(representation theoretic) meaning of sum over even rows of a Young tableau

Question

Think of a Young tableau $R$ as composed by $d$ rows with number of elements $\mu_i:=\mu_i^R$ $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d > \mu_{d+1}=0$ (and $\mu_i =0\, \forall i >d$) and define $\ell(R):=\sum_{i=1}^d \mu_i$. I'd like to know whether there's any representation theoretic meaning, or anything along the lines of the book Symmertic Functions and Hall Polynomials by Macdonald, for the quantities $p$ and $c$, as defined by $$ p(R) = \sum_i \mu_{2i} = (\ell -c)/2$$

Answer 1

For any partition $\lambda $ $\vdash $ n, the RSK (self-) correspondence of any of its tableaux T($\lambda $)*T($\lambda $) produces an involution with cycle structure $\mu $ $\vdash $ n, with transpose $\mu $' = $\{$a,b$\}$, a+b=n

and a=$\sum$ $\lambda _{2 k-1}$ and b= $\sum $$\lambda _{2 k}$

So, the sum over even-indexed parts equals the number of 2-cycles in $\mu$.