论文标题
投票排列的峰值和下降统计数据
The peak and descent statistics over ballot permutations
论文作者
论文摘要
投票排列是一个排列$π$，因此，在$π$的任何前缀中，下降数不超过上升数。通过使用逆转串联图，我们为投票排列的峰值和下降统计量的关节分布（PK，DES）提供了一个公式，并将峰，深度和下降统计量的该分布和关节分布（PK，DP，DES）连接到普通置换率上的普通置换功能。作为推论，我们获得了（i）投票排列的峰值统计量的双变量生成函数的几个公式，（ii）投票排列的下降统计量以及（iii）对普通排列的深度统计量。特别是，我们确认了Spiro的猜想，该猜想发现了投票排列的下降统计量的等分分配以及对奇数排列的下降统计量的类似物。
A ballot permutation is a permutation $π$ such that in any prefix of $π$ the descent number is not more than the ascent number. By using a reversal concatenation map, we give a formula for the joint distribution (pk, des) of the peak and descent statistics over ballot permutations, and connect this distribution and the joint distribution (pk, dp, des) of the peak, depth, and descent statistics over ordinary permutations in terms of generating functions. As corollaries, we obtain several formulas for the bivariate generating function for (i) the peak statistic over ballot permutations,(ii) the descent statistic over ballot permutations, and (iii) the depth statistic over ordinary permutations. In particular, we confirm Spiro's conjecture which finds the equidistribution of the descent statistic for ballot permutations and an analogue of the descent statistic for odd order permutations.
