### $(-1)$-enumeration of self-complementary plane partitions.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

We present a CAT (constant amortized time) algorithm for generating those partitions of n that are in the ice pile model${\text{IPM}}_{k}$(n), a generalization of the sand pile model$\text{SPM}$(n). More precisely, for any fixed integer k, we show that the negative lexicographic ordering naturally identifies a tree structure on the lattice ${\text{IPM}}_{k}$(n): this lets us design an algorithm which generates all the ice piles of ${\text{IPM}}_{k}$(n) in amortized time O(1) and in space O($\sqrt{n}$).

Many links exist between ordinary partitions and partitions with parts in the “gaps”. In this paper, we explore combinatorial explanations for some of these links, along with some natural generalizations. In particular, if we let ${p}_{k,m}(j,n)$ be the number of partitions of n into j parts where each part is ≡ k (mod m), 1 ≤ k ≤ m, and we let $p{*}_{k,m}(j,n)$ be the number of partitions of n into j parts where each part is ≡ k (mod m) with parts of size k in the gaps, then $p{*}_{k,m}(j,n)={p}_{k,m}(j,n)$.