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Séminaire Lotharingien de Combinatoire, 78B.83 (2017), 12 pp.

# Dun Qiu and Jeffrey B. Remmel

# Schur Function Expansions and the Rational Shuffle Conjecture

**Abstract.**
Gorsky and Negut introduced operators *Q*_{m,n}
on symmetric functions
and conjectured that, in the case where *m* and *n* are relatively
prime, the expansion of
*Q*_{m,n}(-1)_{n}
in terms of the fundamental
quasi-symmetric functions are given by polynomials introduced by
Hikita.
Later Bergeron, Garsia, Leven, and Xin extended and refined the
conjectures of Gorsky and Negut to give a combinatorial interpretation
of the coefficients that arise in expansion of
*Q*_{m,n}(-1)_{n} in
terms of the fundamental
quasi-symmetric functions for arbitrary *m* and *n* which
we will call the rational shuffle conjecture. The rational shuffle
conjecture was later proved by Mellit in 2016. The main goal of this
paper is
to study the combinatorics of the coefficients that arise in the Schur
function expansion of
*Q*_{m,n}(-1)_{n}
in the case where *m* or *n*
equals 3.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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