Serie II numero 81 - Dipartimento di Matematica e Informatica

340 G. RESTUCCIA
Proposition 2.2 Let N 2 ≥0 the orthant consisting in all the positive lattice
points (x1,x2). Then
1. The staircase diagram P [m,0] is a complete stair in N 2 ≥0 , between the
points (m, 0) and (0,m) and no step is skipped.
x2
m
m x1
2. The staircase diagram P [i,j] , with i + j = m, 1 ≤ i, j ≤ m − 1, has
the following picture
x2
i
(i,j)
j
m x1
The surface of P [i,j] is a segment of the line x2 = i, starting from the
point (0,i) until the lattice point (i, j). The stair is complete until the
point (0,m) and it has a jump in the lattice point (i, j).
3. The staircase diagram P [0,m] has the line x2 = m as a surface and it
is not finite.
x2
m
Proof: By easy considerations.
We propose a selection procedure only for the staircase diagram P [0,m] .
More precisely, we give:
Definition 2.1 Let P [m,0] be the staircase polytope in the orthant N 2 ≥0 .
Then a selection procedure consists in skipping one or more lattice points
of the complete diagram.
4
x1