SIAM Journal on Discrete Mathematics, Volume 35, Issue 1, Page 205-233, January 2021.
We introduce a class of posets, which includes both ribbon posets (skew shapes) and $d$-complete posets, such that their number of linear extensions is given by a determinant of a matrix whose entries are products of hook lengths. We also give $q$-analogues of this determinantal formula in terms of the major index and inversion statistics. As applications, we give families of tree posets whose numbers of linear extensions are given by generalizations of Euler numbers, we draw relations to Naruse and Okada's positive formulas for the number of linear extensions of skew $d$-complete posets, and we give polynomiality results analogous to those of descent polynomials by Diaz-López, Harris, Insko, Omar, and Sagan.

中文翻译：

---

用钩子长度的行列式计算位姿的线性扩展

SIAM Journal on Discrete Mathematics，第 35 卷，第 1 期，第 205-233 页，2021 年 1 月
。扩展由矩阵的行列式给出，该矩阵的条目是钩子长度的乘积。我们还根据主指数和反演统计量给出了这个行列式公式的 $q$-类似物。作为应用，我们给出了一系列树偏序集，其线性扩展的数量由欧拉数的推广给出，我们绘制了与成濑和冈田关于偏斜$d$-完全偏序集的线性扩展的数量的正公式的关系，并且我们给出了多项式结果类似于 Diaz-López、Harris、Insko、Omar 和 Sagan 的下降多项式。