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تسجيل مستخدم جديدA combinatorial formula for the nabla operator
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We present an LLT-type formula for a general power of the nabla operator applied to the Cauchy product for the modified Macdonald polynomials, and use it to deduce a new proof of the generalized shuffle theorem describing $ abla^k e_n$, and the Elias-Hogancamp formula for $( abla^k p_1^n,e_n)$ as corollaries. We give a direct proof of the theorem by verifying that the LLT expansion satisfies the defining properties of $ abla^k$, such as triangularity in the dominance order, as well as a geometric proof based on a method for counting bundles on $mathbb{P}^1$ due to the second author. These formulas are related to an affine paving of the type A unramified affine Springer fiber studied by Goresky, Kottwitz, and MacPherson, and also to Stanleys chromatic symmetric functions.
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