matroid #2
- n
- 14
- rank
- 5
- bases
- 891
- α(M)
- 8/7 ≈ 1.1429
Description (matrix) GF(3)
[[1,0,1,2,0,1,2,0,1,2,0,1,2,0], [0,1,1,1,1,0,0,0,0,0,0,0,0,0], [0,1,0,0,0,1,1,1,0,0,0,0,0,0], [0,1,0,0,0,0,0,0,1,1,1,0,0,0], [0,1,0,0,0,0,0,0,0,0,0,1,1,1]]
show all 891 bases
[[0,1,2,5,8], [0,1,3,5,8], [1,2,3,5,8], [0,1,4,5,8], [1,2,4,5,8], [1,3,4,5,8], [0,1,2,6,8], [0,1,3,6,8], [1,2,3,6,8], [0,1,4,6,8], [1,2,4,6,8], [1,3,4,6,8], [1,2,5,6,8], [1,3,5,6,8], [1,4,5,6,8], [0,1,2,7,8], [0,1,3,7,8], [1,2,3,7,8], [0,1,4,7,8], [1,2,4,7,8], [1,3,4,7,8], [1,2,5,7,8], [1,3,5,7,8], [1,4,5,7,8], [1,2,6,7,8], [1,3,6,7,8], [1,4,6,7,8], [0,1,2,5,9], [0,1,3,5,9], [1,2,3,5,9], [0,1,4,5,9], [1,2,4,5,9], [1,3,4,5,9], [0,1,2,6,9], [0,1,3,6,9], [1,2,3,6,9], [0,1,4,6,9], [1,2,4,6,9], [1,3,4,6,9], [1,2,5,6,9], [1,3,5,6,9], [1,4,5,6,9], [0,1,2,7,9], [0,1,3,7,9], [1,2,3,7,9], [0,1,4,7,9], [1,2,4,7,9], [1,3,4,7,9], [1,2,5,7,9], [1,3,5,7,9], [1,4,5,7,9], [1,2,6,7,9], [1,3,6,7,9], [1,4,6,7,9], [1,2,5,8,9], [1,3,5,8,9], [1,4,5,8,9], [1,2,6,8,9], [1,3,6,8,9], [1,4,6,8,9], [1,2,7,8,9], [1,3,7,8,9], [1,4,7,8,9], [0,1,2,5,10], [0,1,3,5,10], [1,2,3,5,10], [0,1,4,5,10], [1,2,4,5,10], [1,3,4,5,10], [0,1,2,6,10], [0,1,3,6,10], [1,2,3,6,10], [0,1,4,6,10], [1,2,4,6,10], [1,3,4,6,10], [1,2,5,6,10], [1,3,5,6,10], [1,4,5,6,10], [0,1,2,7,10], [0,1,3,7,10], [1,2,3,7,10], [0,1,4,7,10], [1,2,4,7,10], [1,3,4,7,10], [1,2,5,7,10], [1,3,5,7,10], [1,4,5,7,10], [1,2,6,7,10], [1,3,6,7,10], [1,4,6,7,10], [1,2,5,8,10], [1,3,5,8,10], [1,4,5,8,10], [1,2,6,8,10], [1,3,6,8,10], [1,4,6,8,10], [1,2,7,8,10], [1,3,7,8,10], [1,4,7,8,10], [1,2,5,9,10], [1,3,5,9,10], [1,4,5,9,10], [1,2,6,9,10], [1,3,6,9,10], [1,4,6,9,10], [1,2,7,9,10], [1,3,7,9,10], [1,4,7,9,10], [0,1,2,5,11], [0,1,3,5,11], [1,2,3,5,11], [0,1,4,5,11], [1,2,4,5,11], [1,3,4,5,11], [0,1,2,6,11], [0,1,3,6,11], [1,2,3,6,11], [0,1,4,6,11], [1,2,4,6,11], [1,3,4,6,11], [1,2,5,6,11], [1,3,5,6,11], [1,4,5,6,11], [0,1,2,7,11], [0,1,3,7,11], [1,2,3,7,11], [0,1,4,7,11], [1,2,4,7,11], [1,3,4,7,11], [1,2,5,7,11], [1,3,5,7,11], [1,4,5,7,11], [1,2,6,7,11], [1,3,6,7,11], [1,4,6,7,11], [0,1,2,8,11], [0,1,3,8,11], [1,2,3,8,11], [0,1,4,8,11], [1,2,4,8,11], [1,3,4,8,11], [0,1,5,8,11], [0,2,5,8,11], [1,2,5,8,11], [0,3,5,8,11], [1,3,5,8,11], [2,3,5,8,11], [0,4,5,8,11], [2,4,5,8,11], [3,4,5,8,11], [0,1,6,8,11], [0,2,6,8,11], [1,2,6,8,11], [0,3,6,8,11], [2,3,6,8,11], [0,4,6,8,11], [1,4,6,8,11], [2,4,6,8,11], [3,4,6,8,11], [1,5,6,8,11], [2,5,6,8,11], [3,5,6,8,11], [4,5,6,8,11], [0,1,7,8,11], [0,2,7,8,11], [0,3,7,8,11], [1,3,7,8,11], [2,3,7,8,11], [0,4,7,8,11], [1,4,7,8,11], [2,4,7,8,11], [3,4,7,8,11], [1,5,7,8,11], [2,5,7,8,11], [3,5,7,8,11], [4,5,7,8,11], [1,6,7,8,11], [2,6,7,8,11], [3,6,7,8,11], [4,6,7,8,11], [0,1,2,9,11], [0,1,3,9,11], [1,2,3,9,11], [0,1,4,9,11], [1,2,4,9,11], [1,3,4,9,11], [0,1,5,9,11], [0,2,5,9,11], [1,2,5,9,11], [0,3,5,9,11], [2,3,5,9,11], [0,4,5,9,11], [1,4,5,9,11], [2,4,5,9,11], [3,4,5,9,11], [0,1,6,9,11], [0,2,6,9,11], [0,3,6,9,11], [1,3,6,9,11], [2,3,6,9,11], [0,4,6,9,11], [1,4,6,9,11], [2,4,6,9,11], [3,4,6,9,11], [1,5,6,9,11], [2,5,6,9,11], [3,5,6,9,11], [4,5,6,9,11], [0,1,7,9,11], [0,2,7,9,11], [1,2,7,9,11], [0,3,7,9,11], [1,3,7,9,11], [2,3,7,9,11], [0,4,7,9,11], [2,4,7,9,11], [3,4,7,9,11], [1,5,7,9,11], [2,5,7,9,11], [3,5,7,9,11], [4,5,7,9,11], [1,6,7,9,11], [2,6,7,9,11], [3,6,7,9,11], [4,6,7,9,11], [1,2,8,9,11], [1,3,8,9,11], [1,4,8,9,11], [1,5,8,9,11], [2,5,8,9,11], [3,5,8,9,11], [4,5,8,9,11], [1,6,8,9,11], [2,6,8,9,11], [3,6,8,9,11], [4,6,8,9,11], [1,7,8,9,11], [2,7,8,9,11], [3,7,8,9,11], [4,7,8,9,11], [0,1,2,10,11], [0,1,3,10,11], [1,2,3,10,11], [0,1,4,10,11], [1,2,4,10,11], [1,3,4,10,11], [0,1,5,10,11], [0,2,5,10,11], [0,3,5,10,11], [1,3,5,10,11], [2,3,5,10,11], [0,4,5,10,11], [1,4,5,10,11], [2,4,5,10,11], [3,4,5,10,11], [0,1,6,10,11], [0,2,6,10,11], [1,2,6,10,11], [0,3,6,10,11], [1,3,6,10,11], [2,3,6,10,11], [0,4,6,10,11], [2,4,6,10,11], [3,4,6,10,11], [1,5,6,10,11], [2,5,6,10,11], [3,5,6,10,11], [4,5,6,10,11], [0,1,7,10,11], [0,2,7,10,11], [1,2,7,10,11], [0,3,7,10,11], [2,3,7,10,11], [0,4,7,10,11], [1,4,7,10,11], [2,4,7,10,11], [3,4,7,10,11], [1,5,7,10,11], [2,5,7,10,11], [3,5,7,10,11], [4,5,7,10,11], [1,6,7,10,11], [2,6,7,10,11], [3,6,7,10,11], [4,6,7,10,11], [1,2,8,10,11], [1,3,8,10,11], [1,4,8,10,11], [1,5,8,10,11], [2,5,8,10,11], [3,5,8,10,11], [4,5,8,10,11], [1,6,8,10,11], [2,6,8,10,11], [3,6,8,10,11], [4,6,8,10,11], [1,7,8,10,11], [2,7,8,10,11], [3,7,8,10,11], [4,7,8,10,11], [1,2,9,10,11], [1,3,9,10,11], [1,4,9,10,11], [1,5,9,10,11], [2,5,9,10,11], [3,5,9,10,11], [4,5,9,10,11], [1,6,9,10,11], [2,6,9,10,11], [3,6,9,10,11], [4,6,9,10,11], [1,7,9,10,11], [2,7,9,10,11], [3,7,9,10,11], [4,7,9,10,11], [0,1,2,5,12], [0,1,3,5,12], [1,2,3,5,12], [0,1,4,5,12], [1,2,4,5,12], [1,3,4,5,12], [0,1,2,6,12], [0,1,3,6,12], [1,2,3,6,12], [0,1,4,6,12], [1,2,4,6,12], [1,3,4,6,12], [1,2,5,6,12], [1,3,5,6,12], [1,4,5,6,12], [0,1,2,7,12], [0,1,3,7,12], [1,2,3,7,12], [0,1,4,7,12], [1,2,4,7,12], [1,3,4,7,12], [1,2,5,7,12], [1,3,5,7,12], [1,4,5,7,12], [1,2,6,7,12], [1,3,6,7,12], [1,4,6,7,12], [0,1,2,8,12], [0,1,3,8,12], [1,2,3,8,12], [0,1,4,8,12], [1,2,4,8,12], [1,3,4,8,12], [0,1,5,8,12], [0,2,5,8,12], [1,2,5,8,12], [0,3,5,8,12], [2,3,5,8,12], [0,4,5,8,12], [1,4,5,8,12], [2,4,5,8,12], [3,4,5,8,12], [0,1,6,8,12], [0,2,6,8,12], [0,3,6,8,12], [1,3,6,8,12], [2,3,6,8,12], [0,4,6,8,12], [1,4,6,8,12], [2,4,6,8,12], [3,4,6,8,12], [1,5,6,8,12], [2,5,6,8,12], [3,5,6,8,12], [4,5,6,8,12], [0,1,7,8,12], [0,2,7,8,12], [1,2,7,8,12], [0,3,7,8,12], [1,3,7,8,12], [2,3,7,8,12], [0,4,7,8,12], [2,4,7,8,12], [3,4,7,8,12], [1,5,7,8,12], [2,5,7,8,12], [3,5,7,8,12], [4,5,7,8,12], [1,6,7,8,12], [2,6,7,8,12], [3,6,7,8,12], [4,6,7,8,12], [0,1,2,9,12], [0,1,3,9,12], [1,2,3,9,12], [0,1,4,9,12], [1,2,4,9,12], [1,3,4,9,12], [0,1,5,9,12], [0,2,5,9,12], [0,3,5,9,12], [1,3,5,9,12], [2,3,5,9,12], [0,4,5,9,12], [1,4,5,9,12], [2,4,5,9,12], [3,4,5,9,12], [0,1,6,9,12], [0,2,6,9,12], [1,2,6,9,12], [0,3,6,9,12], [1,3,6,9,12], [2,3,6,9,12], [0,4,6,9,12], [2,4,6,9,12], [3,4,6,9,12], [1,5,6,9,12], [2,5,6,9,12], [3,5,6,9,12], [4,5,6,9,12], [0,1,7,9,12], [0,2,7,9,12], [1,2,7,9,12], [0,3,7,9,12], [2,3,7,9,12], [0,4,7,9,12], [1,4,7,9,12], [2,4,7,9,12], [3,4,7,9,12], [1,5,7,9,12], [2,5,7,9,12], [3,5,7,9,12], [4,5,7,9,12], [1,6,7,9,12], [2,6,7,9,12], [3,6,7,9,12], [4,6,7,9,12], [1,2,8,9,12], [1,3,8,9,12], [1,4,8,9,12], [1,5,8,9,12], [2,5,8,9,12], [3,5,8,9,12], [4,5,8,9,12], [1,6,8,9,12], [2,6,8,9,12], [3,6,8,9,12], [4,6,8,9,12], [1,7,8,9,12], [2,7,8,9,12], [3,7,8,9,12], [4,7,8,9,12], [0,1,2,10,12], [0,1,3,10,12], [1,2,3,10,12], [0,1,4,10,12], [1,2,4,10,12], [1,3,4,10,12], [0,1,5,10,12], [0,2,5,10,12], [1,2,5,10,12], [0,3,5,10,12], [1,3,5,10,12], [2,3,5,10,12], [0,4,5,10,12], [2,4,5,10,12], [3,4,5,10,12], [0,1,6,10,12], [0,2,6,10,12], [1,2,6,10,12], [0,3,6,10,12], [2,3,6,10,12], [0,4,6,10,12], [1,4,6,10,12], [2,4,6,10,12], [3,4,6,10,12], [1,5,6,10,12], [2,5,6,10,12], [3,5,6,10,12], [4,5,6,10,12], [0,1,7,10,12], [0,2,7,10,12], [0,3,7,10,12], [1,3,7,10,12], [2,3,7,10,12], [0,4,7,10,12], [1,4,7,10,12], [2,4,7,10,12], [3,4,7,10,12], [1,5,7,10,12], [2,5,7,10,12], [3,5,7,10,12], [4,5,7,10,12], [1,6,7,10,12], [2,6,7,10,12], [3,6,7,10,12], [4,6,7,10,12], [1,2,8,10,12], [1,3,8,10,12], [1,4,8,10,12], [1,5,8,10,12], [2,5,8,10,12], [3,5,8,10,12], [4,5,8,10,12], [1,6,8,10,12], [2,6,8,10,12], [3,6,8,10,12], [4,6,8,10,12], [1,7,8,10,12], [2,7,8,10,12], [3,7,8,10,12], [4,7,8,10,12], [1,2,9,10,12], [1,3,9,10,12], [1,4,9,10,12], [1,5,9,10,12], [2,5,9,10,12], [3,5,9,10,12], [4,5,9,10,12], [1,6,9,10,12], [2,6,9,10,12], [3,6,9,10,12], [4,6,9,10,12], [1,7,9,10,12], [2,7,9,10,12], [3,7,9,10,12], [4,7,9,10,12], [1,2,5,11,12], [1,3,5,11,12], [1,4,5,11,12], [1,2,6,11,12], [1,3,6,11,12], [1,4,6,11,12], [1,2,7,11,12], [1,3,7,11,12], [1,4,7,11,12], [1,2,8,11,12], [1,3,8,11,12], [1,4,8,11,12], [1,5,8,11,12], [2,5,8,11,12], [3,5,8,11,12], [4,5,8,11,12], [1,6,8,11,12], [2,6,8,11,12], [3,6,8,11,12], [4,6,8,11,12], [1,7,8,11,12], [2,7,8,11,12], [3,7,8,11,12], [4,7,8,11,12], [1,2,9,11,12], [1,3,9,11,12], [1,4,9,11,12], [1,5,9,11,12], [2,5,9,11,12], [3,5,9,11,12], [4,5,9,11,12], [1,6,9,11,12], [2,6,9,11,12], [3,6,9,11,12], [4,6,9,11,12], [1,7,9,11,12], [2,7,9,11,12], [3,7,9,11,12], [4,7,9,11,12], [1,2,10,11,12], [1,3,10,11,12], [1,4,10,11,12], [1,5,10,11,12], [2,5,10,11,12], [3,5,10,11,12], [4,5,10,11,12], [1,6,10,11,12], [2,6,10,11,12], [3,6,10,11,12], [4,6,10,11,12], [1,7,10,11,12], [2,7,10,11,12], [3,7,10,11,12], [4,7,10,11,12], [0,1,2,5,13], [0,1,3,5,13], [1,2,3,5,13], [0,1,4,5,13], [1,2,4,5,13], [1,3,4,5,13], [0,1,2,6,13], [0,1,3,6,13], [1,2,3,6,13], [0,1,4,6,13], [1,2,4,6,13], [1,3,4,6,13], [1,2,5,6,13], [1,3,5,6,13], [1,4,5,6,13], [0,1,2,7,13], [0,1,3,7,13], [1,2,3,7,13], [0,1,4,7,13], [1,2,4,7,13], [1,3,4,7,13], [1,2,5,7,13], [1,3,5,7,13], [1,4,5,7,13], [1,2,6,7,13], [1,3,6,7,13], [1,4,6,7,13], [0,1,2,8,13], [0,1,3,8,13], [1,2,3,8,13], [0,1,4,8,13], [1,2,4,8,13], [1,3,4,8,13], [0,1,5,8,13], [0,2,5,8,13], [0,3,5,8,13], [1,3,5,8,13], [2,3,5,8,13], [0,4,5,8,13], [1,4,5,8,13], [2,4,5,8,13], [3,4,5,8,13], [0,1,6,8,13], [0,2,6,8,13], [1,2,6,8,13], [0,3,6,8,13], [1,3,6,8,13], [2,3,6,8,13], [0,4,6,8,13], [2,4,6,8,13], [3,4,6,8,13], [1,5,6,8,13], [2,5,6,8,13], [3,5,6,8,13], [4,5,6,8,13], [0,1,7,8,13], [0,2,7,8,13], [1,2,7,8,13], [0,3,7,8,13], [2,3,7,8,13], [0,4,7,8,13], [1,4,7,8,13], [2,4,7,8,13], [3,4,7,8,13], [1,5,7,8,13], [2,5,7,8,13], [3,5,7,8,13], [4,5,7,8,13], [1,6,7,8,13], [2,6,7,8,13], [3,6,7,8,13], [4,6,7,8,13], [0,1,2,9,13], [0,1,3,9,13], [1,2,3,9,13], [0,1,4,9,13], [1,2,4,9,13], [1,3,4,9,13], [0,1,5,9,13], [0,2,5,9,13], [1,2,5,9,13], [0,3,5,9,13], [1,3,5,9,13], [2,3,5,9,13], [0,4,5,9,13], [2,4,5,9,13], [3,4,5,9,13], [0,1,6,9,13], [0,2,6,9,13], [1,2,6,9,13], [0,3,6,9,13], [2,3,6,9,13], [0,4,6,9,13], [1,4,6,9,13], [2,4,6,9,13], [3,4,6,9,13], [1,5,6,9,13], [2,5,6,9,13], [3,5,6,9,13], [4,5,6,9,13], [0,1,7,9,13], [0,2,7,9,13], [0,3,7,9,13], [1,3,7,9,13], [2,3,7,9,13], [0,4,7,9,13], [1,4,7,9,13], [2,4,7,9,13], [3,4,7,9,13], [1,5,7,9,13], [2,5,7,9,13], [3,5,7,9,13], [4,5,7,9,13], [1,6,7,9,13], [2,6,7,9,13], [3,6,7,9,13], [4,6,7,9,13], [1,2,8,9,13], [1,3,8,9,13], [1,4,8,9,13], [1,5,8,9,13], [2,5,8,9,13], [3,5,8,9,13], [4,5,8,9,13], [1,6,8,9,13], [2,6,8,9,13], [3,6,8,9,13], [4,6,8,9,13], [1,7,8,9,13], [2,7,8,9,13], [3,7,8,9,13], [4,7,8,9,13], [0,1,2,10,13], [0,1,3,10,13], [1,2,3,10,13], [0,1,4,10,13], [1,2,4,10,13], [1,3,4,10,13], [0,1,5,10,13], [0,2,5,10,13], [1,2,5,10,13], [0,3,5,10,13], [2,3,5,10,13], [0,4,5,10,13], [1,4,5,10,13], [2,4,5,10,13], [3,4,5,10,13], [0,1,6,10,13], [0,2,6,10,13], [0,3,6,10,13], [1,3,6,10,13], [2,3,6,10,13], [0,4,6,10,13], [1,4,6,10,13], [2,4,6,10,13], [3,4,6,10,13], [1,5,6,10,13], [2,5,6,10,13], [3,5,6,10,13], [4,5,6,10,13], [0,1,7,10,13], [0,2,7,10,13], [1,2,7,10,13], [0,3,7,10,13], [1,3,7,10,13], [2,3,7,10,13], [0,4,7,10,13], [2,4,7,10,13], [3,4,7,10,13], [1,5,7,10,13], [2,5,7,10,13], [3,5,7,10,13], [4,5,7,10,13], [1,6,7,10,13], [2,6,7,10,13], [3,6,7,10,13], [4,6,7,10,13], [1,2,8,10,13], [1,3,8,10,13], [1,4,8,10,13], [1,5,8,10,13], [2,5,8,10,13], [3,5,8,10,13], [4,5,8,10,13], [1,6,8,10,13], [2,6,8,10,13], [3,6,8,10,13], [4,6,8,10,13], [1,7,8,10,13], [2,7,8,10,13], [3,7,8,10,13], [4,7,8,10,13], [1,2,9,10,13], [1,3,9,10,13], [1,4,9,10,13], [1,5,9,10,13], [2,5,9,10,13], [3,5,9,10,13], [4,5,9,10,13], [1,6,9,10,13], [2,6,9,10,13], [3,6,9,10,13], [4,6,9,10,13], [1,7,9,10,13], [2,7,9,10,13], [3,7,9,10,13], [4,7,9,10,13], [1,2,5,11,13], [1,3,5,11,13], [1,4,5,11,13], [1,2,6,11,13], [1,3,6,11,13], [1,4,6,11,13], [1,2,7,11,13], [1,3,7,11,13], [1,4,7,11,13], [1,2,8,11,13], [1,3,8,11,13], [1,4,8,11,13], [1,5,8,11,13], [2,5,8,11,13], [3,5,8,11,13], [4,5,8,11,13], [1,6,8,11,13], [2,6,8,11,13], [3,6,8,11,13], [4,6,8,11,13], [1,7,8,11,13], [2,7,8,11,13], [3,7,8,11,13], [4,7,8,11,13], [1,2,9,11,13], [1,3,9,11,13], [1,4,9,11,13], [1,5,9,11,13], [2,5,9,11,13], [3,5,9,11,13], [4,5,9,11,13], [1,6,9,11,13], [2,6,9,11,13], [3,6,9,11,13], [4,6,9,11,13], [1,7,9,11,13], [2,7,9,11,13], [3,7,9,11,13], [4,7,9,11,13], [1,2,10,11,13], [1,3,10,11,13], [1,4,10,11,13], [1,5,10,11,13], [2,5,10,11,13], [3,5,10,11,13], [4,5,10,11,13], [1,6,10,11,13], [2,6,10,11,13], [3,6,10,11,13], [4,6,10,11,13], [1,7,10,11,13], [2,7,10,11,13], [3,7,10,11,13], [4,7,10,11,13], [1,2,5,12,13], [1,3,5,12,13], [1,4,5,12,13], [1,2,6,12,13], [1,3,6,12,13], [1,4,6,12,13], [1,2,7,12,13], [1,3,7,12,13], [1,4,7,12,13], [1,2,8,12,13], [1,3,8,12,13], [1,4,8,12,13], [1,5,8,12,13], [2,5,8,12,13], [3,5,8,12,13], [4,5,8,12,13], [1,6,8,12,13], [2,6,8,12,13], [3,6,8,12,13], [4,6,8,12,13], [1,7,8,12,13], [2,7,8,12,13], [3,7,8,12,13], [4,7,8,12,13], [1,2,9,12,13], [1,3,9,12,13], [1,4,9,12,13], [1,5,9,12,13], [2,5,9,12,13], [3,5,9,12,13], [4,5,9,12,13], [1,6,9,12,13], [2,6,9,12,13], [3,6,9,12,13], [4,6,9,12,13], [1,7,9,12,13], [2,7,9,12,13], [3,7,9,12,13], [4,7,9,12,13], [1,2,10,12,13], [1,3,10,12,13], [1,4,10,12,13], [1,5,10,12,13], [2,5,10,12,13], [3,5,10,12,13], [4,5,10,12,13], [1,6,10,12,13], [2,6,10,12,13], [3,6,10,12,13], [4,6,10,12,13], [1,7,10,12,13], [2,7,10,12,13], [3,7,10,12,13], [4,7,10,12,13]]
Maximizing pair
(0, 1); basis counts at (i, j) = (0, 1):
| j ∈ B | j ∉ B | |
|---|---|---|
| i ∈ B | 108 | 81 |
| i ∉ B | 378 | 324 |
Attribution
The finite-field spike M_3^5 of Huh, Schroter, and Wang (Correlation bounds for fields and matroids, arXiv:1806.02675, Definition 19; discovered June 2018, no AI use reported in that record). Columns over GF(3): the tip e1, the vector e2+e3+e4+e5, and k*e1+e_m for m=2..5, k=1..p. Submitted from the literature.
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