Pfaffian structures and certain solutions to BKP hierarchies I. Sums over partitions

Abstract: We introduce a useful and rather simple class of BKP tau functions which which we shall call "easy tau functions". We consider two versions of BKP hierarchy, one we will call "small BKP hierarchy" (sBKP) related to $O(\infty)$ introduced in Date et al and "large BKP hierarchy" (lBKP) related to $O(2\infty +1)$ introduced in Kac and van de Leur (which is closely related to the large $O(2\infty)$ DKP hierarchy (lDKP) introduced in Jimbo and Miwa). Actually "easy tau functions" of the sBKP hierarchy were already considered in Harnad et al, here we are more interested in the lBKP case and also the mixed small-large BKP tau functions (Kac and van de Leur). Tau functions under consideration are equal to certain sums over partitions and to certain multi-integrals over cone domains. In this way they may be applicable in models of random partitions and models of random matrices. Here is the first part of the paper where sums of Schur and projective Schur functions over partitions are considered.