Computations And Arithmetic Properties Of P-partition Functions

P-partition is defined as a mapping that assigns nonnegative integral entries to the elements of a poset, which is one of the most important objects in enumerative com-binatorics. As a generalization of the ordinary partition function,P-partition function, the counting function of P-partitions, has many applications in symmetric functions, representation theory, linear diophantine equations and inequalities.The main results of this thesis consist of some progress in P-partition functions, including a recursive method for computing the generating functions of P-partitions, and the arithmetic properties of two classes of P-partitions:the broken2-diamond par-titions and multipartitions.This thesis is organized as follows. The first chapter is devoted to an introduction of P-partitions, We first give a brief review of the theory and present the important special cases of P-partitions. Then we recall several classical methods for computing generating functions of P-partitions. Meanwhile, we display some important classical Ramanujan-type congruences of P-partitions and related background.In Chapter2, we present an approach to compute the generating function fP(X) of P-partitions for a given poset P. To this end, we first introduce two kinds of transforma-tions on posets, which are named as deletion and partially linear extension, respectively. We show that the generating function of P-partition of a poset can be expressed in terms of corresponding generating functions of its transformations. In fact, one can compute fp(X) by recursively applying these two transformations. As an application, we consid-er the partially ordinal sum Pn of n copies of a given poset, which generalizes both the direct sum and the ordinal sum. We constructively prove that the sequence{fPn(x)}n≥1satisfies a finite system of recurrence relations with respect to n. Finally, we illustrate our method by giving several examples, including the well-known multi-cube posets and a kind of3-rowed posets which cannot be dealt with by Souza’s "five guidelines"In Chapter3, we focus on the arithmetic properties of the broken/r-diamond par-titions and multipartitions. By constructing appropriate cta-quotients and applying S- turm’s theorem, we discuss the Ramanujan-type congruences of the broken k-diamond partitions. Specially, we give new proofs of six Ramanujan-type congruences of the broken2-diamond partition. Then, we study the arithmetic properties of multipartition-s in the framework of modular forms. Let pr(n) denote the number of r-component multipartitions of n, and let Sγ,λ be the space spanned by η(24z)γφ (24z), where η(z) is the Dedekind’s eta function and φ(z) is a holomorphic modular form in Mλ,(SL2(Z)). We show that the generating function of Pr((mkn+r)/24) with respect to n is congru-ent to a function in the space Sγ,λ modulo mk. As special cases, this relation leads to many well-known congruences, including the Ramanujan congruences of p(n) modu-lo5,7,11, Gandhi’s congruences of P2(n) modulo5and of p8(n) modulo11, as well as a series of new congruences of multipartitions. Furthermore, using the invariance property of Sγ,λ under the Hecke operator T(?)2, we obtain two classes of congruences pertaining to the mk-adic property of pr(n). In the end, we establish an interesting connection between Ramanujan’s conjecture and Newman’s conjecture for multiparti-tions.