The peak and descent statistics over ballot permutations

A ballot permutation is a ? such that in any prefix of the descent number not more than ascent number. By using reversal-concatenation map, we (i) give formula for joint distribution (pk, des) peak and statistics over permutations, (ii) connect this ordinary permutations terms generating functions, (iii) confirm Spiro's conjecture which finds equidistribution statistic an analogue odd order permutations.