PLANE PARTITIONS AMY BECKER, LILLY BETKE-BRUNSWICK, ANNA ZINK Abstract. Throughout our study of the enumeration of plane partitions we make use of bijective proofs to find generating functions. In particular, we con- sider bounded plane partitions, symmetric plane partitions and weak reverse plane partitions. Using the combinatorial interpretations of Schur functions in relation to semistandard Young tableaux, we rely on the properties of sym- metric functions. In our paper we will walk through some of these bijections and present the corresponding generating functions. 1. Introduction In order to understand and interpret plane partitions, it is important to have some background knowledge of various concepts. We will be reviewing the ba- sic ideas behind partitions, semistandard Young tableaux (SSYT), and symmetric functions{ specifically Schur functions{ to develop a strong base for the work we will be doing with plane partitions. We begin with the partition, the more com- monly seen 2-D version of a plane partition. Definition 1.1. For any nonnegative integer n, a partition λ of n is a nonin- 1 P1 creasing sequence fλjgj=1 of nonnegative integers such that j=1 λj = n. If λ is a partition of n then we often write jλj = n. We represent these partitions visually with what we call Ferrers diagrams. There are various styles of Ferrers diagrams, but we will be using the English version, which is left-aligned and stacked from top to bottom. λ1 λ2 λ3 Figure 1: Ferrers diagrams for a few partitions of n=5: λ1 = 4; 1, λ2 = 3; 2, and λ3 = 2; 2; 1. Now, given a partition λ, we can fill it in with entries of numbers or even variables. A semistandard Young tableau is one type of filling of the Ferrers diagram. Definition 1.2. For any partition λ, a semistandard Young tableau (or SSYT) of shape λ is a filling of the Ferrers diagram of λ so that columns are strictly increasing from top to bottom and rows are weakly increasing from left to right. 1 2 AMY BECKER, LILLY BETKE-BRUNSWICK, ANNA ZINK 1 2 2 3 1 2 3 4 2 4 3 4 4 4 Figure 2: Two fillings of the Ferrers diagram of λ = 4; 2; 1 with entries f1; 2; 3; 4g. Given the Ferrers diagram of partition λ, we can see from our above example that there are various ways to fill it to create semistandard Young tableaux given a set of entries. We will let SSYT(λ) denote the set of all such fillings of λ with entries f1; 2; :::; ng. Within SSYT(λ), we can think of each filling or SSYT as cor- responding to a term in a polynomial. In order to create the term, take a SSYT and let each entry i represent the variable xi. Then, multiplying these variables together we let the product be the resulting term. 1 2 x1 x2 2 3 x2 x3 3 x 2 2 =) 3 =) x1x2x3 Figure 3: How to convert a SSYT into a term in a polynomial. The resulting polynomial is the Schur polynomial, sλ(x1; x2; :::; xn), where λ is the partition we are filling in with entries from fx1; x2; :::; xng. Schur polynomials are in fact symmetric functions. Recall: Definition 1.3. A function f(x1; x2; :::; xn) is symmetric whenever it does not change under any permutation of its variables. The Schur function sλ(x1; x2; :::; xn) is a generating function for SSYT(λ). If we look at a term in the Schur function, the coefficient represents the number of ways to fill the partition λ to create a SSYT with the variables in the term. For example if a term in s2;1(x1; x2; :::; xn) is 2x1x2x3 then there are 2 ways to fill in the partition λ = 2; 1 with x1; x2; and x3. We have seen that one way to fill in a Ferrers diagram is with entries that are weakly increasing across rows and strictly increasing down colums. If instead we fill the Ferrers diagram so that the rows are weakly decreasing and the columns strictly decreasing this is referred to as a reverse semistandard Young tableau. A third way to fill in the Ferrers diagram is with weakly decreasing rows and columns. Such a filling is a plane partition. We can think of the numbers in the filling as repre- senting the heights for stacks of blocks placed on each cell of the diagram. PLANE PARTITIONS 3 4 4 3 1 4 3 2 1 3 1 2 1 ) Figure 4: Filling of a Ferrers diagram associated with a plane partition, π More formally: Definition 1.4. A plane partition is an array π = (πij)i;j≥1 of nonnegative integers such that π has finitely many nonzero entries and is weakly decreasing in rows and columns. Note that although a plane partition is an infinite array with finitely many nonzero entries, when writing it we don't include the zero parts{ they are implied. When discussing plane partitions, there are a few important properties that will come up. Definition 1.5. The size of a plane partition, jπj, is the sum of the entries. X jπj = πij: i;j≥1 Informally, we can think of jπj as the total number of blocks in the three- dimensional interpretation of the plane partition. For example, the size of the plane partition in Figure 4 is jπj = 29. Definition 1.6. The shape of a plane partition, sh(π), is the partition whose Ferrers diagram is filled. Definition 1.7. The max of a plane partition, max(π), is the largest part. We can think of max(π) as the height of the tallest stack. This is always the entry of the northwestern-most cell of the diagram. The SSYT and plane partitions represent two different fillings of the Ferrers diagrams. Since sλ is the generating function for SSYT(λ), it would be convenient to relate SSYT to plane partitions in order to make use of Schur functions in our analysis of plane partititions. We will create a correspondence by once more thinking of SSYT as being filled with variables whose subscripts obey the column- strict and row-weak relations. We then let each variable represent a number. In order for this correspondence to result in a plane partition, we let subscript size be inversely related to numeric entry. That is, the variable with smallest subscript equals the largest number, and the variable with the largest subscript equals the smallest number, so that the order is preserved{ in reverse. In our example below we set xi = 10−i. Notice that the resulting plane partition will be column-strict since the SSYT is column-strict. 4 AMY BECKER, LILLY BETKE-BRUNSWICK, ANNA ZINK x1 x1 x2 x3 x5 9 9 8 7 5 x3 x4 x4 x6 7 6 6 4 x4 x6 6 4 x5 x8 5 2 x7 ) 3 2. Plane Partitions Bounded in a Box Now that we have defined plane partitions, it is natural to attempt to enumerate them. Initially, it is convenient to restrict our consideration to a bounded size. We define B(r; c; t) to be the set of all plane partitions with at most r rows and c columns and largest part (height) at most t. We can think of this as restricting our 3D plane partition to an r × c × t box. The following lemma provides a generating function for plane partitions confined to these bounds. Lemma 2.1. r+1 − c X jπj ( 2 ) t+r 3 2 q = q shcr i(q ; :::; q ; q ; q): π⊆B(r;c;t) (Note: hcri is the partition (c; c; c; :::; c) of cr. The Ferrers diagram of hcri is an r × c rectangular grid.) Proof. We establish the following bijection between bounded, column-strict plane partitions with a rectangular shape and ordinary bounded plane partitions. Let µ be a column-strict plane partition of shape hcri with max(µ) ≤ t + r where t, c and r are positive integers. Since each column of µ contains a decreasing list of positive integers, 1 ≤ µr;j < µr−1;j < ::: < µ1;j ≤ t + r. So for all i; j, µi;j > r − i. Let π be the plane partition defined by πi;j = µi;j − (r + 1 − i). When πi;j = 0 we remove that cell from our Ferrers diagram. Consider the following example. − = µ 2 B(r; c; t + r) π 2 B(r; c; t) 9 9 8 7 5 4 4 4 4 4 5 5 4 3 1 7 6 6 4 4 3 3 3 3 3 4 3 3 1 1 6 4 3 2 2 2 2 2 2 2 4 2 1 5 3 1 1 1 1 1 1 1 1 4 2 PLANE PARTITIONS 5 Note that the resulting π is indeed a plane partition. Because we have not changed the relative values across any given row, πi;j − πi;j+1 = µi;j − µi;j+1 ≥ 0, so π too has weakly decreasing rows. Additionally, πi;j −πi+1;j = µi;j −µi+1;j −1 ≥ 0 since µ is column-strict. As such, π is weakly decreasing down columns. Because we have only altered the heights of stacks, sh(π) ⊆ sh(µ). Furthermore, since µi;j ≤ t + r + 1 − i, then πi;j ≤ t for all i; j. Therefore π is in B(r; c; t) The inverse of this correspondence is immediate. We can start with any ordinary 0 0 0 0 plane parition, π , in B(r; c; t) and construct µ such that µi;j = πi;j + (r + 1 − i) for all i and j with 1 ≤ i ≤ r and 1 ≤ j ≤ c.