Statistics on pairs of permutations Original Research Article

Carlitz et al. (1976) were the first to consider statistics on pairs of permutations. Their primary achievement was to count permutation pairs (unrestricted) by common rises. We extend their work by giving recurrence relationships for permutation pairs (unrestricted and restricted) by common rises, inversion number of each, and a new statistic we call the common rise index. The generating functions that arise are expressed in terms of relatives of Jacksonʹs basic Bessel functions. We also indicate how our results for permutation pairs extend to k-tuples of permutations.