Counting signed permutations by their alternating runs

The purpose of this paper is to establish a connection between alternating runs of signed permutations in the hyperoctahedral group and left peaks of permutations in the symmetric group, and then to study some associated enumerative polynomials of signed permutations. Properties of these enumerative polynomials, including combinatorial formulas and recurrence relations are studied. In particular, we deduce a convolution formula connecting the number of alternating runs of permutations in the symmetric group to that in the hyperoctahedral group.