Further common properties of the partially ordered sets R(m, n)={(i, j) ∈ ℤ2: 1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (a rectangle) and T(m, n) = {(i, j)∈ ℤ2: 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).
[1]
Marcel Paul Schützenberger,et al.
La correspondance de Robinson
,
1977
.
[2]
Robert A. Proctor.
Shifted plane partitions of trapezoidal shape
,
1983
.
[3]
I. G. MacDonald,et al.
Symmetric functions and Hall polynomials
,
1979
.
[4]
Dale Raymond Worley,et al.
A theory of shifted Young tableaux
,
1984
.
[5]
RICHARD P. STANLEY,et al.
On the Number of Reduced Decompositions of Elements of Coxeter Groups
,
1984,
Eur. J. Comb..
[6]
R. Stanley,et al.
Combinatorial reciprocity theorems
,
1974
.
[7]
Richard P. Stanley,et al.
Symmetries of plane partitions
,
1986,
J. Comb. Theory A.