Trapezoidal chains and antichains
Further common properties of the partially ordered sets \(R(m,n)=\{(i,j)\in {\mathbb{Z}}^ 2:\) \(1\leq i\leq m\), \(1\leq j\leq n\}\) (a rectangle) and \(T(m,n)=\{(i,j)\in {\mathbb{Z}}^ 2:\) \(1\leq i\leq m\), \(i\leq j\leq m+n-i\}\) (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, theorem, and bijections are also given for the multichains of R(m,n) and T(m,n).
scientific article; zbMATH DE number 4043900