For integers $k \geq 0$ and $d \geq 1$ let $H(k,d)$
be the *Hurwitz* number which, for the purposes
of this posting, will be defined by:

\begin{equation} H(k,d) \, := \ d! \, \sum_{\lambda \, \vdash d} \, \nu_{\scriptscriptstyle T}^k(\lambda) \ \ \text{where} \ \nu_{\scriptscriptstyle T}(\lambda) := \binom{d}{2} \cdot {{\chi^\lambda_{\scriptscriptstyle T}} \over {\dim(\lambda)} } \end{equation}

and where $\chi^\lambda$ is the character value of the irreducible representation $V_\lambda$ of the symmetric group $S_d$ corresponding to the partition $\lambda \vdash d$ evaluated at any representative transposition (taken from the conjugacy class $T$ of all transpositions) and where $\dim(\lambda)$ is the dimension of $V_\lambda$. The Hurwitz number $H(k,d)$ can be interpreted, using the Verlinde formula, as counting the the number of homomorphisms (up to conjugation in $S_d$)

\begin{equation} \rho : \pi_1 \Big( \Bbb{T}^2_k , \, \mathrm{base \, point}\Big) \longrightarrow S_d \end{equation}

where $\Bbb{T}^2_k$ is the 2-torus with $k$ punctures. We can assemble these Hurwitz numbers into the following bivariate generating function

\begin{equation} \begin{array}{ll} H(x;q) &\displaystyle = \ 1 \ + \ \sum_{d \geq 1} \, \sum_{k \geq 0} \, H(k,d) \, {x^k \over {k!}} \, q^d \\ &\displaystyle = \ 1 \ + \ \sum_{\lambda \ne \emptyset} \, q^{|\lambda|} \, \exp \big\{ x \, \nu_{\scriptscriptstyle T}(\lambda) \big\} \end{array} \end{equation}

whose logarithm has a "genus" expansion

\begin{equation} \log H(x;\tau) = F_1(\tau) \ + \ \sum_{g \geq 2} \, F_g(\tau) \, {x^{2g-2} \over {(2g-2)!}} \end{equation}

where we set $q = e^{2\pi i \tau}$ and each $\tau$-series $F_g(\tau)$ is known to be a quasi-modular form.

Now let $t \geq 2$ be an integer and let us consider the following
*$t$-core analogues*:

\begin{equation} \begin{array}{l} \displaystyle H_t(x;q) \, := \ 1 \ + \ \sum_{\stackrel{\scriptstyle \text{$t$-cores}}{\lambda \,\ne \, \emptyset}} \, q^{|\lambda|} \, \exp \ \big\{ x \, \nu_{\scriptscriptstyle T}(\lambda) \big\} \\ \displaystyle F_{g; \, t}(\tau) \, := \ \text{ the coefficient of} \ {x^{2g-2} \over {(2g-2)!}} \ \text{in} \ \log H_t(x;\tau) \end{array} \end{equation}

**Question 1:** Does the generating function $H_t(x;q)$ have
a nice closed expression, e.g. some sort of product formula?

**Question 2:** Does the $\tau$-series $F_{g; \, t}(\tau)$ have any
kind of modular property?

thanks, ines.

**Post Script:**
As a kind of stupid example, consider the case of $2$-cores, which are precisely the *stair-case* partitions. The Murnaghan-Nakayama rule tells us that $\chi^\lambda_{\scriptscriptstyle T}$ can be evaluated recursively as the (signed) sum of dimensions $\dim(\mu)$ of partitions $\mu \vdash |\lambda| -2$ obtained by removing *skew-hooks* of size $2$ from the border of $\lambda$, i.e.

\begin{equation} \chi^\lambda_{\scriptscriptstyle T} \ = \ \sum_{\stackrel{\scriptstyle \lambda \, = \, \mu + \sigma}{\sigma \, \vdash \, 2}} \, \big(-1 \big)^{\#(\sigma)-1} \, \dim(\mu) \end{equation}

where $\#(\sigma)$ is the number of parts of $\sigma$. Of course there are **no** skew-hooks of size $2$ which can be excised from a stair-case partition, so $\chi^\lambda_{\scriptscriptstyle T} = 0$
for any $2$-core partition $\lambda$ and consequently

\begin{equation} \begin{array}{ll} H_2(x \, ;q) &\displaystyle = \ 1 \ + \ \sum_{\stackrel{\scriptstyle \text{$2$-cores}}{\lambda \, \ne \, \emptyset}} \, q^{|\lambda|} \\ &\displaystyle = \ 1 \ + \ \sum_{d \geq 1} \, q^{{1 \over 2}d(d+1)} \\ &\displaystyle = \ \prod_{d \geq 1} \big(1 - q^{2d}\big) \cdot \big(1 + q^d \big) \end{array} \end{equation}

Furthermore

\begin{equation} \begin{array}{ll} \log H_2(x \, ; q) &\displaystyle = \ \sum_{d \geq 1} \, \log(1 - q^{2d}) \, + \, \log(1 + q^d) \\ &\displaystyle = \ F_{1 ; 2}(\tau) \end{array} \end{equation}

The interesting computation begins with $3$-cores.

**Post-Post Script:** One possible approach to the problem may be to
take advantage of the Garvan-Kim-Stanton correspondence (GKS for short) which is a bijection

\begin{equation} \lambda \stackrel{\phi}{\Longleftrightarrow} \vec{n} \end{equation}

between $t$-cores $\lambda$ and integer vectors $\vec{n} = \big(n_0, n_1, \dots, n_{t-1} \big)$ with zero coordinate sum $n_0 + \cdots + n_{t-1} = 0$ such that

\begin{equation} |\lambda| \ = \ {t \over 2} \| \vec{n} \|^2 \, + \, \vec{b} \cdot \vec{n} \end{equation}

where $\vec{b} = \big(0 ,1 , \dots, t-1 \big)$. The trick might be to express the quantity $\nu_{\scriptscriptstyle T}(\lambda)$ in terms of the coordinates of the corresponding GKS-vector $\vec{n}$.

Consider the case of $3$-cores: If my understanding of O. Brunat and R. Nath's *pointed abacus* construction is correct (see https://arxiv.org/pdf/2101.01512.pdf) a $3$-core partition $\lambda$
with GKS-vector $\vec{n}= \big(n_0, n_1, n_2 \big)$ has an *arm*
of length $3p + r$ with residue $0 \leq r \leq 2$
if and only if $n_r$ is positive and $0 \leq p \leq n_r - 1$. Likewise $\lambda$ will have a *leg* of length $3p + 2 - r$ with $0 \leq r \leq 2$ if and only if $n_r$ is negative and $0 \leq p \leq | n_r | - 1 $.
As mentioned in the comments

\begin{equation} \begin{array}{ll} \nu_{\scriptscriptstyle T}(\lambda) &\displaystyle = \ {1 \over 2} \, \sum_{j=1}^k \, \Big(a_j + {1 \over 2} \Big)^2 - \Big(b_j + {1 \over 2} \Big)^2 \\ &\displaystyle = \ \sum_{j=1}^k \, a_j + {1 \over 2} a_j^2 \ - \ \sum_{j=1}^k b_j + {1 \over 2} b_j^2 \end{array} \end{equation}

where $a_j$ and $b_j$ are the respective $j$-th arm and length lengths of the partition $\lambda$. So it should be possible to write $H_3(x \, ; q)$ as a piecewise polynomial function of the GKS-coordinates $n_0$, $n_1$, $n_2$. As illustration consider the situation where $n_0 < 0$ and $n_1 \geq -n_0$ and $n_2 = -n_0 - n_1 \leq 0$ which is one of the of six possible sign configurations of the three GKS-coordinates $n_0$, $n_1$, and $n_2$. By the Brunat-Nath recipe only $n_1$ will contribute arm lengths while $n_0$ and $n_2$ will contribute leg lengths. The arm contribution to $\nu_{\scriptscriptstyle T}(\lambda)$ will be

\begin{equation} \begin{array}{ll} \displaystyle \sum_{j=1}^k a_j + {1 \over 2}a_j^2 &\displaystyle = \ \sum_{p=0}^{n_1 - 1} \, (3p+1) + {1 \over 2}(3p+1)^2 \\ &\displaystyle = \ {1 \over 2} 3n_1 \, + \, 3n_1(n_1-1) \, + \, {3 \over 4}n_1(n_1-1)(2n_1-1) \end{array} \end{equation}

while the leg contribution to $\nu_{\scriptscriptstyle T}(\lambda)$ will be

\begin{equation} \begin{array}{l} \displaystyle \sum_{j=1}^k b_j + {1 \over 2}b_j^2 \ \displaystyle = \ \left\{ \begin{array}{c} \displaystyle \sum_{p=0}^{|n_0| -1} \, (3p + 2) + {1 \over 2}(3p+ 2)^2 \\ + \\ \displaystyle \sum_{p=0}^{|n_2| -1} \, (3p) + {1 \over 2}(3p)^2 \end{array} \right. \\ = \, \left\{ \begin{array}{c} \displaystyle -4n_0 \, + \, {9 \over 2}n_0(n_0+1) \, - \, {3 \over 4}n_0(n_0+1)(2n_0+1) \\ + \\ \displaystyle {3 \over 2}n_2(n_2+1) \, - \, {3 \over 4}n_2(n_2+1)(2n_2+1) \end{array} \right. \\ = \, \left\{ \begin{array}{c} \displaystyle -4n_0 \, + \, {9 \over 2}n_0(n_0+1) \, - \, {3 \over 4}n_0(n_0+1)(2n_0+1) \\ + \\ \displaystyle -{3 \over 2}(n_0+n_1)(1-n_0-n_1) \, + \, {3 \over 4}(n_0+n_1)(1-n_0 -n_1)(1-2n_0 - 2n_1) \end{array} \right. \\ \end{array} \end{equation}

Taking the difference of the arm and leg contributions we get the value of $\nu_{\scriptscriptstyle T}(\lambda)$ namely

\begin{equation} \nu_{\scriptscriptstyle T}(\lambda) \ = \ {1 \over 2}\Big(3n_1^2 - 3n_0^2\big(1 + 3n_1\big) + n_0\big(2 + 3n_1 - 9n_1^2 \big) \Big) \end{equation}

So the lattice points of the cone in $\Bbb{Z}^2$ cut out by the inequalities $n_0 < 0$ and $n_1 \geq -n_0$ and $n_2 = -n_0 - n_1 \leq 0$ make the following contribution to $H_3(x \, ; q)$

\begin{equation} \displaystyle \sum_{n_0 < 0} \sum_{n_1 \geq -n_0} \, q^{3n_0^2 + 3n_1^2 + 3n_0n_1 - 2n_0 - n_1} \exp \Big\{ {x \over 2}\Big(3n_1^2 - 3n_0^2\big(1 + 3n_1\big) + n_0\big(2 + 3n_1 - 9n_1^2 \big) \Big) \Big\} \end{equation}

A similiar calculation can be undertaken for the remaining five cones in $\Bbb{Z}^2$. Does anyone recognize this kind of sum?

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