The combinatorics of symmetric functions and permutation enumeration of the hyperoctahedral group Original Research Article

We study permutation enumeration of the hyperoctahedral group Bn through a combinatorial use of the Bn-analogues of symmetric functions, denoted AB. We define an appropriate homomorphism ζ:AB → Q[x] and remarkably, applying this homomorphism to one of the bases of AB produces polynomials which correspond to enumerating Bn with respect to descents. Applying ζ to a second basis corresponds to enumerating a conjugacy class of Bn with respect to a new descent type statistic which arises naturally.