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Séminaire Lotharingien de Combinatoire, B75c (2016), 6 pp.

# Amitai Regev and Doron Zeilberger

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

**Abstract.**
In a recent article, we noted (and proved) that the sum of the squares
of the characters of the symmetric group,
*\chi*^{\lambda}(*\mu*),
over all shapes *\lambda* with two rows and *n* cells and
*\mu* = 31^{n-3}, equals, surprisingly, to
1/2 of that sum-of-squares taken over all hook shapes with *n*+2
cells and with *\mu* = 321^{n-3}.
In the present note, we show that this is only the tip of a huge
iceberg! We will prove that, if
$\mu$ consists of odd parts and (a possibly empty) string of
*consecutive* powers of 2, namely 2,4,...,2^{t-1}
for *t* >= 1, then
the sum of *\chi*^{\lambda}(*\mu*)^{2}
over all two-rowed shapes
*\lambda* with *n* cells equals exactly 1/2 times
the analogous sum of
*\chi*^{\lambda}(*\mu'*)^{2} over all shapes
*\lambda* of *hook shape* with *n*+2 cells,
where *\mu'* is the partition obtained from $\mu$ by retaining all odd
parts but replacing the string
2,4,...,2^{t-1} by 2^{t}.

Received: October 20, 2015.
Accepted: February 2, 2016.

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