The Catalan Case of Armstrong's Conjecture on Simultaneous Core Partitions

Let λ = (λ1, λ2, . . . , λm) be a partition of size n, i.e., the λi are weakly decreasing positive integers summing to n. We can represent λ by means of its Young (or Ferrers) diagram, which consists of a collection of left-justified rows where row i contains λi cells. To each of these cells B one associates its hook length, that is, the number of cells in the Young diagram of λ that are directly to the right or below B (including B itself). Figure 1 represents the Young diagram of the partition λ = (5, 3, 3, 2) of size 13; the number inside each cell represents its hook length. Let s be a positive integer. We say that λ is an s-core if λ has no hook of length equal to s (or equivalently, equal to a multiple of s). For instance, from Figure 1 we can see that λ = (5, 3, 3, 2) is an s-core for s = 6 and for all s ≥ 9. Finally, λ is an (s, t)-core if it is simultaneously an s-core and a t-core. The theory of (s, t)-cores has been the focus of much interesting research in recent years (see [5, 6, 8] for some of the main results). In particular, when s and t are coprime, there exists only a finite number of (s, t)-core partitions. In fact, there are exactly ( s+t s )