Identifier
Values
[1] => [[1],[]] => ([],1) => 0
[1,1] => [[1,1],[]] => ([],1) => 0
[2] => [[2],[]] => ([],1) => 0
[1,1,1] => [[1,1,1],[]] => ([],1) => 0
[1,2] => [[2,1],[]] => ([],1) => 0
[2,1] => [[2,2],[1]] => ([],1) => 0
[3] => [[3],[]] => ([],1) => 0
[1,1,1,1] => [[1,1,1,1],[]] => ([],1) => 0
[1,1,2] => [[2,1,1],[]] => ([],1) => 0
[1,2,1] => [[2,2,1],[1]] => ([(0,1)],2) => 1
[1,3] => [[3,1],[]] => ([],1) => 0
[2,1,1] => [[2,2,2],[1,1]] => ([],1) => 0
[2,2] => [[3,2],[1]] => ([(0,1)],2) => 1
[3,1] => [[3,3],[2]] => ([],1) => 0
[4] => [[4],[]] => ([],1) => 0
[1,1,1,1,1] => [[1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,2] => [[2,1,1,1],[]] => ([],1) => 0
[1,1,2,1] => [[2,2,1,1],[1]] => ([(0,1)],2) => 1
[1,1,3] => [[3,1,1],[]] => ([],1) => 0
[1,2,1,1] => [[2,2,2,1],[1,1]] => ([(0,1)],2) => 1
[1,2,2] => [[3,2,1],[1]] => ([(0,2),(2,1)],3) => 2
[1,3,1] => [[3,3,1],[2]] => ([(0,1)],2) => 1
[1,4] => [[4,1],[]] => ([],1) => 0
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => ([],1) => 0
[2,1,2] => [[3,2,2],[1,1]] => ([(0,1)],2) => 1
[2,2,1] => [[3,3,2],[2,1]] => ([(0,2),(2,1)],3) => 2
[2,3] => [[4,2],[1]] => ([(0,1)],2) => 1
[3,1,1] => [[3,3,3],[2,2]] => ([],1) => 0
[3,2] => [[4,3],[2]] => ([(0,1)],2) => 1
[4,1] => [[4,4],[3]] => ([],1) => 0
[5] => [[5],[]] => ([],1) => 0
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,1,2] => [[2,1,1,1,1],[]] => ([],1) => 0
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => ([(0,1)],2) => 1
[1,1,1,3] => [[3,1,1,1],[]] => ([],1) => 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => ([(0,2),(2,1)],3) => 2
[1,1,2,2] => [[3,2,1,1],[1]] => ([(0,2),(2,1)],3) => 2
[1,1,3,1] => [[3,3,1,1],[2]] => ([(0,1)],2) => 1
[1,1,4] => [[4,1,1],[]] => ([],1) => 0
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => ([(0,1)],2) => 1
[1,2,1,2] => [[3,2,2,1],[1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,2,1] => [[3,3,2,1],[2,1]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 3
[1,2,3] => [[4,2,1],[1]] => ([(0,2),(2,1)],3) => 2
[1,3,1,1] => [[3,3,3,1],[2,2]] => ([(0,1)],2) => 1
[1,3,2] => [[4,3,1],[2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,4,1] => [[4,4,1],[3]] => ([(0,1)],2) => 1
[1,5] => [[5,1],[]] => ([],1) => 0
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => ([],1) => 0
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => ([(0,1)],2) => 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,1,3] => [[4,2,2],[1,1]] => ([(0,1)],2) => 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => ([(0,2),(2,1)],3) => 2
[2,2,2] => [[4,3,2],[2,1]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
[2,3,1] => [[4,4,2],[3,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,4] => [[5,2],[1]] => ([(0,1)],2) => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => ([],1) => 0
[3,1,2] => [[4,3,3],[2,2]] => ([(0,1)],2) => 1
[3,2,1] => [[4,4,3],[3,2]] => ([(0,2),(2,1)],3) => 2
[3,3] => [[5,3],[2]] => ([(0,2),(2,1)],3) => 2
[4,1,1] => [[4,4,4],[3,3]] => ([],1) => 0
[4,2] => [[5,4],[3]] => ([(0,1)],2) => 1
[5,1] => [[5,5],[4]] => ([],1) => 0
[6] => [[6],[]] => ([],1) => 0
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => ([(0,1)],2) => 1
[1,1,1,1,3] => [[3,1,1,1,1],[]] => ([],1) => 0
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => ([(0,2),(2,1)],3) => 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => ([(0,2),(2,1)],3) => 2
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => ([(0,1)],2) => 1
[1,1,1,4] => [[4,1,1,1],[]] => ([],1) => 0
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => ([(0,2),(2,1)],3) => 2
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 3
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 4
[1,1,2,3] => [[4,2,1,1],[1]] => ([(0,2),(2,1)],3) => 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => ([(0,2),(2,1)],3) => 2
[1,1,3,2] => [[4,3,1,1],[2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,1,4,1] => [[4,4,1,1],[3]] => ([(0,1)],2) => 1
[1,1,5] => [[5,1,1],[]] => ([],1) => 0
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => ([(0,1)],2) => 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 4
[1,2,1,3] => [[4,2,2,1],[1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 4
[1,2,2,2] => [[4,3,2,1],[2,1]] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => 4
[1,2,3,1] => [[4,4,2,1],[3,1]] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 3
[1,2,4] => [[5,2,1],[1]] => ([(0,2),(2,1)],3) => 2
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => ([(0,1)],2) => 1
[1,3,1,2] => [[4,3,3,1],[2,2]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 2
[1,3,2,1] => [[4,4,3,1],[3,2]] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 3
[1,3,3] => [[5,3,1],[2]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
[1,4,1,1] => [[4,4,4,1],[3,3]] => ([(0,1)],2) => 1
[1,4,2] => [[5,4,1],[3]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,5,1] => [[5,5,1],[4]] => ([(0,1)],2) => 1
[1,6] => [[6,1],[]] => ([],1) => 0
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => ([],1) => 0
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => ([(0,1)],2) => 1
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => ([(0,1)],2) => 1
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 3
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 3
>>> Load all 226 entries. <<<
[2,1,3,1] => [[4,4,2,2],[3,1,1]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 2
[2,1,4] => [[5,2,2],[1,1]] => ([(0,1)],2) => 1
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => ([(0,2),(2,1)],3) => 2
[2,2,1,2] => [[4,3,3,2],[2,2,1]] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 3
[2,2,2,1] => [[4,4,3,2],[3,2,1]] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => 4
[2,2,3] => [[5,3,2],[2,1]] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 4
[2,3,1,1] => [[4,4,4,2],[3,3,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,2] => [[5,4,2],[3,1]] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 4
[2,4,1] => [[5,5,2],[4,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,5] => [[6,2],[1]] => ([(0,1)],2) => 1
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => ([],1) => 0
[3,1,1,2] => [[4,3,3,3],[2,2,2]] => ([(0,1)],2) => 1
[3,1,2,1] => [[4,4,3,3],[3,2,2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,1,3] => [[5,3,3],[2,2]] => ([(0,2),(2,1)],3) => 2
[3,2,1,1] => [[4,4,4,3],[3,3,2]] => ([(0,2),(2,1)],3) => 2
[3,2,2] => [[5,4,3],[3,2]] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 4
[3,3,1] => [[5,5,3],[4,2]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
[3,4] => [[6,3],[2]] => ([(0,2),(2,1)],3) => 2
[4,1,1,1] => [[4,4,4,4],[3,3,3]] => ([],1) => 0
[4,1,2] => [[5,4,4],[3,3]] => ([(0,1)],2) => 1
[4,2,1] => [[5,5,4],[4,3]] => ([(0,2),(2,1)],3) => 2
[4,3] => [[6,4],[3]] => ([(0,2),(2,1)],3) => 2
[5,1,1] => [[5,5,5],[4,4]] => ([],1) => 0
[5,2] => [[6,5],[4]] => ([(0,1)],2) => 1
[6,1] => [[6,6],[5]] => ([],1) => 0
[7] => [[7],[]] => ([],1) => 0
[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1],[1]] => ([(0,1)],2) => 1
[1,1,1,1,1,3] => [[3,1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]] => ([(0,2),(2,1)],3) => 2
[1,1,1,1,2,2] => [[3,2,1,1,1,1],[1]] => ([(0,2),(2,1)],3) => 2
[1,1,1,1,4] => [[4,1,1,1,1],[]] => ([],1) => 0
[1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]] => ([(0,3),(2,1),(3,2)],4) => 3
[1,1,1,2,1,2] => [[3,2,2,1,1,1],[1,1]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 3
[1,1,1,2,2,1] => [[3,3,2,1,1,1],[2,1]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 4
[1,1,1,3,2] => [[4,3,1,1,1],[2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,1,1,5] => [[5,1,1,1],[]] => ([],1) => 0
[1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]] => ([(0,2),(2,1)],3) => 2
[1,1,2,1,1,2] => [[3,2,2,2,1,1],[1,1,1]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]] => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => 5
[1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]] => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => 5
[1,1,2,2,2] => [[4,3,2,1,1],[2,1]] => ([(0,6),(1,8),(2,7),(4,7),(5,2),(5,8),(6,1),(6,5),(7,3),(8,4)],9) => 5
[1,1,2,3,1] => [[4,4,2,1,1],[3,1]] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => 4
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7) => 3
[1,1,3,2,1] => [[4,4,3,1,1],[3,2]] => ([(0,5),(1,7),(2,6),(3,6),(4,3),(4,7),(5,1),(5,4),(7,2)],8) => 4
[1,1,3,3] => [[5,3,1,1],[2]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
[1,1,6] => [[6,1,1],[]] => ([],1) => 0
[1,2,1,1,1,1,1] => [[2,2,2,2,2,2,1],[1,1,1,1,1]] => ([(0,1)],2) => 1
[1,2,1,1,1,2] => [[3,2,2,2,2,1],[1,1,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,1,1,2,1] => [[3,3,2,2,2,1],[2,1,1,1]] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 4
[1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]] => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => 5
[1,2,1,3,1] => [[4,4,2,2,1],[3,1,1]] => ([(0,5),(1,7),(2,6),(3,6),(4,3),(4,7),(5,1),(5,4),(7,2)],8) => 4
[1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 4
[1,2,2,3] => [[5,3,2,1],[2,1]] => ([(0,6),(1,8),(3,7),(4,5),(5,1),(5,7),(6,3),(6,4),(7,8),(8,2)],9) => 5
[1,2,3,1,1] => [[4,4,4,2,1],[3,3,1]] => ([(0,5),(1,7),(2,6),(3,6),(4,3),(4,7),(5,1),(5,4),(7,2)],8) => 4
[1,3,1,1,2] => [[4,3,3,3,1],[2,2,2]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 2
[1,3,1,2,1] => [[4,4,3,3,1],[3,2,2]] => ([(0,5),(1,7),(2,6),(3,6),(4,3),(4,7),(5,1),(5,4),(7,2)],8) => 4
[1,3,1,3] => [[5,3,3,1],[2,2]] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7) => 3
[1,3,2,1,1] => [[4,4,4,3,1],[3,3,2]] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => 4
[1,3,3,1] => [[5,5,3,1],[4,2]] => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,2),(5,6),(6,4),(7,8)],9) => 4
[1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]] => ([(0,1)],2) => 1
[1,4,1,2] => [[5,4,4,1],[3,3]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 2
[1,4,2,1] => [[5,5,4,1],[4,3]] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 3
[1,4,3] => [[6,4,1],[3]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[1,7] => [[7,1],[]] => ([],1) => 0
[2,1,1,1,1,2] => [[3,2,2,2,2,2],[1,1,1,1,1]] => ([(0,1)],2) => 1
[2,1,1,1,2,1] => [[3,3,2,2,2,2],[2,1,1,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,1,1,2,1,1] => [[3,3,3,2,2,2],[2,2,1,1,1]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[2,1,1,2,2] => [[4,3,2,2,2],[2,1,1,1]] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 3
[2,1,1,3,1] => [[4,4,2,2,2],[3,1,1,1]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 2
[2,1,2,1,1,1] => [[3,3,3,3,2,2],[2,2,2,1,1]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 3
[2,1,2,1,2] => [[4,3,3,2,2],[2,2,1,1]] => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,2),(5,6),(6,4),(7,8)],9) => 4
[2,1,2,3] => [[5,3,2,2],[2,1,1]] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => 4
[2,1,3,1,1] => [[4,4,4,2,2],[3,3,1,1]] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7) => 3
[2,1,3,2] => [[5,4,2,2],[3,1,1]] => ([(0,3),(0,5),(2,7),(3,6),(4,2),(4,6),(5,4),(6,7),(7,1)],8) => 4
[2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]] => ([(0,2),(2,1)],3) => 2
[2,2,1,1,2] => [[4,3,3,3,2],[2,2,2,1]] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 3
[2,2,1,3] => [[5,3,3,2],[2,2,1]] => ([(0,3),(0,5),(2,7),(3,6),(4,2),(4,6),(5,4),(6,7),(7,1)],8) => 4
[2,2,2,1,1] => [[4,4,4,3,2],[3,3,2,1]] => ([(0,6),(1,8),(2,7),(4,7),(5,2),(5,8),(6,1),(6,5),(7,3),(8,4)],9) => 5
[2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,1,2] => [[5,4,4,2],[3,3,1]] => ([(0,3),(0,5),(2,7),(3,6),(4,2),(4,6),(5,4),(6,7),(7,1)],8) => 4
[2,3,3] => [[6,4,2],[3,1]] => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8) => 5
[2,4,1,1] => [[5,5,5,2],[4,4,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,4,2] => [[6,5,2],[4,1]] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 4
[3,1,1,2,1] => [[4,4,3,3,3],[3,2,2,2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,1,1,3] => [[5,3,3,3],[2,2,2]] => ([(0,2),(2,1)],3) => 2
[3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 3
[3,1,2,2] => [[5,4,3,3],[3,2,2]] => ([(0,3),(0,5),(2,7),(3,6),(4,2),(4,6),(5,4),(6,7),(7,1)],8) => 4
[3,1,3,1] => [[5,5,3,3],[4,2,2]] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7) => 3
[3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]] => ([(0,2),(2,1)],3) => 2
[3,2,1,2] => [[5,4,4,3],[3,3,2]] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => 4
[3,2,2,1] => [[5,5,4,3],[4,3,2]] => ([(0,6),(1,8),(3,7),(4,5),(5,1),(5,7),(6,3),(6,4),(7,8),(8,2)],9) => 5
[3,2,3] => [[6,4,3],[3,2]] => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8) => 5
[3,3,1,1] => [[5,5,5,3],[4,4,2]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
[3,3,2] => [[6,5,3],[4,2]] => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8) => 5
[3,4,1] => [[6,6,3],[5,2]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]] => ([],1) => 0
[4,1,1,2] => [[5,4,4,4],[3,3,3]] => ([(0,1)],2) => 1
[4,1,2,1] => [[5,5,4,4],[4,3,3]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[4,1,3] => [[6,4,4],[3,3]] => ([(0,2),(2,1)],3) => 2
[4,2,1,1] => [[5,5,5,4],[4,4,3]] => ([(0,2),(2,1)],3) => 2
[4,2,2] => [[6,5,4],[4,3]] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 4
[4,3,1] => [[6,6,4],[5,3]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
[4,4] => [[7,4],[3]] => ([(0,3),(2,1),(3,2)],4) => 3
[8] => [[8],[]] => ([],1) => 0
[1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,1,1,1,3] => [[3,1,1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,1,1,4] => [[4,1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,1,5] => [[5,1,1,1,1],[]] => ([],1) => 0
[1,1,1,6] => [[6,1,1,1],[]] => ([],1) => 0
[1,1,7] => [[7,1,1],[]] => ([],1) => 0
[1,8] => [[8,1],[]] => ([],1) => 0
[9] => [[9],[]] => ([],1) => 0
[1,1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,1,1,1,1,3] => [[3,1,1,1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,1,1,1,4] => [[4,1,1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,1,1,5] => [[5,1,1,1,1,1],[]] => ([],1) => 0
[1,1,1,1,6] => [[6,1,1,1,1],[]] => ([],1) => 0
[1,1,1,7] => [[7,1,1,1],[]] => ([],1) => 0
[1,1,8] => [[8,1,1],[]] => ([],1) => 0
[1,9] => [[9,1],[]] => ([],1) => 0
[10] => [[10],[]] => ([],1) => 0
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Description
The dimension of the space of valuations of a lattice.
A valuation, or modular function, on a lattice $L$ is a function $v:L\mapsto\mathbb R$ satisfying
$$ v(a\vee b) + v(a\wedge b) = v(a) + v(b). $$
It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient Mp00196The modular quotient of a lattice..
Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.
Map
dominating sublattice
Description
Return the sublattice of the dominance order induced by the support of the expansion of the skew Schur function into Schur functions.
Consider the expansion of the skew Schur function $s_{\lambda/\mu}=\sum_\nu c^\lambda_{\mu, \nu} s_\nu$ as a linear combination of straight Schur functions.
It is shown in [1] that the subposet of the dominance order whose elements are the partitions $\nu$ with $c^\lambda_{\mu, \nu} > 0$ form a lattice.
The example $\lambda = (5^2,4^2,1)$ and $\mu=(3,2)$ shows that this lattice is not a sublattice of the dominance order.
Map
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.