Gross, Mansour and Tucker introduced the partial-dual orientable genus polynomial and the partial-dual Euler genus polynomial. They computed these two partial-dual genus polynomials of four families of ribbon graphs, posed some research problems and made some conjectures. In this paper, we introduce the notion of signed interlace sequences of bouquets and obtain the partial-dual Euler genus polynomials for all ribbon graphs with the number of edges less than 4 and the partial-dual orientable genus polynomials for all orientable ribbon graphs with the number of edges less than 5 in terms of signed interlace sequences. We check all the conjectures and find a counterexample to the Conjecture 3.1 in their paper: There is no orientable ribbon graph having a non-constant partial-dual genus polynomial with only one non-zero coefficient. Motivated by this counterexample, we further find an infinite family of counterexamples to the conjecture.

Gross，Mansour和Tucker介绍了部分对偶可定向属多项式和部分对偶Euler属多项式。他们计算了四个带状图族的这两个部分对偶属多项式，提出了一些研究问题并做出了一些猜想。在本文中，我们介绍了花束的带符号交织序列的概念，并获得了边数小于4的所有带状图的部分对偶Euler属多项式，以及所有带方向的带状图的部分对偶可定向属多项式就带符号的隔行序列而言，边缘数小于5。我们检查了所有猜想，并在他们的论文中找到了对猜想3.1的反例：没有可定向的带状图，该图具有非恒定的部分对偶属多项式，并且系数只有一个非零。