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Séminaire Lotharingien de Combinatoire, 80B.77 (2018), 12 pp.

# Angela Hicks and Marino Romero

# Delta Operators at *q*=1 and Polyominoes

**Abstract.**
For a symmetric function *G*, the Delta operator Δ_{G} is defined
via its action on modified Macdonald polynomials by setting
Δ_{G}*H*^{~}_{μ}
= *G*[*B*_{μ}],
where *B*_{μ} is a
polynomial in *q* and *t*. Previous work by Haglund, Remmel, Wilson
conjectures a combinatorial interpretation for Δ_{ek}*e*_{n},
generalizing the Shuffle Theorem. Here, we prove combinatorial
interpretations for
Δ_{mλ}*e*_{n}|_{q=1}
and
Δ_{sλ}*e*_{n}|_{q=1},
expressing each as weighted sum
over (parallelogram) polyominoes in a rectangle, and provide an
explicit combinatorial interpretation for their elementary and Schur
function expansions.

Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.

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