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Surprising Relations Between Sums-Of-Squares of Characters of the Symmetric Group Over Two-Rowed Shapes and Over Hook Shapes

In a recent article (arXiv:1507.03499) (joint with Alon Regev) we studied sums of squares of characters Chi(L,M) of the Symmetric Group over shapes L that are two-rowed, and shapes L that are hook shapes, and M is an arbitrary shape that mostly consists of ones, and designed algorithms for closed-form evaluations of each of these. We noted (and proved) that when M is the shape with n cells consisting of 3 followed by n-3 ones, the former sum equals one half time the analogous sum over hook shapes with n+2 cells and M is the partition consisting of 3,2, followed by n-3 ones. Here we show that this is just a tip of an iceberg, and prove (alas, by purely human means) that the former sum with M consisting of all odd parts, and (possibly) a consecutive string of powers of 2, starting at 2, equals one half of the latter sum where M is replaced by a partition where all the odd parts are retained but the consecutive string of powers of 2: 2,4, ..., $2^{t-1}$ is replaced by $2^t$.