Let $\lambda$ be the partition of integer $d$. The Frobenius coordinate of $\lambda$ is given
$$ (a\_1 ,\ldots,a\_{d(\lambda)}|b\_1,\ldots,b\_{d(\lambda)}),$$
where $d(\lambda)$ denote the diagonal of $\lambda$. 
Let $a\_i'= a\_i +\frac12$ and $b\_i'= b\_i +\frac12$ are modified Frobenius coordinates.
By a classical theorem of Frobenius, it states that 


$f(\lambda)=\frac{|C\_{2,1,\ldots,1}|}{dim \lambda}\chi\_{2,1,\ldots,1}^{\lambda}=\frac12\sum\_{i=1}^{d(\lambda)}(a\_i')^2-(b\_i')^2$


Notice that the above example we treat the L.H.S as a function of $\lambda$ as we fix the partition of the form $2,1^{d-2}$


If there is any generalisation of the above theorem? That is if we replace 
$(2,1,\ldots,1)$ with any other partition of $d$ then say $\mu$ then what will be 


$$\frac{|C\_{2^{d/2}}|}{dim \lambda}\chi\_{2^{d/2}}^{\lambda}$$
For example say $\mu$ is of the form $2^{d/2}$


in Frobenius coordinate? If it is already studied can anyone please site the reference. 



Frobenius coordinate expansion of character
There is a generalisation of the character formula, although it is usually stated in terms of contents rather than Frobenius coordinates. Also, it applies to conjugacy classes labelled by partitions $(1^{m\_1} 2^{m\_2} \cdots)$, where $m\_2, m\_3, \ldots$ are fixed and $m\_1$ varies with $n$ (as in the example you gave, where there is one part of size $2$, and $n-2$ parts of size 1). So in particular, it does not apply directly to the partitions $(2^{n/2})$.


For the benefit of the uninitiated, let me state right away that $\frac{|C\_\mu| \chi\_{\mu}^\lambda}{dim(\lambda)}$ is a *central character*. The sum of all elements of $S\_n$ of cycle type $\mu$ is a central element of the group algebra of $S\_n$, and the central character is the scalar by which the central element acts on the irreducible representation labelled by $\lambda$. Central characters are important in modular representation theory, which might provide a little motivation for why the quantity in question is a natural thing to consider.


Suppose that we have a partition $\lambda = (\lambda\_1, \ldots, \lambda\_l)$. Suppose that $d$ is the largest integer such that $\lambda\_d \geq d$ (the *Durfee size* of $\lambda$). Then the Frobenius coordinates of $\lambda$ are $(\alpha\_1, \ldots, \alpha\_d | \beta\_1, \ldots, \beta\_d)$, where $\alpha\_i = \lambda\_i - i$ and $\beta\_i = \lambda\_i^\prime - i$ (here $\lambda^\prime$ is the transpose partition of $\lambda$). So the $\alpha\_i$ and $\beta\_i$ count the number of boxes in past the diagonal in the $i$-th row and $i$-th column respectively (and the number of each is $d$). On the other hand, the contents of a box in row $j$ and column $i$ is $i-j$. It is not difficult to see that the multiset of contents of $\lambda$ consists of $0$ with multiplicity $d$, and $\{1,2,\ldots, \alpha\_i\}$ for each $i$ together with $\{-1, -2, \ldots, -\beta\_i\}$ for each $i$. So in particular, the sum of $r$-th powers of the contents of $\lambda$ can be expressed as polynomials in the $\alpha\_i$ and $\beta\_i$ by [Faulhaber's Formula](https://en.wikipedia.org/wiki/Faulhaber%27s_formula). For example, the sum of the contents ($r=1$) is


$$\sum\_i (1 + 2 + \cdots + \alpha\_i) - (1 + 2 + \cdots + \beta\_i) = \sum\_i {\alpha\_i + 1 \choose 2} - {\beta\_i + 1 \choose 2},$$


which is precisely the quantity appearing in the example you gave. We write $cont(\lambda)$ for the multiset of contents of $\lambda$.


Now, for a partition $\mu = (\mu\_1, \mu\_2, \cdots, \mu\_k)$, let $\bar{\mu} = (\mu\_1 - 1, \mu\_2 - 1, \ldots, \mu\_k - 1)$ be the partition obtained from $\mu$ by subtracting 1 from each nonzero part of $\mu$ (and ignoring trailing zeros). For example, if $\mu = (2,1, \ldots, 1)$ as in your example, then $\bar{\mu} = (1)$.


**Theorem** There exists a family $f\_{\rho}$ of elements of $\mathbb{Q}[n] \otimes \Lambda$ (where $n$ is a polynomial variable) and $\Lambda$ is the ring of symmetric functions with the property that


$$\frac{|C\_\mu| \chi\_{\mu}^\lambda}{dim(\lambda)} = f\_{\bar{\mu}}(|\lambda|, cont(\lambda))$$


By this notation I mean that we evaluate $f\_{\bar{\mu}}$ at $n=|\lambda|$ and its symmetric function variables at $cont(\lambda)$.


It turns out that $f\_{(1)} = p\_1$ (the first power-sum symmetric function) which doesn't actually depend on $n$. Then putting this together, we see


$$\frac{|C\_{(2,1,\ldots,1)}| \chi\_{(2,1,\ldots,1)}^\lambda}{dim(\lambda)} = p\_1(cont(\lambda)) = \sum\_i {\alpha\_i + 1 \choose 2} - {\beta\_i + 1 \choose 2},$$


which recovers the result you stated. In general, you can take $f\_{\bar{\mu}}$, express it as a polynomial in $n$ and the power-sums $p\_1, p\_2, \ldots \in \Lambda$, and upon evaluating at the contents of $\lambda$ you will obtain a polynomial in $n$ and the Frobenius coordinates of $\lambda$. For example,


$$f\_{(2)} = p\_2 - {n \choose 2},$$


which tells us that the central character corresponding to the conjugacy class $(3,1,\ldots, 1)$ may be computed using the formula


$$\left( \sum\_{i} \frac{\alpha\_i(\alpha\_i + 1)(2\alpha\_i + 1)}{6} + \frac{\beta\_i (\beta\_i + 1)(2\beta\_i + 1)}{6} \right) - {|\lambda| \choose 2}.$$


There is a lot more to be said about the $f\_\rho$. For example, they form a $\mathcal{R}$-basis of $\mathcal{R} \otimes \Lambda$, wher $\mathcal{R}$ is the subring of $\mathbb{Q}[n]$ consisting of integer-valued polynomials. But in this post I worked over $\mathbb{Q}$ because I wanted to use power-sum symmetric functions to turn contents-evaluaion into polynomials in Frobenius coordinates.


References you might find helpful (each of which contains an explanation of the above theorem):




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*Corteel, Sylvie; Goupil, Alain; Schaeffer, Gilles*, [**Content evaluation and class symmetric functions**](http://dx.doi.org/10.1016/j.aim.2003.09.010), Adv. Math. 188, No. 2, 315-336 (2004). [ZBL1059.05104](https://zbmath.org/?q=an:1059.05104).




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Section 5.4 of


*Ceccherini-Silberstein, Tullio; Scarabotti, Fabio; Tolli, Filippo*, Representation theory of the symmetric groups. The Okounkov-Vershik approach, character formulas, and partition algebras., Cambridge Studies in Advanced Mathematics 121. Cambridge: Cambridge University Press (ISBN 978-0-521-11817-0/hbk). xv, 412 p. (2010). [ZBL1230.20002](https://zbmath.org/?q=an:1230.20002).


(although this is based on the material of the above paper)




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Section 3 of my own paper (which doesn't appear in the citation engine, so I'm providing an arXiv link instead) [Stable Centres I: Wreath Products](https://arxiv.org/abs/2107.03752)


