1 research outputs found
The peak and descent statistics over ballot permutations
A ballot permutation is a permutation such that in any prefix of
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.Comment: 16 page