The equations going from linear index to (i,j)
index are
i = n - 2 - floor(sqrt(-8*k + 4*n*(n-1)-7)/2.0 - 0.5)
j = k + i + 1 - n*(n-1)/2 + (n-i)*((n-i)-1)/2
The inverse operation, from (i,j)
index to linear index is
k = (n*(n-1)/2) - (n-i)*((n-i)-1)/2 + j - i - 1
Verify in Python with:
from numpy import triu_indices, sqrt
n = 10
for k in range(n*(n-1)/2):
i = n - 2 - int(sqrt(-8*k + 4*n*(n-1)-7)/2.0 - 0.5)
j = k + i + 1 - n*(n-1)/2 + (n-i)*((n-i)-1)/2
assert np.triu_indices(n, k=1)[0][k] == i
assert np.triu_indices(n, k=1)[1][k] == j
for i in range(n):
for j in range(i+1, n):
k = (n*(n-1)/2) - (n-i)*((n-i)-1)/2 + j - i - 1
assert triu_indices(n, k=1)[0][k] == i
assert triu_indices(n, k=1)[1][k] == j