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 ΔG is defined via its action on modified Macdonald polynomials by setting ΔGH~μ = G[Bμ], where Bμ is a polynomial in q and t. Previous work by Haglund, Remmel, Wilson conjectures a combinatorial interpretation for Δeken, generalizing the Shuffle Theorem. Here, we prove combinatorial interpretations for Δmλen|q=1 and Δsλen|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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