Enumeration of Concave Integer Partitions

An integer partition λ ` n corresponds, via its Ferrers diagram, to an artinian monomial ideal I ⊂ C[x, y] with dimC C[x, y]/I = n. If λ corresponds to an integrally closed ideal we call it concave . We study generating functions for the number of concave partitions, unrestricted or with at most r parts. 1. concave partitions By an integer partition λ = (λ1, λ2, λ3, . . . ) we mean a weakly decreasing sequence of nonnegative integers, all but finitely many of which are zero. The non-zero elements are called the parts of the partition. When writing a partition, we often will only write the parts; thus (2, 1, 1, 0, 0, 0, . . . ) may be written as (2, 1, 1). We write r = 〈λ〉 for the number of parts of λ, and n = |λ| = ∑ i λi; equivalently, we write λ ` n if n = |λ|. The set of all partitions is denoted by P , and the set of partitions of n by P(n). We put |P(n)| = p(n). By subscripting any of the above with r we restrict to partitions with at most r parts. We will use the fact that P forms a monoid under component-wise addition. For an integer partition λ ` n we define its Ferrer’s diagram F (λ) = {(i, j) ∈ N i < λj+1 }. In figure 1 the black dots comprise the Ferrer’s diagram of the partition μ = (4, 4, 2, 2). Then F (λ) is a finite order ideal in the partially ordered set (N2,≤), where (a, b) ≤ (c, d) iff a ≤ c and b ≤ d. In fact, integer partitions correspond precisely to finite order ideals in this poset. The complement I(λ) = N \ F (λ) is a monoid ideal in the additive monoid N. Recall that for a monoid ideal I the integral closure Ī is {a `a ∈ I for some ` > 0 } (1) and that I is integrally closed iff it is equal to its integral closure.