Identifier
-
Mp00311:
Plane partitions
—to partition⟶
Integer partitions
St000992: Integer partitions ⟶ ℤ
Values
=>
Cc0025;cc-rep-0
Cc0002;cc-rep
[[1]]=>[1]=>1
[[1],[1]]=>[1,1]=>0
[[2]]=>[2]=>2
[[1,1]]=>[2]=>2
[[1],[1],[1]]=>[1,1,1]=>1
[[2],[1]]=>[2,1]=>1
[[1,1],[1]]=>[2,1]=>1
[[3]]=>[3]=>3
[[2,1]]=>[3]=>3
[[1,1,1]]=>[3]=>3
[[1],[1],[1],[1]]=>[1,1,1,1]=>0
[[2],[1],[1]]=>[2,1,1]=>2
[[2],[2]]=>[2,2]=>0
[[1,1],[1],[1]]=>[2,1,1]=>2
[[1,1],[1,1]]=>[2,2]=>0
[[3],[1]]=>[3,1]=>2
[[2,1],[1]]=>[3,1]=>2
[[1,1,1],[1]]=>[3,1]=>2
[[4]]=>[4]=>4
[[3,1]]=>[4]=>4
[[2,2]]=>[4]=>4
[[2,1,1]]=>[4]=>4
[[1,1,1,1]]=>[4]=>4
[[1],[1],[1],[1],[1]]=>[1,1,1,1,1]=>1
[[2],[1],[1],[1]]=>[2,1,1,1]=>1
[[2],[2],[1]]=>[2,2,1]=>1
[[1,1],[1],[1],[1]]=>[2,1,1,1]=>1
[[1,1],[1,1],[1]]=>[2,2,1]=>1
[[3],[1],[1]]=>[3,1,1]=>3
[[3],[2]]=>[3,2]=>1
[[2,1],[1],[1]]=>[3,1,1]=>3
[[2,1],[2]]=>[3,2]=>1
[[2,1],[1,1]]=>[3,2]=>1
[[1,1,1],[1],[1]]=>[3,1,1]=>3
[[1,1,1],[1,1]]=>[3,2]=>1
[[4],[1]]=>[4,1]=>3
[[3,1],[1]]=>[4,1]=>3
[[2,2],[1]]=>[4,1]=>3
[[2,1,1],[1]]=>[4,1]=>3
[[1,1,1,1],[1]]=>[4,1]=>3
[[5]]=>[5]=>5
[[4,1]]=>[5]=>5
[[3,2]]=>[5]=>5
[[3,1,1]]=>[5]=>5
[[2,2,1]]=>[5]=>5
[[2,1,1,1]]=>[5]=>5
[[1,1,1,1,1]]=>[5]=>5
[[1],[1],[1],[1],[1],[1]]=>[1,1,1,1,1,1]=>0
[[2],[1],[1],[1],[1]]=>[2,1,1,1,1]=>2
[[2],[2],[1],[1]]=>[2,2,1,1]=>0
[[2],[2],[2]]=>[2,2,2]=>2
[[1,1],[1],[1],[1],[1]]=>[2,1,1,1,1]=>2
[[1,1],[1,1],[1],[1]]=>[2,2,1,1]=>0
[[1,1],[1,1],[1,1]]=>[2,2,2]=>2
[[3],[1],[1],[1]]=>[3,1,1,1]=>2
[[3],[2],[1]]=>[3,2,1]=>2
[[3],[3]]=>[3,3]=>0
[[2,1],[1],[1],[1]]=>[3,1,1,1]=>2
[[2,1],[2],[1]]=>[3,2,1]=>2
[[2,1],[1,1],[1]]=>[3,2,1]=>2
[[2,1],[2,1]]=>[3,3]=>0
[[1,1,1],[1],[1],[1]]=>[3,1,1,1]=>2
[[1,1,1],[1,1],[1]]=>[3,2,1]=>2
[[1,1,1],[1,1,1]]=>[3,3]=>0
[[4],[1],[1]]=>[4,1,1]=>4
[[4],[2]]=>[4,2]=>2
[[3,1],[1],[1]]=>[4,1,1]=>4
[[3,1],[2]]=>[4,2]=>2
[[3,1],[1,1]]=>[4,2]=>2
[[2,2],[1],[1]]=>[4,1,1]=>4
[[2,2],[2]]=>[4,2]=>2
[[2,2],[1,1]]=>[4,2]=>2
[[2,1,1],[1],[1]]=>[4,1,1]=>4
[[2,1,1],[2]]=>[4,2]=>2
[[2,1,1],[1,1]]=>[4,2]=>2
[[1,1,1,1],[1],[1]]=>[4,1,1]=>4
[[1,1,1,1],[1,1]]=>[4,2]=>2
[[5],[1]]=>[5,1]=>4
[[4,1],[1]]=>[5,1]=>4
[[3,2],[1]]=>[5,1]=>4
[[3,1,1],[1]]=>[5,1]=>4
[[2,2,1],[1]]=>[5,1]=>4
[[2,1,1,1],[1]]=>[5,1]=>4
[[1,1,1,1,1],[1]]=>[5,1]=>4
[[6]]=>[6]=>6
[[5,1]]=>[6]=>6
[[4,2]]=>[6]=>6
[[4,1,1]]=>[6]=>6
[[3,3]]=>[6]=>6
[[3,2,1]]=>[6]=>6
[[3,1,1,1]]=>[6]=>6
[[2,2,2]]=>[6]=>6
[[2,2,1,1]]=>[6]=>6
[[2,1,1,1,1]]=>[6]=>6
[[1,1,1,1,1,1]]=>[6]=>6
[[1],[1],[1],[1],[1],[1],[1]]=>[1,1,1,1,1,1,1]=>1
[[2],[1],[1],[1],[1],[1]]=>[2,1,1,1,1,1]=>1
[[2],[2],[1],[1],[1]]=>[2,2,1,1,1]=>1
[[2],[2],[2],[1]]=>[2,2,2,1]=>1
[[1,1],[1],[1],[1],[1],[1]]=>[2,1,1,1,1,1]=>1
[[1,1],[1,1],[1],[1],[1]]=>[2,2,1,1,1]=>1
[[1,1],[1,1],[1,1],[1]]=>[2,2,2,1]=>1
[[3],[1],[1],[1],[1]]=>[3,1,1,1,1]=>3
[[3],[2],[1],[1]]=>[3,2,1,1]=>1
[[3],[2],[2]]=>[3,2,2]=>3
[[3],[3],[1]]=>[3,3,1]=>1
[[2,1],[1],[1],[1],[1]]=>[3,1,1,1,1]=>3
[[2,1],[2],[1],[1]]=>[3,2,1,1]=>1
[[2,1],[2],[2]]=>[3,2,2]=>3
[[2,1],[1,1],[1],[1]]=>[3,2,1,1]=>1
[[2,1],[1,1],[1,1]]=>[3,2,2]=>3
[[2,1],[2,1],[1]]=>[3,3,1]=>1
[[1,1,1],[1],[1],[1],[1]]=>[3,1,1,1,1]=>3
[[1,1,1],[1,1],[1],[1]]=>[3,2,1,1]=>1
[[1,1,1],[1,1],[1,1]]=>[3,2,2]=>3
[[1,1,1],[1,1,1],[1]]=>[3,3,1]=>1
[[4],[1],[1],[1]]=>[4,1,1,1]=>3
[[4],[2],[1]]=>[4,2,1]=>3
[[4],[3]]=>[4,3]=>1
[[3,1],[1],[1],[1]]=>[4,1,1,1]=>3
[[3,1],[2],[1]]=>[4,2,1]=>3
[[3,1],[1,1],[1]]=>[4,2,1]=>3
[[3,1],[3]]=>[4,3]=>1
[[3,1],[2,1]]=>[4,3]=>1
[[2,2],[1],[1],[1]]=>[4,1,1,1]=>3
[[2,2],[2],[1]]=>[4,2,1]=>3
[[2,2],[1,1],[1]]=>[4,2,1]=>3
[[2,2],[2,1]]=>[4,3]=>1
[[2,1,1],[1],[1],[1]]=>[4,1,1,1]=>3
[[2,1,1],[2],[1]]=>[4,2,1]=>3
[[2,1,1],[1,1],[1]]=>[4,2,1]=>3
[[2,1,1],[2,1]]=>[4,3]=>1
[[2,1,1],[1,1,1]]=>[4,3]=>1
[[1,1,1,1],[1],[1],[1]]=>[4,1,1,1]=>3
[[1,1,1,1],[1,1],[1]]=>[4,2,1]=>3
[[1,1,1,1],[1,1,1]]=>[4,3]=>1
[[5],[1],[1]]=>[5,1,1]=>5
[[5],[2]]=>[5,2]=>3
[[4,1],[1],[1]]=>[5,1,1]=>5
[[4,1],[2]]=>[5,2]=>3
[[4,1],[1,1]]=>[5,2]=>3
[[3,2],[1],[1]]=>[5,1,1]=>5
[[3,2],[2]]=>[5,2]=>3
[[3,2],[1,1]]=>[5,2]=>3
[[3,1,1],[1],[1]]=>[5,1,1]=>5
[[3,1,1],[2]]=>[5,2]=>3
[[3,1,1],[1,1]]=>[5,2]=>3
[[2,2,1],[1],[1]]=>[5,1,1]=>5
[[2,2,1],[2]]=>[5,2]=>3
[[2,2,1],[1,1]]=>[5,2]=>3
[[2,1,1,1],[1],[1]]=>[5,1,1]=>5
[[2,1,1,1],[2]]=>[5,2]=>3
[[2,1,1,1],[1,1]]=>[5,2]=>3
[[1,1,1,1,1],[1],[1]]=>[5,1,1]=>5
[[1,1,1,1,1],[1,1]]=>[5,2]=>3
[[6],[1]]=>[6,1]=>5
[[5,1],[1]]=>[6,1]=>5
[[4,2],[1]]=>[6,1]=>5
[[4,1,1],[1]]=>[6,1]=>5
[[3,3],[1]]=>[6,1]=>5
[[3,2,1],[1]]=>[6,1]=>5
[[3,1,1,1],[1]]=>[6,1]=>5
[[2,2,2],[1]]=>[6,1]=>5
[[2,2,1,1],[1]]=>[6,1]=>5
[[2,1,1,1,1],[1]]=>[6,1]=>5
[[1,1,1,1,1,1],[1]]=>[6,1]=>5
[[7]]=>[7]=>7
[[6,1]]=>[7]=>7
[[5,2]]=>[7]=>7
[[5,1,1]]=>[7]=>7
[[4,3]]=>[7]=>7
[[4,2,1]]=>[7]=>7
[[4,1,1,1]]=>[7]=>7
[[3,3,1]]=>[7]=>7
[[3,2,2]]=>[7]=>7
[[3,2,1,1]]=>[7]=>7
[[3,1,1,1,1]]=>[7]=>7
[[2,2,2,1]]=>[7]=>7
[[2,2,1,1,1]]=>[7]=>7
[[2,1,1,1,1,1]]=>[7]=>7
[[1,1,1,1,1,1,1]]=>[7]=>7
[[1],[1],[1],[1],[1],[1],[1],[1]]=>[1,1,1,1,1,1,1,1]=>0
[[2],[1],[1],[1],[1],[1],[1]]=>[2,1,1,1,1,1,1]=>2
[[2],[2],[1],[1],[1],[1]]=>[2,2,1,1,1,1]=>0
[[2],[2],[2],[1],[1]]=>[2,2,2,1,1]=>2
[[2],[2],[2],[2]]=>[2,2,2,2]=>0
[[1,1],[1],[1],[1],[1],[1],[1]]=>[2,1,1,1,1,1,1]=>2
[[1,1],[1,1],[1],[1],[1],[1]]=>[2,2,1,1,1,1]=>0
[[1,1],[1,1],[1,1],[1],[1]]=>[2,2,2,1,1]=>2
[[1,1],[1,1],[1,1],[1,1]]=>[2,2,2,2]=>0
[[3],[1],[1],[1],[1],[1]]=>[3,1,1,1,1,1]=>2
[[3],[2],[1],[1],[1]]=>[3,2,1,1,1]=>2
[[3],[2],[2],[1]]=>[3,2,2,1]=>2
[[3],[3],[1],[1]]=>[3,3,1,1]=>0
[[3],[3],[2]]=>[3,3,2]=>2
[[2,1],[1],[1],[1],[1],[1]]=>[3,1,1,1,1,1]=>2
[[2,1],[2],[1],[1],[1]]=>[3,2,1,1,1]=>2
[[2,1],[2],[2],[1]]=>[3,2,2,1]=>2
[[2,1],[1,1],[1],[1],[1]]=>[3,2,1,1,1]=>2
[[2,1],[1,1],[1,1],[1]]=>[3,2,2,1]=>2
[[2,1],[2,1],[1],[1]]=>[3,3,1,1]=>0
[[2,1],[2,1],[2]]=>[3,3,2]=>2
[[2,1],[2,1],[1,1]]=>[3,3,2]=>2
[[1,1,1],[1],[1],[1],[1],[1]]=>[3,1,1,1,1,1]=>2
[[1,1,1],[1,1],[1],[1],[1]]=>[3,2,1,1,1]=>2
[[1,1,1],[1,1],[1,1],[1]]=>[3,2,2,1]=>2
[[1,1,1],[1,1,1],[1],[1]]=>[3,3,1,1]=>0
[[1,1,1],[1,1,1],[1,1]]=>[3,3,2]=>2
[[4],[1],[1],[1],[1]]=>[4,1,1,1,1]=>4
[[4],[2],[1],[1]]=>[4,2,1,1]=>2
[[4],[2],[2]]=>[4,2,2]=>4
[[4],[3],[1]]=>[4,3,1]=>2
[[4],[4]]=>[4,4]=>0
[[3,1],[1],[1],[1],[1]]=>[4,1,1,1,1]=>4
[[3,1],[2],[1],[1]]=>[4,2,1,1]=>2
[[3,1],[2],[2]]=>[4,2,2]=>4
[[3,1],[1,1],[1],[1]]=>[4,2,1,1]=>2
[[3,1],[1,1],[1,1]]=>[4,2,2]=>4
[[3,1],[3],[1]]=>[4,3,1]=>2
[[3,1],[2,1],[1]]=>[4,3,1]=>2
[[3,1],[3,1]]=>[4,4]=>0
[[2,2],[1],[1],[1],[1]]=>[4,1,1,1,1]=>4
[[2,2],[2],[1],[1]]=>[4,2,1,1]=>2
[[2,2],[2],[2]]=>[4,2,2]=>4
[[2,2],[1,1],[1],[1]]=>[4,2,1,1]=>2
[[2,2],[1,1],[1,1]]=>[4,2,2]=>4
[[2,2],[2,1],[1]]=>[4,3,1]=>2
[[2,2],[2,2]]=>[4,4]=>0
[[2,1,1],[1],[1],[1],[1]]=>[4,1,1,1,1]=>4
[[2,1,1],[2],[1],[1]]=>[4,2,1,1]=>2
[[2,1,1],[2],[2]]=>[4,2,2]=>4
[[2,1,1],[1,1],[1],[1]]=>[4,2,1,1]=>2
[[2,1,1],[1,1],[1,1]]=>[4,2,2]=>4
[[2,1,1],[2,1],[1]]=>[4,3,1]=>2
[[2,1,1],[1,1,1],[1]]=>[4,3,1]=>2
[[2,1,1],[2,1,1]]=>[4,4]=>0
[[1,1,1,1],[1],[1],[1],[1]]=>[4,1,1,1,1]=>4
[[1,1,1,1],[1,1],[1],[1]]=>[4,2,1,1]=>2
[[1,1,1,1],[1,1],[1,1]]=>[4,2,2]=>4
[[1,1,1,1],[1,1,1],[1]]=>[4,3,1]=>2
[[1,1,1,1],[1,1,1,1]]=>[4,4]=>0
[[5],[1],[1],[1]]=>[5,1,1,1]=>4
[[5],[2],[1]]=>[5,2,1]=>4
[[5],[3]]=>[5,3]=>2
[[4,1],[1],[1],[1]]=>[5,1,1,1]=>4
[[4,1],[2],[1]]=>[5,2,1]=>4
[[4,1],[1,1],[1]]=>[5,2,1]=>4
[[4,1],[3]]=>[5,3]=>2
[[4,1],[2,1]]=>[5,3]=>2
[[3,2],[1],[1],[1]]=>[5,1,1,1]=>4
[[3,2],[2],[1]]=>[5,2,1]=>4
[[3,2],[1,1],[1]]=>[5,2,1]=>4
[[3,2],[3]]=>[5,3]=>2
[[3,2],[2,1]]=>[5,3]=>2
[[3,1,1],[1],[1],[1]]=>[5,1,1,1]=>4
[[3,1,1],[2],[1]]=>[5,2,1]=>4
[[3,1,1],[1,1],[1]]=>[5,2,1]=>4
[[3,1,1],[3]]=>[5,3]=>2
[[3,1,1],[2,1]]=>[5,3]=>2
[[3,1,1],[1,1,1]]=>[5,3]=>2
[[2,2,1],[1],[1],[1]]=>[5,1,1,1]=>4
[[2,2,1],[2],[1]]=>[5,2,1]=>4
[[2,2,1],[1,1],[1]]=>[5,2,1]=>4
[[2,2,1],[2,1]]=>[5,3]=>2
[[2,2,1],[1,1,1]]=>[5,3]=>2
[[2,1,1,1],[1],[1],[1]]=>[5,1,1,1]=>4
[[2,1,1,1],[2],[1]]=>[5,2,1]=>4
[[2,1,1,1],[1,1],[1]]=>[5,2,1]=>4
[[2,1,1,1],[2,1]]=>[5,3]=>2
[[2,1,1,1],[1,1,1]]=>[5,3]=>2
[[1,1,1,1,1],[1],[1],[1]]=>[5,1,1,1]=>4
[[1,1,1,1,1],[1,1],[1]]=>[5,2,1]=>4
[[1,1,1,1,1],[1,1,1]]=>[5,3]=>2
[[6],[1],[1]]=>[6,1,1]=>6
[[6],[2]]=>[6,2]=>4
[[5,1],[1],[1]]=>[6,1,1]=>6
[[5,1],[2]]=>[6,2]=>4
[[5,1],[1,1]]=>[6,2]=>4
[[4,2],[1],[1]]=>[6,1,1]=>6
[[4,2],[2]]=>[6,2]=>4
[[4,2],[1,1]]=>[6,2]=>4
[[4,1,1],[1],[1]]=>[6,1,1]=>6
[[4,1,1],[2]]=>[6,2]=>4
[[4,1,1],[1,1]]=>[6,2]=>4
[[3,3],[1],[1]]=>[6,1,1]=>6
[[3,3],[2]]=>[6,2]=>4
[[3,3],[1,1]]=>[6,2]=>4
[[3,2,1],[1],[1]]=>[6,1,1]=>6
[[3,2,1],[2]]=>[6,2]=>4
[[3,2,1],[1,1]]=>[6,2]=>4
[[3,1,1,1],[1],[1]]=>[6,1,1]=>6
[[3,1,1,1],[2]]=>[6,2]=>4
[[3,1,1,1],[1,1]]=>[6,2]=>4
[[2,2,2],[1],[1]]=>[6,1,1]=>6
[[2,2,2],[2]]=>[6,2]=>4
[[2,2,2],[1,1]]=>[6,2]=>4
[[2,2,1,1],[1],[1]]=>[6,1,1]=>6
[[2,2,1,1],[2]]=>[6,2]=>4
[[2,2,1,1],[1,1]]=>[6,2]=>4
[[2,1,1,1,1],[1],[1]]=>[6,1,1]=>6
[[2,1,1,1,1],[2]]=>[6,2]=>4
[[2,1,1,1,1],[1,1]]=>[6,2]=>4
[[1,1,1,1,1,1],[1],[1]]=>[6,1,1]=>6
[[1,1,1,1,1,1],[1,1]]=>[6,2]=>4
[[7],[1]]=>[7,1]=>6
[[6,1],[1]]=>[7,1]=>6
[[5,2],[1]]=>[7,1]=>6
[[5,1,1],[1]]=>[7,1]=>6
[[4,3],[1]]=>[7,1]=>6
[[4,2,1],[1]]=>[7,1]=>6
[[4,1,1,1],[1]]=>[7,1]=>6
[[3,3,1],[1]]=>[7,1]=>6
[[3,2,2],[1]]=>[7,1]=>6
[[3,2,1,1],[1]]=>[7,1]=>6
[[3,1,1,1,1],[1]]=>[7,1]=>6
[[2,2,2,1],[1]]=>[7,1]=>6
[[2,2,1,1,1],[1]]=>[7,1]=>6
[[2,1,1,1,1,1],[1]]=>[7,1]=>6
[[1,1,1,1,1,1,1],[1]]=>[7,1]=>6
[[8]]=>[8]=>8
[[7,1]]=>[8]=>8
[[6,2]]=>[8]=>8
[[6,1,1]]=>[8]=>8
[[5,3]]=>[8]=>8
[[5,2,1]]=>[8]=>8
[[5,1,1,1]]=>[8]=>8
[[4,4]]=>[8]=>8
[[4,3,1]]=>[8]=>8
[[4,2,2]]=>[8]=>8
[[4,2,1,1]]=>[8]=>8
[[4,1,1,1,1]]=>[8]=>8
[[3,3,2]]=>[8]=>8
[[3,3,1,1]]=>[8]=>8
[[3,2,2,1]]=>[8]=>8
[[3,2,1,1,1]]=>[8]=>8
[[3,1,1,1,1,1]]=>[8]=>8
[[2,2,2,2]]=>[8]=>8
[[2,2,2,1,1]]=>[8]=>8
[[2,2,1,1,1,1]]=>[8]=>8
[[2,1,1,1,1,1,1]]=>[8]=>8
[[1,1,1,1,1,1,1,1]]=>[8]=>8
[[1],[1],[1],[1],[1],[1],[1],[1],[1]]=>[1,1,1,1,1,1,1,1,1]=>1
[[2],[1],[1],[1],[1],[1],[1],[1]]=>[2,1,1,1,1,1,1,1]=>1
[[2],[2],[1],[1],[1],[1],[1]]=>[2,2,1,1,1,1,1]=>1
[[2],[2],[2],[1],[1],[1]]=>[2,2,2,1,1,1]=>1
[[2],[2],[2],[2],[1]]=>[2,2,2,2,1]=>1
[[1,1],[1],[1],[1],[1],[1],[1],[1]]=>[2,1,1,1,1,1,1,1]=>1
[[1,1],[1,1],[1],[1],[1],[1],[1]]=>[2,2,1,1,1,1,1]=>1
[[1,1],[1,1],[1,1],[1],[1],[1]]=>[2,2,2,1,1,1]=>1
[[1,1],[1,1],[1,1],[1,1],[1]]=>[2,2,2,2,1]=>1
[[3],[1],[1],[1],[1],[1],[1]]=>[3,1,1,1,1,1,1]=>3
[[3],[2],[1],[1],[1],[1]]=>[3,2,1,1,1,1]=>1
[[3],[2],[2],[1],[1]]=>[3,2,2,1,1]=>3
[[3],[2],[2],[2]]=>[3,2,2,2]=>1
[[3],[3],[1],[1],[1]]=>[3,3,1,1,1]=>1
[[3],[3],[2],[1]]=>[3,3,2,1]=>1
[[3],[3],[3]]=>[3,3,3]=>3
[[2,1],[1],[1],[1],[1],[1],[1]]=>[3,1,1,1,1,1,1]=>3
[[2,1],[2],[1],[1],[1],[1]]=>[3,2,1,1,1,1]=>1
[[2,1],[2],[2],[1],[1]]=>[3,2,2,1,1]=>3
[[2,1],[2],[2],[2]]=>[3,2,2,2]=>1
[[2,1],[1,1],[1],[1],[1],[1]]=>[3,2,1,1,1,1]=>1
[[2,1],[1,1],[1,1],[1],[1]]=>[3,2,2,1,1]=>3
[[2,1],[1,1],[1,1],[1,1]]=>[3,2,2,2]=>1
[[2,1],[2,1],[1],[1],[1]]=>[3,3,1,1,1]=>1
[[2,1],[2,1],[2],[1]]=>[3,3,2,1]=>1
[[2,1],[2,1],[1,1],[1]]=>[3,3,2,1]=>1
[[2,1],[2,1],[2,1]]=>[3,3,3]=>3
[[1,1,1],[1],[1],[1],[1],[1],[1]]=>[3,1,1,1,1,1,1]=>3
[[1,1,1],[1,1],[1],[1],[1],[1]]=>[3,2,1,1,1,1]=>1
[[1,1,1],[1,1],[1,1],[1],[1]]=>[3,2,2,1,1]=>3
[[1,1,1],[1,1],[1,1],[1,1]]=>[3,2,2,2]=>1
[[1,1,1],[1,1,1],[1],[1],[1]]=>[3,3,1,1,1]=>1
[[1,1,1],[1,1,1],[1,1],[1]]=>[3,3,2,1]=>1
[[1,1,1],[1,1,1],[1,1,1]]=>[3,3,3]=>3
[[4],[1],[1],[1],[1],[1]]=>[4,1,1,1,1,1]=>3
[[4],[2],[1],[1],[1]]=>[4,2,1,1,1]=>3
[[4],[2],[2],[1]]=>[4,2,2,1]=>3
[[4],[3],[1],[1]]=>[4,3,1,1]=>1
[[4],[3],[2]]=>[4,3,2]=>3
[[4],[4],[1]]=>[4,4,1]=>1
[[3,1],[1],[1],[1],[1],[1]]=>[4,1,1,1,1,1]=>3
[[3,1],[2],[1],[1],[1]]=>[4,2,1,1,1]=>3
[[3,1],[2],[2],[1]]=>[4,2,2,1]=>3
[[3,1],[1,1],[1],[1],[1]]=>[4,2,1,1,1]=>3
[[3,1],[1,1],[1,1],[1]]=>[4,2,2,1]=>3
[[3,1],[3],[1],[1]]=>[4,3,1,1]=>1
[[3,1],[3],[2]]=>[4,3,2]=>3
[[3,1],[2,1],[1],[1]]=>[4,3,1,1]=>1
[[3,1],[2,1],[2]]=>[4,3,2]=>3
[[3,1],[2,1],[1,1]]=>[4,3,2]=>3
[[3,1],[3,1],[1]]=>[4,4,1]=>1
[[2,2],[1],[1],[1],[1],[1]]=>[4,1,1,1,1,1]=>3
[[2,2],[2],[1],[1],[1]]=>[4,2,1,1,1]=>3
[[2,2],[2],[2],[1]]=>[4,2,2,1]=>3
[[2,2],[1,1],[1],[1],[1]]=>[4,2,1,1,1]=>3
[[2,2],[1,1],[1,1],[1]]=>[4,2,2,1]=>3
[[2,2],[2,1],[1],[1]]=>[4,3,1,1]=>1
[[2,2],[2,1],[2]]=>[4,3,2]=>3
[[2,2],[2,1],[1,1]]=>[4,3,2]=>3
[[2,2],[2,2],[1]]=>[4,4,1]=>1
[[2,1,1],[1],[1],[1],[1],[1]]=>[4,1,1,1,1,1]=>3
[[2,1,1],[2],[1],[1],[1]]=>[4,2,1,1,1]=>3
[[2,1,1],[2],[2],[1]]=>[4,2,2,1]=>3
[[2,1,1],[1,1],[1],[1],[1]]=>[4,2,1,1,1]=>3
[[2,1,1],[1,1],[1,1],[1]]=>[4,2,2,1]=>3
[[2,1,1],[2,1],[1],[1]]=>[4,3,1,1]=>1
[[2,1,1],[2,1],[2]]=>[4,3,2]=>3
[[2,1,1],[2,1],[1,1]]=>[4,3,2]=>3
[[2,1,1],[1,1,1],[1],[1]]=>[4,3,1,1]=>1
[[2,1,1],[1,1,1],[1,1]]=>[4,3,2]=>3
[[2,1,1],[2,1,1],[1]]=>[4,4,1]=>1
[[1,1,1,1],[1],[1],[1],[1],[1]]=>[4,1,1,1,1,1]=>3
[[1,1,1,1],[1,1],[1],[1],[1]]=>[4,2,1,1,1]=>3
[[1,1,1,1],[1,1],[1,1],[1]]=>[4,2,2,1]=>3
[[1,1,1,1],[1,1,1],[1],[1]]=>[4,3,1,1]=>1
[[1,1,1,1],[1,1,1],[1,1]]=>[4,3,2]=>3
[[1,1,1,1],[1,1,1,1],[1]]=>[4,4,1]=>1
[[5],[1],[1],[1],[1]]=>[5,1,1,1,1]=>5
[[5],[2],[1],[1]]=>[5,2,1,1]=>3
[[5],[2],[2]]=>[5,2,2]=>5
[[5],[3],[1]]=>[5,3,1]=>3
[[5],[4]]=>[5,4]=>1
[[4,1],[1],[1],[1],[1]]=>[5,1,1,1,1]=>5
[[4,1],[2],[1],[1]]=>[5,2,1,1]=>3
[[4,1],[2],[2]]=>[5,2,2]=>5
[[4,1],[1,1],[1],[1]]=>[5,2,1,1]=>3
[[4,1],[1,1],[1,1]]=>[5,2,2]=>5
[[4,1],[3],[1]]=>[5,3,1]=>3
[[4,1],[2,1],[1]]=>[5,3,1]=>3
[[4,1],[4]]=>[5,4]=>1
[[4,1],[3,1]]=>[5,4]=>1
[[3,2],[1],[1],[1],[1]]=>[5,1,1,1,1]=>5
[[3,2],[2],[1],[1]]=>[5,2,1,1]=>3
[[3,2],[2],[2]]=>[5,2,2]=>5
[[3,2],[1,1],[1],[1]]=>[5,2,1,1]=>3
[[3,2],[1,1],[1,1]]=>[5,2,2]=>5
[[3,2],[3],[1]]=>[5,3,1]=>3
[[3,2],[2,1],[1]]=>[5,3,1]=>3
[[3,2],[3,1]]=>[5,4]=>1
[[3,2],[2,2]]=>[5,4]=>1
[[3,1,1],[1],[1],[1],[1]]=>[5,1,1,1,1]=>5
[[3,1,1],[2],[1],[1]]=>[5,2,1,1]=>3
[[3,1,1],[2],[2]]=>[5,2,2]=>5
[[3,1,1],[1,1],[1],[1]]=>[5,2,1,1]=>3
[[3,1,1],[1,1],[1,1]]=>[5,2,2]=>5
[[3,1,1],[3],[1]]=>[5,3,1]=>3
[[3,1,1],[2,1],[1]]=>[5,3,1]=>3
[[3,1,1],[1,1,1],[1]]=>[5,3,1]=>3
[[3,1,1],[3,1]]=>[5,4]=>1
[[3,1,1],[2,1,1]]=>[5,4]=>1
[[2,2,1],[1],[1],[1],[1]]=>[5,1,1,1,1]=>5
[[2,2,1],[2],[1],[1]]=>[5,2,1,1]=>3
[[2,2,1],[2],[2]]=>[5,2,2]=>5
[[2,2,1],[1,1],[1],[1]]=>[5,2,1,1]=>3
[[2,2,1],[1,1],[1,1]]=>[5,2,2]=>5
[[2,2,1],[2,1],[1]]=>[5,3,1]=>3
[[2,2,1],[1,1,1],[1]]=>[5,3,1]=>3
[[2,2,1],[2,2]]=>[5,4]=>1
[[2,2,1],[2,1,1]]=>[5,4]=>1
[[2,1,1,1],[1],[1],[1],[1]]=>[5,1,1,1,1]=>5
[[2,1,1,1],[2],[1],[1]]=>[5,2,1,1]=>3
[[2,1,1,1],[2],[2]]=>[5,2,2]=>5
[[2,1,1,1],[1,1],[1],[1]]=>[5,2,1,1]=>3
[[2,1,1,1],[1,1],[1,1]]=>[5,2,2]=>5
[[2,1,1,1],[2,1],[1]]=>[5,3,1]=>3
[[2,1,1,1],[1,1,1],[1]]=>[5,3,1]=>3
[[2,1,1,1],[2,1,1]]=>[5,4]=>1
[[2,1,1,1],[1,1,1,1]]=>[5,4]=>1
[[1,1,1,1,1],[1],[1],[1],[1]]=>[5,1,1,1,1]=>5
[[1,1,1,1,1],[1,1],[1],[1]]=>[5,2,1,1]=>3
[[1,1,1,1,1],[1,1],[1,1]]=>[5,2,2]=>5
[[1,1,1,1,1],[1,1,1],[1]]=>[5,3,1]=>3
[[1,1,1,1,1],[1,1,1,1]]=>[5,4]=>1
[[6],[1],[1],[1]]=>[6,1,1,1]=>5
[[6],[2],[1]]=>[6,2,1]=>5
[[6],[3]]=>[6,3]=>3
[[5,1],[1],[1],[1]]=>[6,1,1,1]=>5
[[5,1],[2],[1]]=>[6,2,1]=>5
[[5,1],[1,1],[1]]=>[6,2,1]=>5
[[5,1],[3]]=>[6,3]=>3
[[5,1],[2,1]]=>[6,3]=>3
[[4,2],[1],[1],[1]]=>[6,1,1,1]=>5
[[4,2],[2],[1]]=>[6,2,1]=>5
[[4,2],[1,1],[1]]=>[6,2,1]=>5
[[4,2],[3]]=>[6,3]=>3
[[4,2],[2,1]]=>[6,3]=>3
[[4,1,1],[1],[1],[1]]=>[6,1,1,1]=>5
[[4,1,1],[2],[1]]=>[6,2,1]=>5
[[4,1,1],[1,1],[1]]=>[6,2,1]=>5
[[4,1,1],[3]]=>[6,3]=>3
[[4,1,1],[2,1]]=>[6,3]=>3
[[4,1,1],[1,1,1]]=>[6,3]=>3
[[3,3],[1],[1],[1]]=>[6,1,1,1]=>5
[[3,3],[2],[1]]=>[6,2,1]=>5
[[3,3],[1,1],[1]]=>[6,2,1]=>5
[[3,3],[3]]=>[6,3]=>3
[[3,3],[2,1]]=>[6,3]=>3
[[3,2,1],[1],[1],[1]]=>[6,1,1,1]=>5
[[3,2,1],[2],[1]]=>[6,2,1]=>5
[[3,2,1],[1,1],[1]]=>[6,2,1]=>5
[[3,2,1],[3]]=>[6,3]=>3
[[3,2,1],[2,1]]=>[6,3]=>3
[[3,2,1],[1,1,1]]=>[6,3]=>3
[[3,1,1,1],[1],[1],[1]]=>[6,1,1,1]=>5
[[3,1,1,1],[2],[1]]=>[6,2,1]=>5
[[3,1,1,1],[1,1],[1]]=>[6,2,1]=>5
[[3,1,1,1],[3]]=>[6,3]=>3
[[3,1,1,1],[2,1]]=>[6,3]=>3
[[3,1,1,1],[1,1,1]]=>[6,3]=>3
[[2,2,2],[1],[1],[1]]=>[6,1,1,1]=>5
[[2,2,2],[2],[1]]=>[6,2,1]=>5
[[2,2,2],[1,1],[1]]=>[6,2,1]=>5
[[2,2,2],[2,1]]=>[6,3]=>3
[[2,2,2],[1,1,1]]=>[6,3]=>3
[[2,2,1,1],[1],[1],[1]]=>[6,1,1,1]=>5
[[2,2,1,1],[2],[1]]=>[6,2,1]=>5
[[2,2,1,1],[1,1],[1]]=>[6,2,1]=>5
[[2,2,1,1],[2,1]]=>[6,3]=>3
[[2,2,1,1],[1,1,1]]=>[6,3]=>3
[[2,1,1,1,1],[1],[1],[1]]=>[6,1,1,1]=>5
[[2,1,1,1,1],[2],[1]]=>[6,2,1]=>5
[[2,1,1,1,1],[1,1],[1]]=>[6,2,1]=>5
[[2,1,1,1,1],[2,1]]=>[6,3]=>3
[[2,1,1,1,1],[1,1,1]]=>[6,3]=>3
[[1,1,1,1,1,1],[1],[1],[1]]=>[6,1,1,1]=>5
[[1,1,1,1,1,1],[1,1],[1]]=>[6,2,1]=>5
[[1,1,1,1,1,1],[1,1,1]]=>[6,3]=>3
[[7],[1],[1]]=>[7,1,1]=>7
[[7],[2]]=>[7,2]=>5
[[6,1],[1],[1]]=>[7,1,1]=>7
[[6,1],[2]]=>[7,2]=>5
[[6,1],[1,1]]=>[7,2]=>5
[[5,2],[1],[1]]=>[7,1,1]=>7
[[5,2],[2]]=>[7,2]=>5
[[5,2],[1,1]]=>[7,2]=>5
[[5,1,1],[1],[1]]=>[7,1,1]=>7
[[5,1,1],[2]]=>[7,2]=>5
[[5,1,1],[1,1]]=>[7,2]=>5
[[4,3],[1],[1]]=>[7,1,1]=>7
[[4,3],[2]]=>[7,2]=>5
[[4,3],[1,1]]=>[7,2]=>5
[[4,2,1],[1],[1]]=>[7,1,1]=>7
[[4,2,1],[2]]=>[7,2]=>5
[[4,2,1],[1,1]]=>[7,2]=>5
[[4,1,1,1],[1],[1]]=>[7,1,1]=>7
[[4,1,1,1],[2]]=>[7,2]=>5
[[4,1,1,1],[1,1]]=>[7,2]=>5
[[3,3,1],[1],[1]]=>[7,1,1]=>7
[[3,3,1],[2]]=>[7,2]=>5
[[3,3,1],[1,1]]=>[7,2]=>5
[[3,2,2],[1],[1]]=>[7,1,1]=>7
[[3,2,2],[2]]=>[7,2]=>5
[[3,2,2],[1,1]]=>[7,2]=>5
[[3,2,1,1],[1],[1]]=>[7,1,1]=>7
[[3,2,1,1],[2]]=>[7,2]=>5
[[3,2,1,1],[1,1]]=>[7,2]=>5
[[3,1,1,1,1],[1],[1]]=>[7,1,1]=>7
[[3,1,1,1,1],[2]]=>[7,2]=>5
[[3,1,1,1,1],[1,1]]=>[7,2]=>5
[[2,2,2,1],[1],[1]]=>[7,1,1]=>7
[[2,2,2,1],[2]]=>[7,2]=>5
[[2,2,2,1],[1,1]]=>[7,2]=>5
[[2,2,1,1,1],[1],[1]]=>[7,1,1]=>7
[[2,2,1,1,1],[2]]=>[7,2]=>5
[[2,2,1,1,1],[1,1]]=>[7,2]=>5
[[2,1,1,1,1,1],[1],[1]]=>[7,1,1]=>7
[[2,1,1,1,1,1],[2]]=>[7,2]=>5
[[2,1,1,1,1,1],[1,1]]=>[7,2]=>5
[[1,1,1,1,1,1,1],[1],[1]]=>[7,1,1]=>7
[[1,1,1,1,1,1,1],[1,1]]=>[7,2]=>5
[[8],[1]]=>[8,1]=>7
[[7,1],[1]]=>[8,1]=>7
[[6,2],[1]]=>[8,1]=>7
[[6,1,1],[1]]=>[8,1]=>7
[[5,3],[1]]=>[8,1]=>7
[[5,2,1],[1]]=>[8,1]=>7
[[5,1,1,1],[1]]=>[8,1]=>7
[[4,4],[1]]=>[8,1]=>7
[[4,3,1],[1]]=>[8,1]=>7
[[4,2,2],[1]]=>[8,1]=>7
[[4,2,1,1],[1]]=>[8,1]=>7
[[4,1,1,1,1],[1]]=>[8,1]=>7
[[3,3,2],[1]]=>[8,1]=>7
[[3,3,1,1],[1]]=>[8,1]=>7
[[3,2,2,1],[1]]=>[8,1]=>7
[[3,2,1,1,1],[1]]=>[8,1]=>7
[[3,1,1,1,1,1],[1]]=>[8,1]=>7
[[2,2,2,2],[1]]=>[8,1]=>7
[[2,2,2,1,1],[1]]=>[8,1]=>7
[[2,2,1,1,1,1],[1]]=>[8,1]=>7
[[2,1,1,1,1,1,1],[1]]=>[8,1]=>7
[[1,1,1,1,1,1,1,1],[1]]=>[8,1]=>7
[[9]]=>[9]=>9
[[8,1]]=>[9]=>9
[[7,2]]=>[9]=>9
[[7,1,1]]=>[9]=>9
[[6,3]]=>[9]=>9
[[6,2,1]]=>[9]=>9
[[6,1,1,1]]=>[9]=>9
[[5,4]]=>[9]=>9
[[5,3,1]]=>[9]=>9
[[5,2,2]]=>[9]=>9
[[5,2,1,1]]=>[9]=>9
[[5,1,1,1,1]]=>[9]=>9
[[4,4,1]]=>[9]=>9
[[4,3,2]]=>[9]=>9
[[4,3,1,1]]=>[9]=>9
[[4,2,2,1]]=>[9]=>9
[[4,2,1,1,1]]=>[9]=>9
[[4,1,1,1,1,1]]=>[9]=>9
[[3,3,3]]=>[9]=>9
[[3,3,2,1]]=>[9]=>9
[[3,3,1,1,1]]=>[9]=>9
[[3,2,2,2]]=>[9]=>9
[[3,2,2,1,1]]=>[9]=>9
[[3,2,1,1,1,1]]=>[9]=>9
[[3,1,1,1,1,1,1]]=>[9]=>9
[[2,2,2,2,1]]=>[9]=>9
[[2,2,2,1,1,1]]=>[9]=>9
[[2,2,1,1,1,1,1]]=>[9]=>9
[[2,1,1,1,1,1,1,1]]=>[9]=>9
[[1,1,1,1,1,1,1,1,1]]=>[9]=>9
[[1],[1],[1],[1],[1],[1],[1],[1],[1],[1]]=>[1,1,1,1,1,1,1,1,1,1]=>0
[[2],[1],[1],[1],[1],[1],[1],[1],[1]]=>[2,1,1,1,1,1,1,1,1]=>2
[[2],[2],[1],[1],[1],[1],[1],[1]]=>[2,2,1,1,1,1,1,1]=>0
[[2],[2],[2],[1],[1],[1],[1]]=>[2,2,2,1,1,1,1]=>2
[[2],[2],[2],[2],[1],[1]]=>[2,2,2,2,1,1]=>0
[[2],[2],[2],[2],[2]]=>[2,2,2,2,2]=>2
[[1,1],[1],[1],[1],[1],[1],[1],[1],[1]]=>[2,1,1,1,1,1,1,1,1]=>2
[[1,1],[1,1],[1],[1],[1],[1],[1],[1]]=>[2,2,1,1,1,1,1,1]=>0
[[1,1],[1,1],[1,1],[1],[1],[1],[1]]=>[2,2,2,1,1,1,1]=>2
[[1,1],[1,1],[1,1],[1,1],[1],[1]]=>[2,2,2,2,1,1]=>0
[[1,1],[1,1],[1,1],[1,1],[1,1]]=>[2,2,2,2,2]=>2
[[3],[1],[1],[1],[1],[1],[1],[1]]=>[3,1,1,1,1,1,1,1]=>2
[[3],[2],[1],[1],[1],[1],[1]]=>[3,2,1,1,1,1,1]=>2
[[3],[2],[2],[1],[1],[1]]=>[3,2,2,1,1,1]=>2
[[3],[2],[2],[2],[1]]=>[3,2,2,2,1]=>2
[[3],[3],[1],[1],[1],[1]]=>[3,3,1,1,1,1]=>0
[[3],[3],[2],[1],[1]]=>[3,3,2,1,1]=>2
[[3],[3],[2],[2]]=>[3,3,2,2]=>0
[[3],[3],[3],[1]]=>[3,3,3,1]=>2
[[2,1],[1],[1],[1],[1],[1],[1],[1]]=>[3,1,1,1,1,1,1,1]=>2
[[2,1],[2],[1],[1],[1],[1],[1]]=>[3,2,1,1,1,1,1]=>2
[[2,1],[2],[2],[1],[1],[1]]=>[3,2,2,1,1,1]=>2
[[2,1],[2],[2],[2],[1]]=>[3,2,2,2,1]=>2
[[2,1],[1,1],[1],[1],[1],[1],[1]]=>[3,2,1,1,1,1,1]=>2
[[2,1],[1,1],[1,1],[1],[1],[1]]=>[3,2,2,1,1,1]=>2
[[2,1],[1,1],[1,1],[1,1],[1]]=>[3,2,2,2,1]=>2
[[2,1],[2,1],[1],[1],[1],[1]]=>[3,3,1,1,1,1]=>0
[[2,1],[2,1],[2],[1],[1]]=>[3,3,2,1,1]=>2
[[2,1],[2,1],[2],[2]]=>[3,3,2,2]=>0
[[2,1],[2,1],[1,1],[1],[1]]=>[3,3,2,1,1]=>2
[[2,1],[2,1],[1,1],[1,1]]=>[3,3,2,2]=>0
[[2,1],[2,1],[2,1],[1]]=>[3,3,3,1]=>2
[[1,1,1],[1],[1],[1],[1],[1],[1],[1]]=>[3,1,1,1,1,1,1,1]=>2
[[1,1,1],[1,1],[1],[1],[1],[1],[1]]=>[3,2,1,1,1,1,1]=>2
[[1,1,1],[1,1],[1,1],[1],[1],[1]]=>[3,2,2,1,1,1]=>2
[[1,1,1],[1,1],[1,1],[1,1],[1]]=>[3,2,2,2,1]=>2
[[1,1,1],[1,1,1],[1],[1],[1],[1]]=>[3,3,1,1,1,1]=>0
[[1,1,1],[1,1,1],[1,1],[1],[1]]=>[3,3,2,1,1]=>2
[[1,1,1],[1,1,1],[1,1],[1,1]]=>[3,3,2,2]=>0
[[1,1,1],[1,1,1],[1,1,1],[1]]=>[3,3,3,1]=>2
[[4],[1],[1],[1],[1],[1],[1]]=>[4,1,1,1,1,1,1]=>4
[[4],[2],[1],[1],[1],[1]]=>[4,2,1,1,1,1]=>2
[[4],[2],[2],[1],[1]]=>[4,2,2,1,1]=>4
[[4],[2],[2],[2]]=>[4,2,2,2]=>2
[[4],[3],[1],[1],[1]]=>[4,3,1,1,1]=>2
[[4],[3],[2],[1]]=>[4,3,2,1]=>2
[[4],[3],[3]]=>[4,3,3]=>4
[[4],[4],[1],[1]]=>[4,4,1,1]=>0
[[4],[4],[2]]=>[4,4,2]=>2
[[3,1],[1],[1],[1],[1],[1],[1]]=>[4,1,1,1,1,1,1]=>4
[[3,1],[2],[1],[1],[1],[1]]=>[4,2,1,1,1,1]=>2
[[3,1],[2],[2],[1],[1]]=>[4,2,2,1,1]=>4
[[3,1],[2],[2],[2]]=>[4,2,2,2]=>2
[[3,1],[1,1],[1],[1],[1],[1]]=>[4,2,1,1,1,1]=>2
[[3,1],[1,1],[1,1],[1],[1]]=>[4,2,2,1,1]=>4
[[3,1],[1,1],[1,1],[1,1]]=>[4,2,2,2]=>2
[[3,1],[3],[1],[1],[1]]=>[4,3,1,1,1]=>2
[[3,1],[3],[2],[1]]=>[4,3,2,1]=>2
[[3,1],[3],[3]]=>[4,3,3]=>4
[[3,1],[2,1],[1],[1],[1]]=>[4,3,1,1,1]=>2
[[3,1],[2,1],[2],[1]]=>[4,3,2,1]=>2
[[3,1],[2,1],[1,1],[1]]=>[4,3,2,1]=>2
[[3,1],[2,1],[2,1]]=>[4,3,3]=>4
[[3,1],[3,1],[1],[1]]=>[4,4,1,1]=>0
[[3,1],[3,1],[2]]=>[4,4,2]=>2
[[3,1],[3,1],[1,1]]=>[4,4,2]=>2
[[2,2],[1],[1],[1],[1],[1],[1]]=>[4,1,1,1,1,1,1]=>4
[[2,2],[2],[1],[1],[1],[1]]=>[4,2,1,1,1,1]=>2
[[2,2],[2],[2],[1],[1]]=>[4,2,2,1,1]=>4
[[2,2],[2],[2],[2]]=>[4,2,2,2]=>2
[[2,2],[1,1],[1],[1],[1],[1]]=>[4,2,1,1,1,1]=>2
[[2,2],[1,1],[1,1],[1],[1]]=>[4,2,2,1,1]=>4
[[2,2],[1,1],[1,1],[1,1]]=>[4,2,2,2]=>2
[[2,2],[2,1],[1],[1],[1]]=>[4,3,1,1,1]=>2
[[2,2],[2,1],[2],[1]]=>[4,3,2,1]=>2
[[2,2],[2,1],[1,1],[1]]=>[4,3,2,1]=>2
[[2,2],[2,1],[2,1]]=>[4,3,3]=>4
[[2,2],[2,2],[1],[1]]=>[4,4,1,1]=>0
[[2,2],[2,2],[2]]=>[4,4,2]=>2
[[2,2],[2,2],[1,1]]=>[4,4,2]=>2
[[2,1,1],[1],[1],[1],[1],[1],[1]]=>[4,1,1,1,1,1,1]=>4
[[2,1,1],[2],[1],[1],[1],[1]]=>[4,2,1,1,1,1]=>2
[[2,1,1],[2],[2],[1],[1]]=>[4,2,2,1,1]=>4
[[2,1,1],[2],[2],[2]]=>[4,2,2,2]=>2
[[2,1,1],[1,1],[1],[1],[1],[1]]=>[4,2,1,1,1,1]=>2
[[2,1,1],[1,1],[1,1],[1],[1]]=>[4,2,2,1,1]=>4
[[2,1,1],[1,1],[1,1],[1,1]]=>[4,2,2,2]=>2
[[2,1,1],[2,1],[1],[1],[1]]=>[4,3,1,1,1]=>2
[[2,1,1],[2,1],[2],[1]]=>[4,3,2,1]=>2
[[2,1,1],[2,1],[1,1],[1]]=>[4,3,2,1]=>2
[[2,1,1],[2,1],[2,1]]=>[4,3,3]=>4
[[2,1,1],[1,1,1],[1],[1],[1]]=>[4,3,1,1,1]=>2
[[2,1,1],[1,1,1],[1,1],[1]]=>[4,3,2,1]=>2
[[2,1,1],[1,1,1],[1,1,1]]=>[4,3,3]=>4
[[2,1,1],[2,1,1],[1],[1]]=>[4,4,1,1]=>0
[[2,1,1],[2,1,1],[2]]=>[4,4,2]=>2
[[2,1,1],[2,1,1],[1,1]]=>[4,4,2]=>2
[[1,1,1,1],[1],[1],[1],[1],[1],[1]]=>[4,1,1,1,1,1,1]=>4
[[1,1,1,1],[1,1],[1],[1],[1],[1]]=>[4,2,1,1,1,1]=>2
[[1,1,1,1],[1,1],[1,1],[1],[1]]=>[4,2,2,1,1]=>4
[[1,1,1,1],[1,1],[1,1],[1,1]]=>[4,2,2,2]=>2
[[1,1,1,1],[1,1,1],[1],[1],[1]]=>[4,3,1,1,1]=>2
[[1,1,1,1],[1,1,1],[1,1],[1]]=>[4,3,2,1]=>2
[[1,1,1,1],[1,1,1],[1,1,1]]=>[4,3,3]=>4
[[1,1,1,1],[1,1,1,1],[1],[1]]=>[4,4,1,1]=>0
[[1,1,1,1],[1,1,1,1],[1,1]]=>[4,4,2]=>2
[[5],[1],[1],[1],[1],[1]]=>[5,1,1,1,1,1]=>4
[[5],[2],[1],[1],[1]]=>[5,2,1,1,1]=>4
[[5],[2],[2],[1]]=>[5,2,2,1]=>4
[[5],[3],[1],[1]]=>[5,3,1,1]=>2
[[5],[3],[2]]=>[5,3,2]=>4
[[5],[4],[1]]=>[5,4,1]=>2
[[5],[5]]=>[5,5]=>0
[[4,1],[1],[1],[1],[1],[1]]=>[5,1,1,1,1,1]=>4
[[4,1],[2],[1],[1],[1]]=>[5,2,1,1,1]=>4
[[4,1],[2],[2],[1]]=>[5,2,2,1]=>4
[[4,1],[1,1],[1],[1],[1]]=>[5,2,1,1,1]=>4
[[4,1],[1,1],[1,1],[1]]=>[5,2,2,1]=>4
[[4,1],[3],[1],[1]]=>[5,3,1,1]=>2
[[4,1],[3],[2]]=>[5,3,2]=>4
[[4,1],[2,1],[1],[1]]=>[5,3,1,1]=>2
[[4,1],[2,1],[2]]=>[5,3,2]=>4
[[4,1],[2,1],[1,1]]=>[5,3,2]=>4
[[4,1],[4],[1]]=>[5,4,1]=>2
[[4,1],[3,1],[1]]=>[5,4,1]=>2
[[4,1],[4,1]]=>[5,5]=>0
[[3,2],[1],[1],[1],[1],[1]]=>[5,1,1,1,1,1]=>4
[[3,2],[2],[1],[1],[1]]=>[5,2,1,1,1]=>4
[[3,2],[2],[2],[1]]=>[5,2,2,1]=>4
[[3,2],[1,1],[1],[1],[1]]=>[5,2,1,1,1]=>4
[[3,2],[1,1],[1,1],[1]]=>[5,2,2,1]=>4
[[3,2],[3],[1],[1]]=>[5,3,1,1]=>2
[[3,2],[3],[2]]=>[5,3,2]=>4
[[3,2],[2,1],[1],[1]]=>[5,3,1,1]=>2
[[3,2],[2,1],[2]]=>[5,3,2]=>4
[[3,2],[2,1],[1,1]]=>[5,3,2]=>4
[[3,2],[3,1],[1]]=>[5,4,1]=>2
[[3,2],[2,2],[1]]=>[5,4,1]=>2
[[3,2],[3,2]]=>[5,5]=>0
[[3,1,1],[1],[1],[1],[1],[1]]=>[5,1,1,1,1,1]=>4
[[3,1,1],[2],[1],[1],[1]]=>[5,2,1,1,1]=>4
[[3,1,1],[2],[2],[1]]=>[5,2,2,1]=>4
[[3,1,1],[1,1],[1],[1],[1]]=>[5,2,1,1,1]=>4
[[3,1,1],[1,1],[1,1],[1]]=>[5,2,2,1]=>4
[[3,1,1],[3],[1],[1]]=>[5,3,1,1]=>2
[[3,1,1],[3],[2]]=>[5,3,2]=>4
[[3,1,1],[2,1],[1],[1]]=>[5,3,1,1]=>2
[[3,1,1],[2,1],[2]]=>[5,3,2]=>4
[[3,1,1],[2,1],[1,1]]=>[5,3,2]=>4
[[3,1,1],[1,1,1],[1],[1]]=>[5,3,1,1]=>2
[[3,1,1],[1,1,1],[1,1]]=>[5,3,2]=>4
[[3,1,1],[3,1],[1]]=>[5,4,1]=>2
[[3,1,1],[2,1,1],[1]]=>[5,4,1]=>2
[[3,1,1],[3,1,1]]=>[5,5]=>0
[[2,2,1],[1],[1],[1],[1],[1]]=>[5,1,1,1,1,1]=>4
[[2,2,1],[2],[1],[1],[1]]=>[5,2,1,1,1]=>4
[[2,2,1],[2],[2],[1]]=>[5,2,2,1]=>4
[[2,2,1],[1,1],[1],[1],[1]]=>[5,2,1,1,1]=>4
[[2,2,1],[1,1],[1,1],[1]]=>[5,2,2,1]=>4
[[2,2,1],[2,1],[1],[1]]=>[5,3,1,1]=>2
[[2,2,1],[2,1],[2]]=>[5,3,2]=>4
[[2,2,1],[2,1],[1,1]]=>[5,3,2]=>4
[[2,2,1],[1,1,1],[1],[1]]=>[5,3,1,1]=>2
[[2,2,1],[1,1,1],[1,1]]=>[5,3,2]=>4
[[2,2,1],[2,2],[1]]=>[5,4,1]=>2
[[2,2,1],[2,1,1],[1]]=>[5,4,1]=>2
[[2,2,1],[2,2,1]]=>[5,5]=>0
[[2,1,1,1],[1],[1],[1],[1],[1]]=>[5,1,1,1,1,1]=>4
[[2,1,1,1],[2],[1],[1],[1]]=>[5,2,1,1,1]=>4
[[2,1,1,1],[2],[2],[1]]=>[5,2,2,1]=>4
[[2,1,1,1],[1,1],[1],[1],[1]]=>[5,2,1,1,1]=>4
[[2,1,1,1],[1,1],[1,1],[1]]=>[5,2,2,1]=>4
[[2,1,1,1],[2,1],[1],[1]]=>[5,3,1,1]=>2
[[2,1,1,1],[2,1],[2]]=>[5,3,2]=>4
[[2,1,1,1],[2,1],[1,1]]=>[5,3,2]=>4
[[2,1,1,1],[1,1,1],[1],[1]]=>[5,3,1,1]=>2
[[2,1,1,1],[1,1,1],[1,1]]=>[5,3,2]=>4
[[2,1,1,1],[2,1,1],[1]]=>[5,4,1]=>2
[[2,1,1,1],[1,1,1,1],[1]]=>[5,4,1]=>2
[[2,1,1,1],[2,1,1,1]]=>[5,5]=>0
[[1,1,1,1,1],[1],[1],[1],[1],[1]]=>[5,1,1,1,1,1]=>4
[[1,1,1,1,1],[1,1],[1],[1],[1]]=>[5,2,1,1,1]=>4
[[1,1,1,1,1],[1,1],[1,1],[1]]=>[5,2,2,1]=>4
[[1,1,1,1,1],[1,1,1],[1],[1]]=>[5,3,1,1]=>2
[[1,1,1,1,1],[1,1,1],[1,1]]=>[5,3,2]=>4
[[1,1,1,1,1],[1,1,1,1],[1]]=>[5,4,1]=>2
[[1,1,1,1,1],[1,1,1,1,1]]=>[5,5]=>0
[[6],[1],[1],[1],[1]]=>[6,1,1,1,1]=>6
[[6],[2],[1],[1]]=>[6,2,1,1]=>4
[[6],[2],[2]]=>[6,2,2]=>6
[[6],[3],[1]]=>[6,3,1]=>4
[[6],[4]]=>[6,4]=>2
[[5,1],[1],[1],[1],[1]]=>[6,1,1,1,1]=>6
[[5,1],[2],[1],[1]]=>[6,2,1,1]=>4
[[5,1],[2],[2]]=>[6,2,2]=>6
[[5,1],[1,1],[1],[1]]=>[6,2,1,1]=>4
[[5,1],[1,1],[1,1]]=>[6,2,2]=>6
[[5,1],[3],[1]]=>[6,3,1]=>4
[[5,1],[2,1],[1]]=>[6,3,1]=>4
[[5,1],[4]]=>[6,4]=>2
[[5,1],[3,1]]=>[6,4]=>2
[[4,2],[1],[1],[1],[1]]=>[6,1,1,1,1]=>6
[[4,2],[2],[1],[1]]=>[6,2,1,1]=>4
[[4,2],[2],[2]]=>[6,2,2]=>6
[[4,2],[1,1],[1],[1]]=>[6,2,1,1]=>4
[[4,2],[1,1],[1,1]]=>[6,2,2]=>6
[[4,2],[3],[1]]=>[6,3,1]=>4
[[4,2],[2,1],[1]]=>[6,3,1]=>4
[[4,2],[4]]=>[6,4]=>2
[[4,2],[3,1]]=>[6,4]=>2
[[4,2],[2,2]]=>[6,4]=>2
[[4,1,1],[1],[1],[1],[1]]=>[6,1,1,1,1]=>6
[[4,1,1],[2],[1],[1]]=>[6,2,1,1]=>4
[[4,1,1],[2],[2]]=>[6,2,2]=>6
[[4,1,1],[1,1],[1],[1]]=>[6,2,1,1]=>4
[[4,1,1],[1,1],[1,1]]=>[6,2,2]=>6
[[4,1,1],[3],[1]]=>[6,3,1]=>4
[[4,1,1],[2,1],[1]]=>[6,3,1]=>4
[[4,1,1],[1,1,1],[1]]=>[6,3,1]=>4
[[4,1,1],[4]]=>[6,4]=>2
[[4,1,1],[3,1]]=>[6,4]=>2
[[4,1,1],[2,1,1]]=>[6,4]=>2
[[3,3],[1],[1],[1],[1]]=>[6,1,1,1,1]=>6
[[3,3],[2],[1],[1]]=>[6,2,1,1]=>4
[[3,3],[2],[2]]=>[6,2,2]=>6
[[3,3],[1,1],[1],[1]]=>[6,2,1,1]=>4
[[3,3],[1,1],[1,1]]=>[6,2,2]=>6
[[3,3],[3],[1]]=>[6,3,1]=>4
[[3,3],[2,1],[1]]=>[6,3,1]=>4
[[3,3],[3,1]]=>[6,4]=>2
[[3,3],[2,2]]=>[6,4]=>2
[[3,2,1],[1],[1],[1],[1]]=>[6,1,1,1,1]=>6
[[3,2,1],[2],[1],[1]]=>[6,2,1,1]=>4
[[3,2,1],[2],[2]]=>[6,2,2]=>6
[[3,2,1],[1,1],[1],[1]]=>[6,2,1,1]=>4
[[3,2,1],[1,1],[1,1]]=>[6,2,2]=>6
[[3,2,1],[3],[1]]=>[6,3,1]=>4
[[3,2,1],[2,1],[1]]=>[6,3,1]=>4
[[3,2,1],[1,1,1],[1]]=>[6,3,1]=>4
[[3,2,1],[3,1]]=>[6,4]=>2
[[3,2,1],[2,2]]=>[6,4]=>2
[[3,2,1],[2,1,1]]=>[6,4]=>2
[[3,1,1,1],[1],[1],[1],[1]]=>[6,1,1,1,1]=>6
[[3,1,1,1],[2],[1],[1]]=>[6,2,1,1]=>4
[[3,1,1,1],[2],[2]]=>[6,2,2]=>6
[[3,1,1,1],[1,1],[1],[1]]=>[6,2,1,1]=>4
[[3,1,1,1],[1,1],[1,1]]=>[6,2,2]=>6
[[3,1,1,1],[3],[1]]=>[6,3,1]=>4
[[3,1,1,1],[2,1],[1]]=>[6,3,1]=>4
[[3,1,1,1],[1,1,1],[1]]=>[6,3,1]=>4
[[3,1,1,1],[3,1]]=>[6,4]=>2
[[3,1,1,1],[2,1,1]]=>[6,4]=>2
[[3,1,1,1],[1,1,1,1]]=>[6,4]=>2
[[2,2,2],[1],[1],[1],[1]]=>[6,1,1,1,1]=>6
[[2,2,2],[2],[1],[1]]=>[6,2,1,1]=>4
[[2,2,2],[2],[2]]=>[6,2,2]=>6
[[2,2,2],[1,1],[1],[1]]=>[6,2,1,1]=>4
[[2,2,2],[1,1],[1,1]]=>[6,2,2]=>6
[[2,2,2],[2,1],[1]]=>[6,3,1]=>4
[[2,2,2],[1,1,1],[1]]=>[6,3,1]=>4
[[2,2,2],[2,2]]=>[6,4]=>2
[[2,2,2],[2,1,1]]=>[6,4]=>2
[[2,2,1,1],[1],[1],[1],[1]]=>[6,1,1,1,1]=>6
[[2,2,1,1],[2],[1],[1]]=>[6,2,1,1]=>4
[[2,2,1,1],[2],[2]]=>[6,2,2]=>6
[[2,2,1,1],[1,1],[1],[1]]=>[6,2,1,1]=>4
[[2,2,1,1],[1,1],[1,1]]=>[6,2,2]=>6
[[2,2,1,1],[2,1],[1]]=>[6,3,1]=>4
[[2,2,1,1],[1,1,1],[1]]=>[6,3,1]=>4
[[2,2,1,1],[2,2]]=>[6,4]=>2
[[2,2,1,1],[2,1,1]]=>[6,4]=>2
[[2,2,1,1],[1,1,1,1]]=>[6,4]=>2
[[2,1,1,1,1],[1],[1],[1],[1]]=>[6,1,1,1,1]=>6
[[2,1,1,1,1],[2],[1],[1]]=>[6,2,1,1]=>4
[[2,1,1,1,1],[2],[2]]=>[6,2,2]=>6
[[2,1,1,1,1],[1,1],[1],[1]]=>[6,2,1,1]=>4
[[2,1,1,1,1],[1,1],[1,1]]=>[6,2,2]=>6
[[2,1,1,1,1],[2,1],[1]]=>[6,3,1]=>4
[[2,1,1,1,1],[1,1,1],[1]]=>[6,3,1]=>4
[[2,1,1,1,1],[2,1,1]]=>[6,4]=>2
[[2,1,1,1,1],[1,1,1,1]]=>[6,4]=>2
[[1,1,1,1,1,1],[1],[1],[1],[1]]=>[6,1,1,1,1]=>6
[[1,1,1,1,1,1],[1,1],[1],[1]]=>[6,2,1,1]=>4
[[1,1,1,1,1,1],[1,1],[1,1]]=>[6,2,2]=>6
[[1,1,1,1,1,1],[1,1,1],[1]]=>[6,3,1]=>4
[[1,1,1,1,1,1],[1,1,1,1]]=>[6,4]=>2
[[7],[1],[1],[1]]=>[7,1,1,1]=>6
[[7],[2],[1]]=>[7,2,1]=>6
[[7],[3]]=>[7,3]=>4
[[6,1],[1],[1],[1]]=>[7,1,1,1]=>6
[[6,1],[2],[1]]=>[7,2,1]=>6
[[6,1],[1,1],[1]]=>[7,2,1]=>6
[[6,1],[3]]=>[7,3]=>4
[[6,1],[2,1]]=>[7,3]=>4
[[5,2],[1],[1],[1]]=>[7,1,1,1]=>6
[[5,2],[2],[1]]=>[7,2,1]=>6
[[5,2],[1,1],[1]]=>[7,2,1]=>6
[[5,2],[3]]=>[7,3]=>4
[[5,2],[2,1]]=>[7,3]=>4
[[5,1,1],[1],[1],[1]]=>[7,1,1,1]=>6
[[5,1,1],[2],[1]]=>[7,2,1]=>6
[[5,1,1],[1,1],[1]]=>[7,2,1]=>6
[[5,1,1],[3]]=>[7,3]=>4
[[5,1,1],[2,1]]=>[7,3]=>4
[[5,1,1],[1,1,1]]=>[7,3]=>4
[[4,3],[1],[1],[1]]=>[7,1,1,1]=>6
[[4,3],[2],[1]]=>[7,2,1]=>6
[[4,3],[1,1],[1]]=>[7,2,1]=>6
[[4,3],[3]]=>[7,3]=>4
[[4,3],[2,1]]=>[7,3]=>4
[[4,2,1],[1],[1],[1]]=>[7,1,1,1]=>6
[[4,2,1],[2],[1]]=>[7,2,1]=>6
[[4,2,1],[1,1],[1]]=>[7,2,1]=>6
[[4,2,1],[3]]=>[7,3]=>4
[[4,2,1],[2,1]]=>[7,3]=>4
[[4,2,1],[1,1,1]]=>[7,3]=>4
[[4,1,1,1],[1],[1],[1]]=>[7,1,1,1]=>6
[[4,1,1,1],[2],[1]]=>[7,2,1]=>6
[[4,1,1,1],[1,1],[1]]=>[7,2,1]=>6
[[4,1,1,1],[3]]=>[7,3]=>4
[[4,1,1,1],[2,1]]=>[7,3]=>4
[[4,1,1,1],[1,1,1]]=>[7,3]=>4
[[3,3,1],[1],[1],[1]]=>[7,1,1,1]=>6
[[3,3,1],[2],[1]]=>[7,2,1]=>6
[[3,3,1],[1,1],[1]]=>[7,2,1]=>6
[[3,3,1],[3]]=>[7,3]=>4
[[3,3,1],[2,1]]=>[7,3]=>4
[[3,3,1],[1,1,1]]=>[7,3]=>4
[[3,2,2],[1],[1],[1]]=>[7,1,1,1]=>6
[[3,2,2],[2],[1]]=>[7,2,1]=>6
[[3,2,2],[1,1],[1]]=>[7,2,1]=>6
[[3,2,2],[3]]=>[7,3]=>4
[[3,2,2],[2,1]]=>[7,3]=>4
[[3,2,2],[1,1,1]]=>[7,3]=>4
[[3,2,1,1],[1],[1],[1]]=>[7,1,1,1]=>6
[[3,2,1,1],[2],[1]]=>[7,2,1]=>6
[[3,2,1,1],[1,1],[1]]=>[7,2,1]=>6
[[3,2,1,1],[3]]=>[7,3]=>4
[[3,2,1,1],[2,1]]=>[7,3]=>4
[[3,2,1,1],[1,1,1]]=>[7,3]=>4
[[3,1,1,1,1],[1],[1],[1]]=>[7,1,1,1]=>6
[[3,1,1,1,1],[2],[1]]=>[7,2,1]=>6
[[3,1,1,1,1],[1,1],[1]]=>[7,2,1]=>6
[[3,1,1,1,1],[3]]=>[7,3]=>4
[[3,1,1,1,1],[2,1]]=>[7,3]=>4
[[3,1,1,1,1],[1,1,1]]=>[7,3]=>4
[[2,2,2,1],[1],[1],[1]]=>[7,1,1,1]=>6
[[2,2,2,1],[2],[1]]=>[7,2,1]=>6
[[2,2,2,1],[1,1],[1]]=>[7,2,1]=>6
[[2,2,2,1],[2,1]]=>[7,3]=>4
[[2,2,2,1],[1,1,1]]=>[7,3]=>4
[[2,2,1,1,1],[1],[1],[1]]=>[7,1,1,1]=>6
[[2,2,1,1,1],[2],[1]]=>[7,2,1]=>6
[[2,2,1,1,1],[1,1],[1]]=>[7,2,1]=>6
[[2,2,1,1,1],[2,1]]=>[7,3]=>4
[[2,2,1,1,1],[1,1,1]]=>[7,3]=>4
[[2,1,1,1,1,1],[1],[1],[1]]=>[7,1,1,1]=>6
[[2,1,1,1,1,1],[2],[1]]=>[7,2,1]=>6
[[2,1,1,1,1,1],[1,1],[1]]=>[7,2,1]=>6
[[2,1,1,1,1,1],[2,1]]=>[7,3]=>4
[[2,1,1,1,1,1],[1,1,1]]=>[7,3]=>4
[[1,1,1,1,1,1,1],[1],[1],[1]]=>[7,1,1,1]=>6
[[1,1,1,1,1,1,1],[1,1],[1]]=>[7,2,1]=>6
[[1,1,1,1,1,1,1],[1,1,1]]=>[7,3]=>4
[[8],[1],[1]]=>[8,1,1]=>8
[[8],[2]]=>[8,2]=>6
[[7,1],[1],[1]]=>[8,1,1]=>8
[[7,1],[2]]=>[8,2]=>6
[[7,1],[1,1]]=>[8,2]=>6
[[6,2],[1],[1]]=>[8,1,1]=>8
[[6,2],[2]]=>[8,2]=>6
[[6,2],[1,1]]=>[8,2]=>6
[[6,1,1],[1],[1]]=>[8,1,1]=>8
[[6,1,1],[2]]=>[8,2]=>6
[[6,1,1],[1,1]]=>[8,2]=>6
[[5,3],[1],[1]]=>[8,1,1]=>8
[[5,3],[2]]=>[8,2]=>6
[[5,3],[1,1]]=>[8,2]=>6
[[5,2,1],[1],[1]]=>[8,1,1]=>8
[[5,2,1],[2]]=>[8,2]=>6
[[5,2,1],[1,1]]=>[8,2]=>6
[[5,1,1,1],[1],[1]]=>[8,1,1]=>8
[[5,1,1,1],[2]]=>[8,2]=>6
[[5,1,1,1],[1,1]]=>[8,2]=>6
[[4,4],[1],[1]]=>[8,1,1]=>8
[[4,4],[2]]=>[8,2]=>6
[[4,4],[1,1]]=>[8,2]=>6
[[4,3,1],[1],[1]]=>[8,1,1]=>8
[[4,3,1],[2]]=>[8,2]=>6
[[4,3,1],[1,1]]=>[8,2]=>6
[[4,2,2],[1],[1]]=>[8,1,1]=>8
[[4,2,2],[2]]=>[8,2]=>6
[[4,2,2],[1,1]]=>[8,2]=>6
[[4,2,1,1],[1],[1]]=>[8,1,1]=>8
[[4,2,1,1],[2]]=>[8,2]=>6
[[4,2,1,1],[1,1]]=>[8,2]=>6
[[4,1,1,1,1],[1],[1]]=>[8,1,1]=>8
[[4,1,1,1,1],[2]]=>[8,2]=>6
[[4,1,1,1,1],[1,1]]=>[8,2]=>6
[[3,3,2],[1],[1]]=>[8,1,1]=>8
[[3,3,2],[2]]=>[8,2]=>6
[[3,3,2],[1,1]]=>[8,2]=>6
[[3,3,1,1],[1],[1]]=>[8,1,1]=>8
[[3,3,1,1],[2]]=>[8,2]=>6
[[3,3,1,1],[1,1]]=>[8,2]=>6
[[3,2,2,1],[1],[1]]=>[8,1,1]=>8
[[3,2,2,1],[2]]=>[8,2]=>6
[[3,2,2,1],[1,1]]=>[8,2]=>6
[[3,2,1,1,1],[1],[1]]=>[8,1,1]=>8
[[3,2,1,1,1],[2]]=>[8,2]=>6
[[3,2,1,1,1],[1,1]]=>[8,2]=>6
[[3,1,1,1,1,1],[1],[1]]=>[8,1,1]=>8
[[3,1,1,1,1,1],[2]]=>[8,2]=>6
[[3,1,1,1,1,1],[1,1]]=>[8,2]=>6
[[2,2,2,2],[1],[1]]=>[8,1,1]=>8
[[2,2,2,2],[2]]=>[8,2]=>6
[[2,2,2,2],[1,1]]=>[8,2]=>6
[[2,2,2,1,1],[1],[1]]=>[8,1,1]=>8
[[2,2,2,1,1],[2]]=>[8,2]=>6
[[2,2,2,1,1],[1,1]]=>[8,2]=>6
[[2,2,1,1,1,1],[1],[1]]=>[8,1,1]=>8
[[2,2,1,1,1,1],[2]]=>[8,2]=>6
[[2,2,1,1,1,1],[1,1]]=>[8,2]=>6
[[2,1,1,1,1,1,1],[1],[1]]=>[8,1,1]=>8
[[2,1,1,1,1,1,1],[2]]=>[8,2]=>6
[[2,1,1,1,1,1,1],[1,1]]=>[8,2]=>6
[[1,1,1,1,1,1,1,1],[1],[1]]=>[8,1,1]=>8
[[1,1,1,1,1,1,1,1],[1,1]]=>[8,2]=>6
[[9],[1]]=>[9,1]=>8
[[8,1],[1]]=>[9,1]=>8
[[7,2],[1]]=>[9,1]=>8
[[7,1,1],[1]]=>[9,1]=>8
[[6,3],[1]]=>[9,1]=>8
[[6,2,1],[1]]=>[9,1]=>8
[[6,1,1,1],[1]]=>[9,1]=>8
[[5,4],[1]]=>[9,1]=>8
[[5,3,1],[1]]=>[9,1]=>8
[[5,2,2],[1]]=>[9,1]=>8
[[5,2,1,1],[1]]=>[9,1]=>8
[[5,1,1,1,1],[1]]=>[9,1]=>8
[[4,4,1],[1]]=>[9,1]=>8
[[4,3,2],[1]]=>[9,1]=>8
[[4,3,1,1],[1]]=>[9,1]=>8
[[4,2,2,1],[1]]=>[9,1]=>8
[[4,2,1,1,1],[1]]=>[9,1]=>8
[[4,1,1,1,1,1],[1]]=>[9,1]=>8
[[3,3,3],[1]]=>[9,1]=>8
[[3,3,2,1],[1]]=>[9,1]=>8
[[3,3,1,1,1],[1]]=>[9,1]=>8
[[3,2,2,2],[1]]=>[9,1]=>8
[[3,2,2,1,1],[1]]=>[9,1]=>8
[[3,2,1,1,1,1],[1]]=>[9,1]=>8
[[3,1,1,1,1,1,1],[1]]=>[9,1]=>8
[[2,2,2,2,1],[1]]=>[9,1]=>8
[[2,2,2,1,1,1],[1]]=>[9,1]=>8
[[2,2,1,1,1,1,1],[1]]=>[9,1]=>8
[[2,1,1,1,1,1,1,1],[1]]=>[9,1]=>8
[[1,1,1,1,1,1,1,1,1],[1]]=>[9,1]=>8
[[10]]=>[10]=>10
[[9,1]]=>[10]=>10
[[8,2]]=>[10]=>10
[[8,1,1]]=>[10]=>10
[[7,3]]=>[10]=>10
[[7,2,1]]=>[10]=>10
[[7,1,1,1]]=>[10]=>10
[[6,4]]=>[10]=>10
[[6,3,1]]=>[10]=>10
[[6,2,2]]=>[10]=>10
[[6,2,1,1]]=>[10]=>10
[[6,1,1,1,1]]=>[10]=>10
[[5,5]]=>[10]=>10
[[5,4,1]]=>[10]=>10
[[5,3,2]]=>[10]=>10
[[5,3,1,1]]=>[10]=>10
[[5,2,2,1]]=>[10]=>10
[[5,2,1,1,1]]=>[10]=>10
[[5,1,1,1,1,1]]=>[10]=>10
[[4,4,2]]=>[10]=>10
[[4,4,1,1]]=>[10]=>10
[[4,3,3]]=>[10]=>10
[[4,3,2,1]]=>[10]=>10
[[4,3,1,1,1]]=>[10]=>10
[[4,2,2,2]]=>[10]=>10
[[4,2,2,1,1]]=>[10]=>10
[[4,2,1,1,1,1]]=>[10]=>10
[[4,1,1,1,1,1,1]]=>[10]=>10
[[3,3,3,1]]=>[10]=>10
[[3,3,2,2]]=>[10]=>10
[[3,3,2,1,1]]=>[10]=>10
[[3,3,1,1,1,1]]=>[10]=>10
[[3,2,2,2,1]]=>[10]=>10
[[3,2,2,1,1,1]]=>[10]=>10
[[3,2,1,1,1,1,1]]=>[10]=>10
[[3,1,1,1,1,1,1,1]]=>[10]=>10
[[2,2,2,2,2]]=>[10]=>10
[[2,2,2,2,1,1]]=>[10]=>10
[[2,2,2,1,1,1,1]]=>[10]=>10
[[2,2,1,1,1,1,1,1]]=>[10]=>10
[[2,1,1,1,1,1,1,1,1]]=>[10]=>10
[[1,1,1,1,1,1,1,1,1,1]]=>[10]=>10
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Description
The alternating sum of the parts of an integer partition.
For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$, this is $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$, this is $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
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Description
The underlying integer partition of a plane partition.
This is the partition whose parts are the sums of the individual rows of the plane partition.
This is the partition whose parts are the sums of the individual rows of the plane partition.
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