On the roots of the Poupard and Kreweras polynomials

Weyl group $q$-Kreweras numbers and cyclic sieving

The paper concerns a definition for $q$-Kreweras numbers for finite Weyl groups $W$, refining the $q$-Catalan numbers for $W$, and arising from work of the second author. We give explicit formulas in all types for the $q$-Kreweras numbers. In the classical types $A, B, C$, we also record formulas for the $q$-Narayana numbers and in the process show that the formulas depend only on the Weyl group (that is, they coincide in types $B$ and $C$). In addition we verify that in the classical types $A,B,C,D$ that the $q$-Kreweras numbers obey the expected cyclic sieving phenomena when evaluated at app…

单位圆上的交错：通过系数倒数

Interlacing on the Unit Circle via Coefficientwise Reciprocals

本文证明具有负实数零点的回文多项式之系数倒数在单位圆上严格交错，基于有限Blaschke乘积，并应用于二项式、欧拉、Narayana等多项式族。

Counting genus one partitions and permutations

We prove the conjecture by M. Yip stating that counting genus one partitions by the number of their elements and parts yields, up to a shift of indices, the same array of numbers as counting genus one rooted hypermonopoles. Our proof involves representing each genus one permutation by a four-colored noncrossing partition. This representation may be selected in a unique way for permutations containing no trivial cycles. The conclusion follows from a general generating function formula that holds for any class of permutations that is closed under the removal and reinsertion of trivial cycles. Ou…