Trapezoidal Chains and Antichains

Abstract

Further common properties of the partially ordered sets R ( m, n )={( i, j ) ∈ ℤ 2 : 1 ⩽ i ⩽ m , 1 ⩽ j ⩽ n } (a rectangle) and T ( m, n ) = {( i, j )∈ ℤ 2 : 1 ⩽ i ⩽ j ⩽ m + n - 1} (a trapezoid) are established. Weights are attached to the antichains of R( m, n ) and T( m, n )in such a way that the sum of the weights of the antichains in R( m, n ) and T( m, n ) are the same. A proof is given which exploits connections among antichains in T( m, n ), the theory of Schur functions, and self-complementary tableaux. A bijection is given between the antichains of R ( m, n ) and T( m, n ) which uses jeu de taquin . Similar weights, theorems, and bijections are also given for the multichains of R( m, n ) and T( m, n ).