Hilbert-Poincaré series of parity binomial edge ideals and permanental ideals of complete graph

Abstract: We give an explicit formula for the Hilbert-Poincar\'{e} series of the parity binomial edge ideal of a complete graph $K_{n}$ or equivalently for the ideal generated by all $2\times 2$-permanents of a $2\times n$-matrix. It follows that the depth and Castelnuovo-Mumford regularity of these ideals are independent of $n$.