2020/09/13 by David G. L. Wang, Wang, David G. L., Tongyuan Zhao +1
Agricultural and Biological Sciences · Computer Science · Mathematics · #05A05 #05A15 #05A19 #11B37 #Advanced Combinatorial Mathematics #Bayesian Methods and Mixture Models #Botanical Research and Chemistry #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2009.05973
openalex publication_date 2020/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.