Rim Hook

We consider the partial order on partitions of integers deened by removal of rim hooks of a given length. The isomorphism between this poset and a product of Young's lattices leads to rim hook versions of Schensted correspondences. Analogous results are given for shifted shapes. A shape, or Young diagram, is a nite order ideal of the lattice P 2 = f(k; l) : k; l 1g. Each shape = f(i; j) : 1 j i g can be identiied with the corresponding partition = (1 2 : : :). Shapes form the Young lattice Y (see, e.g., St86]). In drawing the shapes, we use the so-called \English notation;" for example, the partition (5; 3; 1; 1) is represented by the shape : We denote by # the number of boxes in a shape. The i'th diagonal of a shape is the set f(k; l) : l ? k = ig. A rim hook is a contiguous strip of boxes in P 2 , which has at most one box on each diagonal. Throughout the paper a positive integer r is xed, and all the rim hooks contain exactly r boxes, with the exception of Deenition 3.4(3).