The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias, Proudfoot and Wakefield. The properties of these polynomials need to be further explored. In this paper we prove that the Kazhdan-Lusztig polynomials of fan matroids coincide with Motzkin polynomials, which was conjectured by Gedeon. As a byproduct, we determine the Kazhdan-Lusztig polynomials of graphic matroids of squares of paths. We further obtain explicit formulas of the Kazhdan-Lusztig polynomials of wheel matroids and whirl matroids. We prove the real-rootedness of the Kazhdan-Lusztig polynomials of these matroids, thus providing positive evidence for a conjecture due to Gedeon, Proudfoot and Young. Based on the results on the Kazhdan-Lusztig polynomials, we also determine the Z-polynomials of fan matroids, wheel matroids and whirl matroids, and prove their real-rootedness, thus providing further evidence in support of a conjecture of Proudfoot, Xu, and Young.

Elias、Proudfoot 和 Wakefield 介绍了拟阵的 Kazhdan-Lusztig 多项式。这些多项式的性质需要进一步探索。在本文中，我们证明了扇拟阵的 Kazhdan-Lusztig 多项式与 Gedeon 猜想的 Motzkin 多项式一致。作为副产品，我们确定了路径正方形图形拟阵的 Kazhdan-Lusztig 多项式。我们进一步得到了车轮拟阵和回旋拟阵的 Kazhdan-Lusztig 多项式的显式公式。我们证明了这些拟阵的 Kazhdan-Lusztig 多项式的实根性，从而为 Gedeon、Proudfoot 和 Young 的猜想提供了积极的证据。基于 Kazhdan-Lusztig 多项式的结果，我们还确定了Z-扇拟阵、轮拟阵和回旋拟阵的多项式，并证明它们的实根性，从而为Proudfoot、Xu和Young的猜想提供进一步的证据。