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

By Amitai Regev and Doron Zeilberger

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Written: Oct. 20, 2015
Appeared Séminaire Lotharingien de Combinatoire v. 75 (2016), B75c.

In the interesting article it was noted (and proved) that the sum of the squares of the characters of the symmetric group
Σ (χλ(μ) )2
over all shapes λ with two rows and n cells and μ=31n-3 equals, surprisingly, to 1/2 of that sum taken over all hook shapes with n+2 cells and with μ equals 321n-3.

[This was first conjectured, by Alon Regev, using the OEIS.]

But that is only the tip of an iceberg! First we asked Shalosh to search for other such miracles, and then we were able to conjecture that whenever μ consists of only odd parts, then the sum over two-rowed shapes with n cells is equal to 1/2 of the sum over hook shapes with n+2 cells and where 2 is appended to μ. More generally, when μ consists of odd parts and consecutive powers of 2: 2,22, ..., 2r-1, then the corresponding sum over hook-shapes is where μ is replaced by μ' that is obtained from μ by keeping the odd parts and replacing the string of powers of 2 by 2r   .

The original conjecture was made (by Alon Regev) thanks to the OEIS! So, without the OEIS, this paper (like so many other ones) would not have come to be.

Added Oct. 27, 2016: Christine Bessenrodt discovered (and proved) a lovely refinement