The center of the twisted Heisenberg category, factorial Schur Q-functions, and transition functions on the Schur graph

We establish an isomorphism between the center of the twisted Heisenberg category and the subalgebra Gamma of the symmetric functions generated by odd power sums. We give a graphical description of the factorial Schur Q-functions and inhomogeneous power sums as closed diagrams in the twisted Heisenberg category and show that the bubble generators of the center correspond to two sets of generators of Gamma which encode data related to up/down transition functions on the Schur graph. Finally, we describe an action of the trace of the twisted Heisenberg category, the W-algebra W- subset of W1+infinity, on Gamma.