Open Journal of Discrete Mathematics, 2011, 1, 108-115 doi:10.4236/ojdm.2011.13014 Published Online October 2011 (http://www.SciRP.org/journal/ojdm) Copyright © 2011 SciRes. OJDM Bijections between Lattice Paths and Plane Partitions Mateus Alegri 1 , Eduardo Henrique de Mattos Brietzke 2 , José Plínio de Oliveira Santos 3* , Robson da Silva 4 1 Departamento de Matemática, UFS, Campus Itabaiana, Itabaiana, Brazil 2 Instituto de Matemática, UFRGS, Porto Alegre, Brazil 3 IMECC-UNICAMP, Campinas, Brazil 4 ICE-UNIFEI, Itajubá, Brazil E-mail: * josepli@ime.unicamp.br Received July 4, 2011; revised August 5, 2011; accepted August 28, 2011 Abstract By using lattice paths in the three-dimensional space we obtain bijectively an interpretation for the overparti- tions of a positive integer n in terms of a set of plane partitions of n. We also exhibit two bijections between unrestricted partitions of n and different subsets of plane partitions of n. Keywords: Lattice Paths, Plane Partitions, Partitions, q-Analog 1. Introduction In [1], one of us, Santos, in a joint work with Mondek and Ribeiro, introduced a new combinatorial interpreta- tion for partitions in terms of two-line matrices. In that paper a new way of representing, as two-line matrices, unrestricted partitions and several identities from Slater’s list [2] including Rogers-Ramanujan Identities, and Lebesgue’s Partition Identity is described. In [3] we were able to provide a number of bijective proofs for several identities based on the two-line matrix representation, including a new bijective proof for the Lebesgue Identity, as well as a combinatorial proof for an identity related to three-quadrant Ferrers graphs, given by Andrews (see [4]). The possibility of associating a partition of n to a two-line matrix and the two-line matrix to a lattice path from the origin to the line x + y – n was mentioned in [1]. This construction can be done in many different ways. Each one of those provides us with a family of polyno- mials that are, each one, a q-analog for the partition func- tion. In order to construct these families it was employed the same idea of counting the lattice path according to the area limited by the path and the x-axis, as done by Pólya. Our main goal in the present paper is to associate a three-line matrix to a path in the 3-dimensional space, in order to build a volume, which is going to correspond to a plane partition. In this way it is possible to construct bijections from certain classes of plane partitions to overpartitions or unrestricted partitions. In some cases mock theta functions appear as the generating functions for these matrices (see [5]). 2. Background In this section we remember a few definitions and state a theorem that is the base for constructing the bijections. Definition 2.1 A partition of a positive integer n is a collection of positive integers 1 2 s        such that 1 2 = s n        . Each i  is called a part of the partition. Definition 2.2 An overpartition is a partition where the first occurrence of a part can be overlined. Example 2.1 There are 14 overpartitions of 4: 4,4,3 1,3 1,3 1, 3 1, 2 2,2 2, 2 1 1, 2 1 1,2 1 1,2 1 1,1 1 1 1,1 1 1 1                     Definition 2.3 A Plane partition π is a left-justified array of positive integers   , , 1 π ij ij  such that , π ij    , 1 1, max π , π ij i j   . We say that π is a plane partition of n if , , 1 = π ij ij n   . Now, if we stack , π ij cubes over the position , ij of each part of a plane partition π , we obtain a geomet- rical figure in the first octant of the 3-dimensional space, which represents π in the same way a Ferrers diagram represents a partition. Example 2.2 Below (shown in Figures 1 and 2) we have a plane partitions of 43 and its standard geometri- cal representation.