Séminaire Lotharingien de Combinatoire, B75c (2016), 6 pp.
Amitai Regev and Doron Zeilberger
Between Sums-Of-Squares of Characters of the Symmetric Group
Over Two-Rowed Shapes and Over Hook Shapes
In a recent article, we noted (and proved) that the sum of the squares
of the characters of the symmetric group,
over all shapes \lambda with two rows and n cells and
\mu = 31n-3, equals, surprisingly, to
1/2 of that sum-of-squares taken over all hook shapes with n+2
cells and with \mu = 321n-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,...,2t-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,...,2t-1 by 2t.
Received: October 20, 2015.
Accepted: February 2, 2016.
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