of the set Z. Observe that , define an equivalence relation on (n) by writing α ∼ β if Proof , 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 The following result not only computes the connected product of the moment functions S k , but also is one of the main technical results needed to prove Theorem 1.1.
where f A = a∈A f a and * denotes the convolution product (11).
(ii) If f 1 (x) = x, then ∂g = f 1 ∂g withg given by S k 2 ,f 2 | . . . | S k n ,f n = S |k|,g . Remark 3.5. 5 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 (23) and (18) respectively), we have μ(α, 1) Then, given m ∈ N n , Z = Z(m) and A ∈ α| Z , the function λ → m A k A f A (r m A (λ)) is supported on {m 1 }, whereas for A ∈ α| Z c the function λ → m A k A f A (r m A (λ) ) is supported on N\{m 1 }. Hence, by Lemma 3.5.2(ii) we find that (34) equals Instead of writing the second factor as a product of functions which are all supported on a singleton set, we make the following observation.
As α| Z = β| Z and α| Z c = β| Z c , the only dependence on α in the above equation is in μ(α, 1). By construction Z(m) is non-empty. Hence, by Lemma 3.5.3 we have that if Z = [n] then for all β ∈ E(m) we have α∈[β] μ(α, 1) = 0. This implies that we can restrict the first sum in (35) to m ∈ N n for which m i = m j for all i, j, that is, Applying Lemma 3.5.2iii (α) − 1 times and using (12), we obtain the desired result.
In case A 1 = {1} (i.e., |A 1 | ≥ 2), one finds by (13) that By (12) and (14), 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 generalizes 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 3.5.4(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. Proof By Theorem 3.5.4(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 (16) 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 it 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 (38) is positive, so the polynomial p is strictly of degree |d| − 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.  These double moment functions give rise to three different graded algebras, which turn out to be quasimodular (see page 1).  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 * are different algebras, as follows from the observation that f (λ) = (−1) k f (λ ) for all f ∈ * k , which follows by writing a shifted symmetric polynomial as a symmetric polynomial in the Frobenius coordinates. This does not hold for all f ∈ T , as can easily be checked numerically. On the other hand, it is not true that f (λ) = ±f (λ ) for all f ∈ T , as Q 2 = T 1,1 with Q k defined by Eq. (2). More precisely, one has Namely, if f ∈ T ∩ * , consider a strict partition λ (i.e., a partition for which 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 ∈ * , 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 (39) for all pairs (k, l) = (k i , l i ).

Theorem 4.2.1 The algebra T is closed under the induced product.
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 3.5.4 and Lemma 3.5.7, we have that an expression of the form: Hence, by using this proposition recursively, we can replace the pointwise products in T k,l and T k ,l by a linear combination of connected products of double moment 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.

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 It suffices to prove the second part, as from the stated bases statement (40) 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 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 (41) 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 (41). Moreover, we conclude that k min should be constant among all T α in (41). 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 nonzero 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 3.5.6 we find that Therefore, T k,l ∈ Sym (S) for all k ≥ l. Moreover, by applying Möbius inversion on Eq. (23), 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.

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 reordering of the factors in T k 1 ,l 1 · · · T k n ,l n due to commutativity of the products to be the same element. Then, we have: The elements T k 1 ,l 1 · · · T k n ,l n 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 Assign to T k,l weight k + l. This defines a weight filtering on T with respect to the pointwise product. Consider the subspace of elements of weight at most 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 depth. Here, by depth we mean the unique filtering under the pointwise product for which every T k,l has depth 1, usually called the total depth.
Trivially, every element of weight 0 or depth 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 depth 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 k n ,l n − T k ,l | T k n ,l n .
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 Proposition 3.2.7 and Theorem 3.5.4 it follows that the depth of T k ,l | T k n ,l n 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 . 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 Sect. 5 we extend this weight operator to an sl 2 -triple acting on T , so that T becomes an sl 2 -algebra.

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 F n (u 1 , . . . u n , v 1 , . . . v n ) = k,l T k 1 ,l 1 · · · T k n ,l n q u k 1 1 · · · u k n n v l 1 −1 1 · · · v l n −1 n k 1 ! · · · k n !(l 1 − 1)! · · · (l n − 1)! 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 k n ,l n q t k 1 ,l 1 · · · t k n ,l n .
The following result (together with (43)) expresses these functions in terms of the Jacobi theta series (see (8)).
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 [20,3]. The expression for follows immediately from this result.

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: Let S m be the set of strict partitions not containing m as a part. Then, we can rewrite (45) as since μ(λ ∪ (m)) = −μ(λ) for λ ∈ S m , so that for r ≥ 2 the coefficient of u r m u λ cancels in pairs. 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 3.5.4.
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 (44).

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 2triple 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: Proof We already observed that the first of the three equality holds and the second is the homogeneity statement. Hence, it suffices to prove the last statement. Using (9) we find that for a ≥ 0, b ≥ 2 one has Hence, 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 .
Note that the formula (46) 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 (9), we find that where 1 (−1)! should taken to be 0. This is a telescoping sum, vanishing identically.

Remark 5.3.3
The above bracket makes the algebra T into a Rankin-Cohen algebra, 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 ). We say A is a Rankin-Cohen algebra if there are bilinear operations [ , ] n : A k ⊗ A l → A k+l+2n (k, l, n > 0) which satisfy all the algebraic identities satisfied by the Rankin-Cohen brackets on M [21].

A restricted sl 2 -action
Theorem 5.2.2 does not make S into an sl 2 -algebra. Namely, D does not preserve S. However, if we allow ourselves to deform the sl 2 -triple (D, d, W ) as in [18], we can define an sl 2 -action on S. This action, however, does not make S into an sl 2 -algebra, as the deformed operators are not derivations. The operator taking the role of d is the operator s : S k → S k−2 defined by The operator D is replaced by multiplication with S 2 . Proof Observe that As s and S 2 decrease, respectively, increasing the weight by 2, the claim follows.
Proof The first of the three equalities in (47) 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 On the other hand, observe that if f is of weight |l| − 2 (I) + 2, Eq. (9) yields Therefore, which by the above reasoning is exactly equal to sS k q .
6 Relating the two products

The structure constants
In Theorem 3.5.4, we deduced that In the particular case that f 1 = .
. . = f n is the identity function, we saw in Corollary 3.5.6 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 3.5.7, ∂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 polynomials. 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 3.5.6 is the statement More generally, by Theorem 3.5.4(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 ≥0 , we introduce the following numbers: which by [2, Proposition A.10] satisfy 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: Proof The first two equations, of which the former is the well-known expansion of the Faulhaber polynomials, follow by considering the corresponding generating series. In order to prove (ii), we let n ∈ N and consider As the generating series of the Bernoulli numbers ∞ j=0 B j z j j! = z(e z − 1) −1 implies that Since B l 1 ,l 2 i vanishes for odd i if l 1 , l 2 > 1, this proves the second equation. The third equation follows from the first by noting that 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 up to full symmetrization, i.e., summing over all σ ∈ S 3 with l i replaced by l σ (i) . In general, given α ∈ (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: Note that the latter formula is written in an asymmetric way, but (by associativity of the convolution product) is symmetric in the l i .

From the pointwise product to the induced product
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 Eq. (23), as given by Eq. (42), i.e., However, as every factor T k A 1 ,l A 1 | T k A

Related functions on partitions
We apply our results to interesting functions on partitions.

Hook-length moments
First of all, we focus on the hook-length moments H k [5,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! |C ν | 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 [23,Eqn (45)] where the sum ν n is over all partitions of size n and χ λ (ρ) denotes 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 [5, part III] to show equality between M(S k ) and (49) 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-lengths, 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 Fig. 1 The Young diagrams corresponding to the border strip tableaux of type (2, 1, 2) within λ = (4, 2, 1, 1) 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 (Fig. 1).
The Murnaghan-Nakayama rule (recursively) expresses the characters of the symmetric groups in terms the heights of border strip tableau. Namely, if ρ ⊆ ρ (both ρ and ρ considered as multisets) where ρ − ρ denotes the difference of (multi)sets. Of particular interest are the cases ρ = ρ and ρ = (ρ 1 ), yielding a direct or recursive combinatorial formula for χ λ (ρ), respectively: and where the latter sum is over all borders strips γ of λ of length ρ 1 . The skew character χ λ/ν (ρ ) is defined by (|λ/ν| = |ρ |) so that To conclude, we have the following definitions of the hook-length moments: (iv) H k = M(S k ).

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). Let Observe that this product converges since r a (m) = 0 for all but finitely many values of a.
Let U be the algebra generated by the U k,l . Generalize the hook-length moments in Definition 7.1.1(ii) by the following notion: Let X be the vector space spanned by all the X k,l . Define a filtration on X by assigning to X k,l degree |k| + |l|.

Remark 7.2.2
Observe that for n = 1 and l = 1, the sum restricts to a sum over all border strips γ of λ and for such a border strip γ the factor χ γ (m) 2 equals 1 and z m equals m.
As the set of hook-lengths is in bijection with the set of all border strip lengths, one has that − B k 2k + X k,1 = H k+1 .
Denote by n j the Stirling numbers of the second kind (i.e., the number of elements in (n) of length j).
Proof For the first property, we use the known identity As F l (x) and x j are the unique polynomials with constant term equal to zero and such that ∂F l (x) = x l−1 and ∂ x j = x−1 j−1 , respectively, we find and their derivatives of weight ≤ , where to D r G k we assign the weight k + 2r. Now, Proposition 7.2.3 implies the following result: Theorem 7.2.5 For all f ∈ X ≤k , one has f q ∈ C ≤k .
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. Mutatis mutandis in either of three approaches in Sect. 3.1, we find 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 (51) 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.

Moments of other partition invariants
So far we provided many examples of functions on partitions in * and T related to the representation theory of the symmetric group. Now, we see that many purely combinatorial notions lead to different bases for S. We compare these bases to corresponding bases of * . Most of these bases take the following form. Suppose 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) M k (s)(λ) = i∈I s i (λ) k − s i (∅) k .
For example, let the functions p, q for the index set N be given by Then, by definition, 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. Proof As the Faulhaber polynomials F k with k odd form a basis for the space of all odd polynomials, the functions 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, (52) 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).