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Séminaire Lotharingien de Combinatoire, B77c (2017), 40 pp.

# Jonah Blasiak

# Kronecker Coefficients For One Hook Shape

**Abstract.**
We give a positive combinatorial formula for the Kronecker
coefficient *g*_{\lambda \mu(d) \nu}
for any partitions *\lambda*, *\nu* of *n*
and hook shape *\mu*(*d*) := (*n-d*,1^{d}).
Our main tool is Haiman's *mixed insertion*. This is a
generalization of Schensted insertion to *colored words*, words
in the alphabet of barred letters 1^{-}, 2^{-}, ... and
unbarred letters 1, 2, ...
We define the set of *colored Yamanouchi tableaux of content
\lambda and total color d* (CYT_{\lambda,d}) to be the
set of mixed insertion tableaux of colored words *w* with exactly
*d*
barred letters and such that *w*^{blft} is a Yamanouchi word of
content *\lambda*, where *w*^{blft}
is the ordinary word formed from
*w* by shuffling its barred letters to the left and then removing
their bars.
We prove that *g*_{\lambda
\mu(d) \nu}
is equal to the number of CYT_{\lambda,d}
of shape *\nu* with unbarred southwest corner.

Received: December 1, 2016.
Accepted: December 2, 2016.
Final version: December 6, 2016.

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