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The combinatorics of the HMZ operators applied to Schur functions

Jeffrey B. Remmel, Meesue Yoo
2012 Journal of Combinatorics  
Haglund, Morse, and Zabrocki [12] introduced a family of symmetric function operators {B m } m≥1 and {C m } m≥1 which are closely related to operators of Jing [18] . Hanglund, Morse, and Zabrocki used these operators to refine the shuffle conjecture of Haglund, Haiman, Loehr, Remmel and Ulyanov [9] which gives a combinatorial interpretation of the coefficient of the monomial symmetric function in the Frobenius image of the character generating function of the ring of diagonal harmonics. In this
more » ... harmonics. In this paper, we give combinatorial interpretations of the coefficients that arise in Schur function expansion of B m s λ [X] and C m s λ [X] where s λ [X] is the Schur function associated to the partition λ. We then use such combinatorial interpretations to give a new recursion for the Kostka-Foulkes polynomials K λ,μ (q). Dedicated to Adriano Garsia on the occasion of his 84-th birthday. where X = x 1 + x 2 + · · · , Ω[zX] = i 1 1−zxi , and Ω[− zX] = i (1 + zx i ).
doi:10.4310/joc.2012.v3.n3.a6 fatcat:oimf6gqm4vdhdo7poqwpe37cgi