Identifier
Mp00233:
Dyck paths
—skew partition⟶
Skew partitions
Mp00187: Skew partitions —conjugate⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00187: Skew partitions —conjugate⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Images
=>
Cc0005;cc-rep-0Cc0028;cc-rep-1Cc0028;cc-rep-2Cc0028;cc-rep-3
[1,0]=>[[1],[]]=>[[1],[]]=>[[1],[]]
[1,0,1,0]=>[[1,1],[]]=>[[2],[]]=>[[2],[]]
[1,1,0,0]=>[[2],[]]=>[[1,1],[]]=>[[1,1],[]]
[1,0,1,0,1,0]=>[[1,1,1],[]]=>[[3],[]]=>[[3],[]]
[1,0,1,1,0,0]=>[[2,1],[]]=>[[2,1],[]]=>[[2,2],[1]]
[1,1,0,0,1,0]=>[[2,2],[1]]=>[[2,2],[1]]=>[[2,1],[]]
[1,1,0,1,0,0]=>[[3],[]]=>[[1,1,1],[]]=>[[1,1,1],[]]
[1,1,1,0,0,0]=>[[2,2],[]]=>[[2,2],[]]=>[[2,2],[]]
[1,0,1,0,1,0,1,0]=>[[1,1,1,1],[]]=>[[4],[]]=>[[4],[]]
[1,0,1,0,1,1,0,0]=>[[2,1,1],[]]=>[[3,1],[]]=>[[3,3],[2]]
[1,0,1,1,0,0,1,0]=>[[2,2,1],[1]]=>[[3,2],[1]]=>[[3,2],[1]]
[1,0,1,1,0,1,0,0]=>[[3,1],[]]=>[[2,1,1],[]]=>[[2,2,2],[1,1]]
[1,0,1,1,1,0,0,0]=>[[2,2,1],[]]=>[[3,2],[]]=>[[3,3],[1]]
[1,1,0,0,1,0,1,0]=>[[2,2,2],[1,1]]=>[[3,3],[2]]=>[[3,1],[]]
[1,1,0,0,1,1,0,0]=>[[3,2],[1]]=>[[2,2,1],[1]]=>[[2,2,1],[1]]
[1,1,0,1,0,0,1,0]=>[[3,3],[2]]=>[[2,2,2],[1,1]]=>[[2,1,1],[]]
[1,1,0,1,0,1,0,0]=>[[4],[]]=>[[1,1,1,1],[]]=>[[1,1,1,1],[]]
[1,1,0,1,1,0,0,0]=>[[3,3],[1]]=>[[2,2,2],[1]]=>[[2,2,1],[]]
[1,1,1,0,0,0,1,0]=>[[2,2,2],[1]]=>[[3,3],[1]]=>[[3,2],[]]
[1,1,1,0,0,1,0,0]=>[[3,2],[]]=>[[2,2,1],[]]=>[[2,2,2],[1]]
[1,1,1,0,1,0,0,0]=>[[2,2,2],[]]=>[[3,3],[]]=>[[3,3],[]]
[1,1,1,1,0,0,0,0]=>[[3,3],[]]=>[[2,2,2],[]]=>[[2,2,2],[]]
[1,0,1,0,1,0,1,0,1,0]=>[[1,1,1,1,1],[]]=>[[5],[]]=>[[5],[]]
[1,0,1,0,1,0,1,1,0,0]=>[[2,1,1,1],[]]=>[[4,1],[]]=>[[4,4],[3]]
[1,0,1,0,1,1,0,0,1,0]=>[[2,2,1,1],[1]]=>[[4,2],[1]]=>[[4,3],[2]]
[1,0,1,0,1,1,0,1,0,0]=>[[3,1,1],[]]=>[[3,1,1],[]]=>[[3,3,3],[2,2]]
[1,0,1,0,1,1,1,0,0,0]=>[[2,2,1,1],[]]=>[[4,2],[]]=>[[4,4],[2]]
[1,0,1,1,0,0,1,0,1,0]=>[[2,2,2,1],[1,1]]=>[[4,3],[2]]=>[[4,2],[1]]
[1,0,1,1,0,0,1,1,0,0]=>[[3,2,1],[1]]=>[[3,2,1],[1]]=>[[3,3,2],[2,1]]
[1,0,1,1,0,1,0,0,1,0]=>[[3,3,1],[2]]=>[[3,2,2],[1,1]]=>[[3,2,2],[1,1]]
[1,0,1,1,0,1,0,1,0,0]=>[[4,1],[]]=>[[2,1,1,1],[]]=>[[2,2,2,2],[1,1,1]]
[1,0,1,1,0,1,1,0,0,0]=>[[3,3,1],[1]]=>[[3,2,2],[1]]=>[[3,3,2],[1,1]]
[1,0,1,1,1,0,0,0,1,0]=>[[2,2,2,1],[1]]=>[[4,3],[1]]=>[[4,3],[1]]
[1,0,1,1,1,0,0,1,0,0]=>[[3,2,1],[]]=>[[3,2,1],[]]=>[[3,3,3],[2,1]]
[1,0,1,1,1,0,1,0,0,0]=>[[2,2,2,1],[]]=>[[4,3],[]]=>[[4,4],[1]]
[1,0,1,1,1,1,0,0,0,0]=>[[3,3,1],[]]=>[[3,2,2],[]]=>[[3,3,3],[1,1]]
[1,1,0,0,1,0,1,0,1,0]=>[[2,2,2,2],[1,1,1]]=>[[4,4],[3]]=>[[4,1],[]]
[1,1,0,0,1,0,1,1,0,0]=>[[3,2,2],[1,1]]=>[[3,3,1],[2]]=>[[3,3,1],[2]]
[1,1,0,0,1,1,0,0,1,0]=>[[3,3,2],[2,1]]=>[[3,3,2],[2,1]]=>[[3,2,1],[1]]
[1,1,0,0,1,1,0,1,0,0]=>[[4,2],[1]]=>[[2,2,1,1],[1]]=>[[2,2,2,1],[1,1]]
[1,1,0,0,1,1,1,0,0,0]=>[[3,3,2],[1,1]]=>[[3,3,2],[2]]=>[[3,3,1],[1]]
[1,1,0,1,0,0,1,0,1,0]=>[[3,3,3],[2,2]]=>[[3,3,3],[2,2]]=>[[3,1,1],[]]
[1,1,0,1,0,0,1,1,0,0]=>[[4,3],[2]]=>[[2,2,2,1],[1,1]]=>[[2,2,1,1],[1]]
[1,1,0,1,0,1,0,0,1,0]=>[[4,4],[3]]=>[[2,2,2,2],[1,1,1]]=>[[2,1,1,1],[]]
[1,1,0,1,0,1,0,1,0,0]=>[[5],[]]=>[[1,1,1,1,1],[]]=>[[1,1,1,1,1],[]]
[1,1,0,1,0,1,1,0,0,0]=>[[4,4],[2]]=>[[2,2,2,2],[1,1]]=>[[2,2,1,1],[]]
[1,1,0,1,1,0,0,0,1,0]=>[[3,3,3],[2,1]]=>[[3,3,3],[2,1]]=>[[3,2,1],[]]
[1,1,0,1,1,0,0,1,0,0]=>[[4,3],[1]]=>[[2,2,2,1],[1]]=>[[2,2,2,1],[1]]
[1,1,0,1,1,0,1,0,0,0]=>[[3,3,3],[1,1]]=>[[3,3,3],[2]]=>[[3,3,1],[]]
[1,1,0,1,1,1,0,0,0,0]=>[[4,4],[1]]=>[[2,2,2,2],[1]]=>[[2,2,2,1],[]]
[1,1,1,0,0,0,1,0,1,0]=>[[2,2,2,2],[1,1]]=>[[4,4],[2]]=>[[4,2],[]]
[1,1,1,0,0,0,1,1,0,0]=>[[3,2,2],[1]]=>[[3,3,1],[1]]=>[[3,3,2],[2]]
[1,1,1,0,0,1,0,0,1,0]=>[[3,3,2],[2]]=>[[3,3,2],[1,1]]=>[[3,2,2],[1]]
[1,1,1,0,0,1,0,1,0,0]=>[[4,2],[]]=>[[2,2,1,1],[]]=>[[2,2,2,2],[1,1]]
[1,1,1,0,0,1,1,0,0,0]=>[[3,3,2],[1]]=>[[3,3,2],[1]]=>[[3,3,2],[1]]
[1,1,1,0,1,0,0,0,1,0]=>[[2,2,2,2],[1]]=>[[4,4],[1]]=>[[4,3],[]]
[1,1,1,0,1,0,0,1,0,0]=>[[3,2,2],[]]=>[[3,3,1],[]]=>[[3,3,3],[2]]
[1,1,1,0,1,0,1,0,0,0]=>[[2,2,2,2],[]]=>[[4,4],[]]=>[[4,4],[]]
[1,1,1,0,1,1,0,0,0,0]=>[[3,3,2],[]]=>[[3,3,2],[]]=>[[3,3,3],[1]]
[1,1,1,1,0,0,0,0,1,0]=>[[3,3,3],[2]]=>[[3,3,3],[1,1]]=>[[3,2,2],[]]
[1,1,1,1,0,0,0,1,0,0]=>[[4,3],[]]=>[[2,2,2,1],[]]=>[[2,2,2,2],[1]]
[1,1,1,1,0,0,1,0,0,0]=>[[3,3,3],[1]]=>[[3,3,3],[1]]=>[[3,3,2],[]]
[1,1,1,1,0,1,0,0,0,0]=>[[4,4],[]]=>[[2,2,2,2],[]]=>[[2,2,2,2],[]]
[1,1,1,1,1,0,0,0,0,0]=>[[3,3,3],[]]=>[[3,3,3],[]]=>[[3,3,3],[]]
[1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,1,1,1,1,1],[]]=>[[6],[]]=>[[6],[]]
[1,0,1,0,1,0,1,0,1,1,0,0]=>[[2,1,1,1,1],[]]=>[[5,1],[]]=>[[5,5],[4]]
[1,0,1,0,1,0,1,1,0,0,1,0]=>[[2,2,1,1,1],[1]]=>[[5,2],[1]]=>[[5,4],[3]]
[1,0,1,0,1,0,1,1,0,1,0,0]=>[[3,1,1,1],[]]=>[[4,1,1],[]]=>[[4,4,4],[3,3]]
[1,0,1,0,1,0,1,1,1,0,0,0]=>[[2,2,1,1,1],[]]=>[[5,2],[]]=>[[5,5],[3]]
[1,0,1,0,1,1,0,0,1,0,1,0]=>[[2,2,2,1,1],[1,1]]=>[[5,3],[2]]=>[[5,3],[2]]
[1,0,1,0,1,1,0,0,1,1,0,0]=>[[3,2,1,1],[1]]=>[[4,2,1],[1]]=>[[4,4,3],[3,2]]
[1,0,1,0,1,1,0,1,0,0,1,0]=>[[3,3,1,1],[2]]=>[[4,2,2],[1,1]]=>[[4,3,3],[2,2]]
[1,0,1,0,1,1,0,1,0,1,0,0]=>[[4,1,1],[]]=>[[3,1,1,1],[]]=>[[3,3,3,3],[2,2,2]]
[1,0,1,0,1,1,0,1,1,0,0,0]=>[[3,3,1,1],[1]]=>[[4,2,2],[1]]=>[[4,4,3],[2,2]]
[1,0,1,0,1,1,1,0,0,0,1,0]=>[[2,2,2,1,1],[1]]=>[[5,3],[1]]=>[[5,4],[2]]
[1,0,1,0,1,1,1,0,0,1,0,0]=>[[3,2,1,1],[]]=>[[4,2,1],[]]=>[[4,4,4],[3,2]]
[1,0,1,0,1,1,1,0,1,0,0,0]=>[[2,2,2,1,1],[]]=>[[5,3],[]]=>[[5,5],[2]]
[1,0,1,0,1,1,1,1,0,0,0,0]=>[[3,3,1,1],[]]=>[[4,2,2],[]]=>[[4,4,4],[2,2]]
[1,0,1,1,0,0,1,0,1,0,1,0]=>[[2,2,2,2,1],[1,1,1]]=>[[5,4],[3]]=>[[5,2],[1]]
[1,0,1,1,0,0,1,0,1,1,0,0]=>[[3,2,2,1],[1,1]]=>[[4,3,1],[2]]=>[[4,4,2],[3,1]]
[1,0,1,1,0,0,1,1,0,0,1,0]=>[[3,3,2,1],[2,1]]=>[[4,3,2],[2,1]]=>[[4,3,2],[2,1]]
[1,0,1,1,0,0,1,1,0,1,0,0]=>[[4,2,1],[1]]=>[[3,2,1,1],[1]]=>[[3,3,3,2],[2,2,1]]
[1,0,1,1,0,0,1,1,1,0,0,0]=>[[3,3,2,1],[1,1]]=>[[4,3,2],[2]]=>[[4,4,2],[2,1]]
[1,0,1,1,0,1,0,0,1,0,1,0]=>[[3,3,3,1],[2,2]]=>[[4,3,3],[2,2]]=>[[4,2,2],[1,1]]
[1,0,1,1,0,1,0,0,1,1,0,0]=>[[4,3,1],[2]]=>[[3,2,2,1],[1,1]]=>[[3,3,2,2],[2,1,1]]
[1,0,1,1,0,1,0,1,0,0,1,0]=>[[4,4,1],[3]]=>[[3,2,2,2],[1,1,1]]=>[[3,2,2,2],[1,1,1]]
[1,0,1,1,0,1,0,1,0,1,0,0]=>[[5,1],[]]=>[[2,1,1,1,1],[]]=>[[2,2,2,2,2],[1,1,1,1]]
[1,0,1,1,0,1,0,1,1,0,0,0]=>[[4,4,1],[2]]=>[[3,2,2,2],[1,1]]=>[[3,3,2,2],[1,1,1]]
[1,0,1,1,0,1,1,0,0,0,1,0]=>[[3,3,3,1],[2,1]]=>[[4,3,3],[2,1]]=>[[4,3,2],[1,1]]
[1,0,1,1,0,1,1,0,0,1,0,0]=>[[4,3,1],[1]]=>[[3,2,2,1],[1]]=>[[3,3,3,2],[2,1,1]]
[1,0,1,1,0,1,1,0,1,0,0,0]=>[[3,3,3,1],[1,1]]=>[[4,3,3],[2]]=>[[4,4,2],[1,1]]
[1,0,1,1,0,1,1,1,0,0,0,0]=>[[4,4,1],[1]]=>[[3,2,2,2],[1]]=>[[3,3,3,2],[1,1,1]]
[1,0,1,1,1,0,0,0,1,0,1,0]=>[[2,2,2,2,1],[1,1]]=>[[5,4],[2]]=>[[5,3],[1]]
[1,0,1,1,1,0,0,0,1,1,0,0]=>[[3,2,2,1],[1]]=>[[4,3,1],[1]]=>[[4,4,3],[3,1]]
[1,0,1,1,1,0,0,1,0,0,1,0]=>[[3,3,2,1],[2]]=>[[4,3,2],[1,1]]=>[[4,3,3],[2,1]]
[1,0,1,1,1,0,0,1,0,1,0,0]=>[[4,2,1],[]]=>[[3,2,1,1],[]]=>[[3,3,3,3],[2,2,1]]
[1,0,1,1,1,0,0,1,1,0,0,0]=>[[3,3,2,1],[1]]=>[[4,3,2],[1]]=>[[4,4,3],[2,1]]
[1,0,1,1,1,0,1,0,0,0,1,0]=>[[2,2,2,2,1],[1]]=>[[5,4],[1]]=>[[5,4],[1]]
[1,0,1,1,1,0,1,0,0,1,0,0]=>[[3,2,2,1],[]]=>[[4,3,1],[]]=>[[4,4,4],[3,1]]
[1,0,1,1,1,0,1,0,1,0,0,0]=>[[2,2,2,2,1],[]]=>[[5,4],[]]=>[[5,5],[1]]
[1,0,1,1,1,0,1,1,0,0,0,0]=>[[3,3,2,1],[]]=>[[4,3,2],[]]=>[[4,4,4],[2,1]]
[1,0,1,1,1,1,0,0,0,0,1,0]=>[[3,3,3,1],[2]]=>[[4,3,3],[1,1]]=>[[4,3,3],[1,1]]
[1,0,1,1,1,1,0,0,0,1,0,0]=>[[4,3,1],[]]=>[[3,2,2,1],[]]=>[[3,3,3,3],[2,1,1]]
[1,0,1,1,1,1,0,0,1,0,0,0]=>[[3,3,3,1],[1]]=>[[4,3,3],[1]]=>[[4,4,3],[1,1]]
[1,0,1,1,1,1,0,1,0,0,0,0]=>[[4,4,1],[]]=>[[3,2,2,2],[]]=>[[3,3,3,3],[1,1,1]]
[1,0,1,1,1,1,1,0,0,0,0,0]=>[[3,3,3,1],[]]=>[[4,3,3],[]]=>[[4,4,4],[1,1]]
[1,1,0,0,1,0,1,0,1,0,1,0]=>[[2,2,2,2,2],[1,1,1,1]]=>[[5,5],[4]]=>[[5,1],[]]
[1,1,0,0,1,0,1,0,1,1,0,0]=>[[3,2,2,2],[1,1,1]]=>[[4,4,1],[3]]=>[[4,4,1],[3]]
[1,1,0,0,1,0,1,1,0,0,1,0]=>[[3,3,2,2],[2,1,1]]=>[[4,4,2],[3,1]]=>[[4,3,1],[2]]
[1,1,0,0,1,0,1,1,0,1,0,0]=>[[4,2,2],[1,1]]=>[[3,3,1,1],[2]]=>[[3,3,3,1],[2,2]]
[1,1,0,0,1,0,1,1,1,0,0,0]=>[[3,3,2,2],[1,1,1]]=>[[4,4,2],[3]]=>[[4,4,1],[2]]
[1,1,0,0,1,1,0,0,1,0,1,0]=>[[3,3,3,2],[2,2,1]]=>[[4,4,3],[3,2]]=>[[4,2,1],[1]]
[1,1,0,0,1,1,0,0,1,1,0,0]=>[[4,3,2],[2,1]]=>[[3,3,2,1],[2,1]]=>[[3,3,2,1],[2,1]]
[1,1,0,0,1,1,0,1,0,0,1,0]=>[[4,4,2],[3,1]]=>[[3,3,2,2],[2,1,1]]=>[[3,2,2,1],[1,1]]
[1,1,0,0,1,1,0,1,0,1,0,0]=>[[5,2],[1]]=>[[2,2,1,1,1],[1]]=>[[2,2,2,2,1],[1,1,1]]
[1,1,0,0,1,1,0,1,1,0,0,0]=>[[4,4,2],[2,1]]=>[[3,3,2,2],[2,1]]=>[[3,3,2,1],[1,1]]
[1,1,0,0,1,1,1,0,0,0,1,0]=>[[3,3,3,2],[2,1,1]]=>[[4,4,3],[3,1]]=>[[4,3,1],[1]]
[1,1,0,0,1,1,1,0,0,1,0,0]=>[[4,3,2],[1,1]]=>[[3,3,2,1],[2]]=>[[3,3,3,1],[2,1]]
[1,1,0,0,1,1,1,0,1,0,0,0]=>[[3,3,3,2],[1,1,1]]=>[[4,4,3],[3]]=>[[4,4,1],[1]]
[1,1,0,0,1,1,1,1,0,0,0,0]=>[[4,4,2],[1,1]]=>[[3,3,2,2],[2]]=>[[3,3,3,1],[1,1]]
[1,1,0,1,0,0,1,0,1,0,1,0]=>[[3,3,3,3],[2,2,2]]=>[[4,4,4],[3,3]]=>[[4,1,1],[]]
[1,1,0,1,0,0,1,0,1,1,0,0]=>[[4,3,3],[2,2]]=>[[3,3,3,1],[2,2]]=>[[3,3,1,1],[2]]
[1,1,0,1,0,0,1,1,0,0,1,0]=>[[4,4,3],[3,2]]=>[[3,3,3,2],[2,2,1]]=>[[3,2,1,1],[1]]
[1,1,0,1,0,0,1,1,0,1,0,0]=>[[5,3],[2]]=>[[2,2,2,1,1],[1,1]]=>[[2,2,2,1,1],[1,1]]
[1,1,0,1,0,0,1,1,1,0,0,0]=>[[4,4,3],[2,2]]=>[[3,3,3,2],[2,2]]=>[[3,3,1,1],[1]]
[1,1,0,1,0,1,0,0,1,0,1,0]=>[[4,4,4],[3,3]]=>[[3,3,3,3],[2,2,2]]=>[[3,1,1,1],[]]
[1,1,0,1,0,1,0,0,1,1,0,0]=>[[5,4],[3]]=>[[2,2,2,2,1],[1,1,1]]=>[[2,2,1,1,1],[1]]
[1,1,0,1,0,1,0,1,0,0,1,0]=>[[5,5],[4]]=>[[2,2,2,2,2],[1,1,1,1]]=>[[2,1,1,1,1],[]]
[1,1,0,1,0,1,0,1,0,1,0,0]=>[[6],[]]=>[[1,1,1,1,1,1],[]]=>[[1,1,1,1,1,1],[]]
[1,1,0,1,0,1,0,1,1,0,0,0]=>[[5,5],[3]]=>[[2,2,2,2,2],[1,1,1]]=>[[2,2,1,1,1],[]]
[1,1,0,1,0,1,1,0,0,0,1,0]=>[[4,4,4],[3,2]]=>[[3,3,3,3],[2,2,1]]=>[[3,2,1,1],[]]
[1,1,0,1,0,1,1,0,0,1,0,0]=>[[5,4],[2]]=>[[2,2,2,2,1],[1,1]]=>[[2,2,2,1,1],[1]]
[1,1,0,1,0,1,1,0,1,0,0,0]=>[[4,4,4],[2,2]]=>[[3,3,3,3],[2,2]]=>[[3,3,1,1],[]]
[1,1,0,1,0,1,1,1,0,0,0,0]=>[[5,5],[2]]=>[[2,2,2,2,2],[1,1]]=>[[2,2,2,1,1],[]]
[1,1,0,1,1,0,0,0,1,0,1,0]=>[[3,3,3,3],[2,2,1]]=>[[4,4,4],[3,2]]=>[[4,2,1],[]]
[1,1,0,1,1,0,0,0,1,1,0,0]=>[[4,3,3],[2,1]]=>[[3,3,3,1],[2,1]]=>[[3,3,2,1],[2]]
[1,1,0,1,1,0,0,1,0,0,1,0]=>[[4,4,3],[3,1]]=>[[3,3,3,2],[2,1,1]]=>[[3,2,2,1],[1]]
[1,1,0,1,1,0,0,1,0,1,0,0]=>[[5,3],[1]]=>[[2,2,2,1,1],[1]]=>[[2,2,2,2,1],[1,1]]
[1,1,0,1,1,0,0,1,1,0,0,0]=>[[4,4,3],[2,1]]=>[[3,3,3,2],[2,1]]=>[[3,3,2,1],[1]]
[1,1,0,1,1,0,1,0,0,0,1,0]=>[[3,3,3,3],[2,1,1]]=>[[4,4,4],[3,1]]=>[[4,3,1],[]]
[1,1,0,1,1,0,1,0,0,1,0,0]=>[[4,3,3],[1,1]]=>[[3,3,3,1],[2]]=>[[3,3,3,1],[2]]
[1,1,0,1,1,0,1,0,1,0,0,0]=>[[3,3,3,3],[1,1,1]]=>[[4,4,4],[3]]=>[[4,4,1],[]]
[1,1,0,1,1,0,1,1,0,0,0,0]=>[[4,4,3],[1,1]]=>[[3,3,3,2],[2]]=>[[3,3,3,1],[1]]
[1,1,0,1,1,1,0,0,0,0,1,0]=>[[4,4,4],[3,1]]=>[[3,3,3,3],[2,1,1]]=>[[3,2,2,1],[]]
[1,1,0,1,1,1,0,0,0,1,0,0]=>[[5,4],[1]]=>[[2,2,2,2,1],[1]]=>[[2,2,2,2,1],[1]]
[1,1,0,1,1,1,0,0,1,0,0,0]=>[[4,4,4],[2,1]]=>[[3,3,3,3],[2,1]]=>[[3,3,2,1],[]]
[1,1,0,1,1,1,0,1,0,0,0,0]=>[[5,5],[1]]=>[[2,2,2,2,2],[1]]=>[[2,2,2,2,1],[]]
[1,1,0,1,1,1,1,0,0,0,0,0]=>[[4,4,4],[1,1]]=>[[3,3,3,3],[2]]=>[[3,3,3,1],[]]
[1,1,1,0,0,0,1,0,1,0,1,0]=>[[2,2,2,2,2],[1,1,1]]=>[[5,5],[3]]=>[[5,2],[]]
[1,1,1,0,0,0,1,0,1,1,0,0]=>[[3,2,2,2],[1,1]]=>[[4,4,1],[2]]=>[[4,4,2],[3]]
[1,1,1,0,0,0,1,1,0,0,1,0]=>[[3,3,2,2],[2,1]]=>[[4,4,2],[2,1]]=>[[4,3,2],[2]]
[1,1,1,0,0,0,1,1,0,1,0,0]=>[[4,2,2],[1]]=>[[3,3,1,1],[1]]=>[[3,3,3,2],[2,2]]
[1,1,1,0,0,0,1,1,1,0,0,0]=>[[3,3,2,2],[1,1]]=>[[4,4,2],[2]]=>[[4,4,2],[2]]
[1,1,1,0,0,1,0,0,1,0,1,0]=>[[3,3,3,2],[2,2]]=>[[4,4,3],[2,2]]=>[[4,2,2],[1]]
[1,1,1,0,0,1,0,0,1,1,0,0]=>[[4,3,2],[2]]=>[[3,3,2,1],[1,1]]=>[[3,3,2,2],[2,1]]
[1,1,1,0,0,1,0,1,0,0,1,0]=>[[4,4,2],[3]]=>[[3,3,2,2],[1,1,1]]=>[[3,2,2,2],[1,1]]
[1,1,1,0,0,1,0,1,0,1,0,0]=>[[5,2],[]]=>[[2,2,1,1,1],[]]=>[[2,2,2,2,2],[1,1,1]]
[1,1,1,0,0,1,0,1,1,0,0,0]=>[[4,4,2],[2]]=>[[3,3,2,2],[1,1]]=>[[3,3,2,2],[1,1]]
[1,1,1,0,0,1,1,0,0,0,1,0]=>[[3,3,3,2],[2,1]]=>[[4,4,3],[2,1]]=>[[4,3,2],[1]]
[1,1,1,0,0,1,1,0,0,1,0,0]=>[[4,3,2],[1]]=>[[3,3,2,1],[1]]=>[[3,3,3,2],[2,1]]
[1,1,1,0,0,1,1,0,1,0,0,0]=>[[3,3,3,2],[1,1]]=>[[4,4,3],[2]]=>[[4,4,2],[1]]
[1,1,1,0,0,1,1,1,0,0,0,0]=>[[4,4,2],[1]]=>[[3,3,2,2],[1]]=>[[3,3,3,2],[1,1]]
[1,1,1,0,1,0,0,0,1,0,1,0]=>[[2,2,2,2,2],[1,1]]=>[[5,5],[2]]=>[[5,3],[]]
[1,1,1,0,1,0,0,0,1,1,0,0]=>[[3,2,2,2],[1]]=>[[4,4,1],[1]]=>[[4,4,3],[3]]
[1,1,1,0,1,0,0,1,0,0,1,0]=>[[3,3,2,2],[2]]=>[[4,4,2],[1,1]]=>[[4,3,3],[2]]
[1,1,1,0,1,0,0,1,0,1,0,0]=>[[4,2,2],[]]=>[[3,3,1,1],[]]=>[[3,3,3,3],[2,2]]
[1,1,1,0,1,0,0,1,1,0,0,0]=>[[3,3,2,2],[1]]=>[[4,4,2],[1]]=>[[4,4,3],[2]]
[1,1,1,0,1,0,1,0,0,0,1,0]=>[[2,2,2,2,2],[1]]=>[[5,5],[1]]=>[[5,4],[]]
[1,1,1,0,1,0,1,0,0,1,0,0]=>[[3,2,2,2],[]]=>[[4,4,1],[]]=>[[4,4,4],[3]]
[1,1,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2],[]]=>[[5,5],[]]=>[[5,5],[]]
[1,1,1,0,1,0,1,1,0,0,0,0]=>[[3,3,2,2],[]]=>[[4,4,2],[]]=>[[4,4,4],[2]]
[1,1,1,0,1,1,0,0,0,0,1,0]=>[[3,3,3,2],[2]]=>[[4,4,3],[1,1]]=>[[4,3,3],[1]]
[1,1,1,0,1,1,0,0,0,1,0,0]=>[[4,3,2],[]]=>[[3,3,2,1],[]]=>[[3,3,3,3],[2,1]]
[1,1,1,0,1,1,0,0,1,0,0,0]=>[[3,3,3,2],[1]]=>[[4,4,3],[1]]=>[[4,4,3],[1]]
[1,1,1,0,1,1,0,1,0,0,0,0]=>[[4,4,2],[]]=>[[3,3,2,2],[]]=>[[3,3,3,3],[1,1]]
[1,1,1,0,1,1,1,0,0,0,0,0]=>[[3,3,3,2],[]]=>[[4,4,3],[]]=>[[4,4,4],[1]]
[1,1,1,1,0,0,0,0,1,0,1,0]=>[[3,3,3,3],[2,2]]=>[[4,4,4],[2,2]]=>[[4,2,2],[]]
[1,1,1,1,0,0,0,0,1,1,0,0]=>[[4,3,3],[2]]=>[[3,3,3,1],[1,1]]=>[[3,3,2,2],[2]]
[1,1,1,1,0,0,0,1,0,0,1,0]=>[[4,4,3],[3]]=>[[3,3,3,2],[1,1,1]]=>[[3,2,2,2],[1]]
[1,1,1,1,0,0,0,1,0,1,0,0]=>[[5,3],[]]=>[[2,2,2,1,1],[]]=>[[2,2,2,2,2],[1,1]]
[1,1,1,1,0,0,0,1,1,0,0,0]=>[[4,4,3],[2]]=>[[3,3,3,2],[1,1]]=>[[3,3,2,2],[1]]
[1,1,1,1,0,0,1,0,0,0,1,0]=>[[3,3,3,3],[2,1]]=>[[4,4,4],[2,1]]=>[[4,3,2],[]]
[1,1,1,1,0,0,1,0,0,1,0,0]=>[[4,3,3],[1]]=>[[3,3,3,1],[1]]=>[[3,3,3,2],[2]]
[1,1,1,1,0,0,1,0,1,0,0,0]=>[[3,3,3,3],[1,1]]=>[[4,4,4],[2]]=>[[4,4,2],[]]
[1,1,1,1,0,0,1,1,0,0,0,0]=>[[4,4,3],[1]]=>[[3,3,3,2],[1]]=>[[3,3,3,2],[1]]
[1,1,1,1,0,1,0,0,0,0,1,0]=>[[4,4,4],[3]]=>[[3,3,3,3],[1,1,1]]=>[[3,2,2,2],[]]
[1,1,1,1,0,1,0,0,0,1,0,0]=>[[5,4],[]]=>[[2,2,2,2,1],[]]=>[[2,2,2,2,2],[1]]
[1,1,1,1,0,1,0,0,1,0,0,0]=>[[4,4,4],[2]]=>[[3,3,3,3],[1,1]]=>[[3,3,2,2],[]]
[1,1,1,1,0,1,0,1,0,0,0,0]=>[[5,5],[]]=>[[2,2,2,2,2],[]]=>[[2,2,2,2,2],[]]
[1,1,1,1,0,1,1,0,0,0,0,0]=>[[4,4,4],[1]]=>[[3,3,3,3],[1]]=>[[3,3,3,2],[]]
[1,1,1,1,1,0,0,0,0,0,1,0]=>[[3,3,3,3],[2]]=>[[4,4,4],[1,1]]=>[[4,3,3],[]]
[1,1,1,1,1,0,0,0,0,1,0,0]=>[[4,3,3],[]]=>[[3,3,3,1],[]]=>[[3,3,3,3],[2]]
[1,1,1,1,1,0,0,0,1,0,0,0]=>[[3,3,3,3],[1]]=>[[4,4,4],[1]]=>[[4,4,3],[]]
[1,1,1,1,1,0,0,1,0,0,0,0]=>[[4,4,3],[]]=>[[3,3,3,2],[]]=>[[3,3,3,3],[1]]
[1,1,1,1,1,0,1,0,0,0,0,0]=>[[3,3,3,3],[]]=>[[4,4,4],[]]=>[[4,4,4],[]]
[1,1,1,1,1,1,0,0,0,0,0,0]=>[[4,4,4],[]]=>[[3,3,3,3],[]]=>[[3,3,3,3],[]]
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,1,1,1,1,1,1],[]]=>[[7],[]]=>[[7],[]]
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[2,1,1,1,1,1],[]]=>[[6,1],[]]=>[[6,6],[5]]
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]=>[[2,2,1,1,1,1],[1]]=>[[6,2],[1]]=>[[6,5],[4]]
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[[3,1,1,1,1],[]]=>[[5,1,1],[]]=>[[5,5,5],[4,4]]
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[2,2,1,1,1,1],[]]=>[[6,2],[]]=>[[6,6],[4]]
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]=>[[2,2,2,1,1,1],[1,1]]=>[[6,3],[2]]=>[[6,4],[3]]
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]=>[[3,2,1,1,1],[1]]=>[[5,2,1],[1]]=>[[5,5,4],[4,3]]
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]=>[[3,3,1,1,1],[2]]=>[[5,2,2],[1,1]]=>[[5,4,4],[3,3]]
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[[4,1,1,1],[]]=>[[4,1,1,1],[]]=>[[4,4,4,4],[3,3,3]]
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[[3,2,1,1,1],[]]=>[[5,2,1],[]]=>[[5,5,5],[4,3]]
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]=>[[2,2,2,1,1,1],[]]=>[[6,3],[]]=>[[6,6],[3]]
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[[3,3,1,1,1],[]]=>[[5,2,2],[]]=>[[5,5,5],[3,3]]
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]=>[[2,2,2,2,1,1],[1,1,1]]=>[[6,4],[3]]=>[[6,3],[2]]
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]=>[[3,2,2,1,1],[1,1]]=>[[5,3,1],[2]]=>[[5,5,3],[4,2]]
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]=>[[3,3,2,1,1],[2,1]]=>[[5,3,2],[2,1]]=>[[5,4,3],[3,2]]
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]=>[[4,2,1,1],[1]]=>[[4,2,1,1],[1]]=>[[4,4,4,3],[3,3,2]]
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]=>[[3,3,3,1,1],[2,2]]=>[[5,3,3],[2,2]]=>[[5,3,3],[2,2]]
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]=>[[4,3,1,1],[2]]=>[[4,2,2,1],[1,1]]=>[[4,4,3,3],[3,2,2]]
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]=>[[4,4,1,1],[3]]=>[[4,2,2,2],[1,1,1]]=>[[4,3,3,3],[2,2,2]]
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[[5,1,1],[]]=>[[3,1,1,1,1],[]]=>[[3,3,3,3,3],[2,2,2,2]]
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[[4,2,1,1],[]]=>[[4,2,1,1],[]]=>[[4,4,4,4],[3,3,2]]
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]=>[[3,2,2,1,1],[]]=>[[5,3,1],[]]=>[[5,5,5],[4,2]]
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]=>[[2,2,2,2,1,1],[]]=>[[6,4],[]]=>[[6,6],[2]]
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]=>[[3,3,2,1,1],[]]=>[[5,3,2],[]]=>[[5,5,5],[3,2]]
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]=>[[4,3,1,1],[]]=>[[4,2,2,1],[]]=>[[4,4,4,4],[3,2,2]]
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]=>[[4,4,1,1],[]]=>[[4,2,2,2],[]]=>[[4,4,4,4],[2,2,2]]
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[[3,3,3,1,1],[]]=>[[5,3,3],[]]=>[[5,5,5],[2,2]]
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]=>[[2,2,2,2,2,1],[1,1,1,1]]=>[[6,5],[4]]=>[[6,2],[1]]
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]=>[[3,2,2,2,1],[1,1,1]]=>[[5,4,1],[3]]=>[[5,5,2],[4,1]]
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]=>[[3,3,2,2,1],[2,1,1]]=>[[5,4,2],[3,1]]=>[[5,4,2],[3,1]]
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]=>[[4,2,2,1],[1,1]]=>[[4,3,1,1],[2]]=>[[4,4,4,2],[3,3,1]]
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]=>[[3,3,3,2,1],[2,2,1]]=>[[5,4,3],[3,2]]=>[[5,3,2],[2,1]]
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[[4,3,2,1],[2,1]]=>[[4,3,2,1],[2,1]]=>[[4,4,3,2],[3,2,1]]
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]=>[[4,4,2,1],[3,1]]=>[[4,3,2,2],[2,1,1]]=>[[4,3,3,2],[2,2,1]]
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]=>[[5,2,1],[1]]=>[[3,2,1,1,1],[1]]=>[[3,3,3,3,2],[2,2,2,1]]
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]=>[[3,3,3,3,1],[2,2,2]]=>[[5,4,4],[3,3]]=>[[5,2,2],[1,1]]
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]=>[[4,3,3,1],[2,2]]=>[[4,3,3,1],[2,2]]=>[[4,4,2,2],[3,1,1]]
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]=>[[4,4,3,1],[3,2]]=>[[4,3,3,2],[2,2,1]]=>[[4,3,2,2],[2,1,1]]
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]=>[[5,3,1],[2]]=>[[3,2,2,1,1],[1,1]]=>[[3,3,3,2,2],[2,2,1,1]]
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]=>[[4,4,4,1],[3,3]]=>[[4,3,3,3],[2,2,2]]=>[[4,2,2,2],[1,1,1]]
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]=>[[5,4,1],[3]]=>[[3,2,2,2,1],[1,1,1]]=>[[3,3,2,2,2],[2,1,1,1]]
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]=>[[5,5,1],[4]]=>[[3,2,2,2,2],[1,1,1,1]]=>[[3,2,2,2,2],[1,1,1,1]]
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[[6,1],[]]=>[[2,1,1,1,1,1],[]]=>[[2,2,2,2,2,2],[1,1,1,1,1]]
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[[5,2,1],[]]=>[[3,2,1,1,1],[]]=>[[3,3,3,3,3],[2,2,2,1]]
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]=>[[4,2,2,1],[]]=>[[4,3,1,1],[]]=>[[4,4,4,4],[3,3,1]]
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]=>[[3,2,2,2,1],[]]=>[[5,4,1],[]]=>[[5,5,5],[4,1]]
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2,1],[]]=>[[6,5],[]]=>[[6,6],[1]]
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]=>[[3,3,2,2,1],[]]=>[[5,4,2],[]]=>[[5,5,5],[3,1]]
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]=>[[4,3,2,1],[]]=>[[4,3,2,1],[]]=>[[4,4,4,4],[3,2,1]]
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]=>[[4,4,2,1],[]]=>[[4,3,2,2],[]]=>[[4,4,4,4],[2,2,1]]
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]=>[[3,3,3,2,1],[]]=>[[5,4,3],[]]=>[[5,5,5],[2,1]]
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]=>[[5,3,1],[]]=>[[3,2,2,1,1],[]]=>[[3,3,3,3,3],[2,2,1,1]]
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]=>[[5,4,1],[]]=>[[3,2,2,2,1],[]]=>[[3,3,3,3,3],[2,1,1,1]]
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]=>[[5,5,1],[]]=>[[3,2,2,2,2],[]]=>[[3,3,3,3,3],[1,1,1,1]]
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]=>[[4,3,3,1],[]]=>[[4,3,3,1],[]]=>[[4,4,4,4],[3,1,1]]
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]=>[[4,4,3,1],[]]=>[[4,3,3,2],[]]=>[[4,4,4,4],[2,1,1]]
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]=>[[3,3,3,3,1],[]]=>[[5,4,4],[]]=>[[5,5,5],[1,1]]
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]=>[[4,4,4,1],[]]=>[[4,3,3,3],[]]=>[[4,4,4,4],[1,1,1]]
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]=>[[2,2,2,2,2,2],[1,1,1,1,1]]=>[[6,6],[5]]=>[[6,1],[]]
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]=>[[3,2,2,2,2],[1,1,1,1]]=>[[5,5,1],[4]]=>[[5,5,1],[4]]
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]=>[[3,3,2,2,2],[2,1,1,1]]=>[[5,5,2],[4,1]]=>[[5,4,1],[3]]
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]=>[[4,2,2,2],[1,1,1]]=>[[4,4,1,1],[3]]=>[[4,4,4,1],[3,3]]
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]=>[[3,3,3,2,2],[2,2,1,1]]=>[[5,5,3],[4,2]]=>[[5,3,1],[2]]
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]=>[[4,3,2,2],[2,1,1]]=>[[4,4,2,1],[3,1]]=>[[4,4,3,1],[3,2]]
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]=>[[4,4,2,2],[3,1,1]]=>[[4,4,2,2],[3,1,1]]=>[[4,3,3,1],[2,2]]
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]=>[[5,2,2],[1,1]]=>[[3,3,1,1,1],[2]]=>[[3,3,3,3,1],[2,2,2]]
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]=>[[3,3,3,3,2],[2,2,2,1]]=>[[5,5,4],[4,3]]=>[[5,2,1],[1]]
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]=>[[4,3,3,2],[2,2,1]]=>[[4,4,3,1],[3,2]]=>[[4,4,2,1],[3,1]]
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]=>[[4,4,3,2],[3,2,1]]=>[[4,4,3,2],[3,2,1]]=>[[4,3,2,1],[2,1]]
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]=>[[5,3,2],[2,1]]=>[[3,3,2,1,1],[2,1]]=>[[3,3,3,2,1],[2,2,1]]
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]=>[[4,4,4,2],[3,3,1]]=>[[4,4,3,3],[3,2,2]]=>[[4,2,2,1],[1,1]]
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]=>[[5,4,2],[3,1]]=>[[3,3,2,2,1],[2,1,1]]=>[[3,3,2,2,1],[2,1,1]]
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]=>[[5,5,2],[4,1]]=>[[3,3,2,2,2],[2,1,1,1]]=>[[3,2,2,2,1],[1,1,1]]
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]=>[[6,2],[1]]=>[[2,2,1,1,1,1],[1]]=>[[2,2,2,2,2,1],[1,1,1,1]]
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]=>[[3,3,3,3,3],[2,2,2,2]]=>[[5,5,5],[4,4]]=>[[5,1,1],[]]
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]=>[[4,3,3,3],[2,2,2]]=>[[4,4,4,1],[3,3]]=>[[4,4,1,1],[3]]
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]=>[[4,4,3,3],[3,2,2]]=>[[4,4,4,2],[3,3,1]]=>[[4,3,1,1],[2]]
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]=>[[5,3,3],[2,2]]=>[[3,3,3,1,1],[2,2]]=>[[3,3,3,1,1],[2,2]]
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]=>[[4,4,4,3],[3,3,2]]=>[[4,4,4,3],[3,3,2]]=>[[4,2,1,1],[1]]
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]=>[[5,4,3],[3,2]]=>[[3,3,3,2,1],[2,2,1]]=>[[3,3,2,1,1],[2,1]]
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]=>[[5,5,3],[4,2]]=>[[3,3,3,2,2],[2,2,1,1]]=>[[3,2,2,1,1],[1,1]]
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]=>[[6,3],[2]]=>[[2,2,2,1,1,1],[1,1]]=>[[2,2,2,2,1,1],[1,1,1]]
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]=>[[4,4,4,4],[3,3,3]]=>[[4,4,4,4],[3,3,3]]=>[[4,1,1,1],[]]
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]=>[[5,4,4],[3,3]]=>[[3,3,3,3,1],[2,2,2]]=>[[3,3,1,1,1],[2]]
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]=>[[5,5,4],[4,3]]=>[[3,3,3,3,2],[2,2,2,1]]=>[[3,2,1,1,1],[1]]
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]=>[[6,4],[3]]=>[[2,2,2,2,1,1],[1,1,1]]=>[[2,2,2,1,1,1],[1,1]]
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]=>[[5,5,5],[4,4]]=>[[3,3,3,3,3],[2,2,2,2]]=>[[3,1,1,1,1],[]]
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]=>[[6,5],[4]]=>[[2,2,2,2,2,1],[1,1,1,1]]=>[[2,2,1,1,1,1],[1]]
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]=>[[6,6],[5]]=>[[2,2,2,2,2,2],[1,1,1,1,1]]=>[[2,1,1,1,1,1],[]]
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7],[]]=>[[1,1,1,1,1,1,1],[]]=>[[1,1,1,1,1,1,1],[]]
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[[6,2],[]]=>[[2,2,1,1,1,1],[]]=>[[2,2,2,2,2,2],[1,1,1,1]]
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>[[5,2,2],[]]=>[[3,3,1,1,1],[]]=>[[3,3,3,3,3],[2,2,2]]
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]=>[[4,2,2,2],[]]=>[[4,4,1,1],[]]=>[[4,4,4,4],[3,3]]
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]=>[[3,2,2,2,2],[]]=>[[5,5,1],[]]=>[[5,5,5],[4]]
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2,2],[]]=>[[6,6],[]]=>[[6,6],[]]
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]=>[[3,3,2,2,2],[]]=>[[5,5,2],[]]=>[[5,5,5],[3]]
[1,1,1,0,1,0,1,1,0,0,0,1,0,0]=>[[4,3,2,2],[]]=>[[4,4,2,1],[]]=>[[4,4,4,4],[3,2]]
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]=>[[4,4,2,2],[]]=>[[4,4,2,2],[]]=>[[4,4,4,4],[2,2]]
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]=>[[3,3,3,2,2],[]]=>[[5,5,3],[]]=>[[5,5,5],[2]]
[1,1,1,0,1,1,0,0,0,1,0,1,0,0]=>[[5,3,2],[]]=>[[3,3,2,1,1],[]]=>[[3,3,3,3,3],[2,2,1]]
[1,1,1,0,1,1,0,1,0,0,0,1,0,0]=>[[5,4,2],[]]=>[[3,3,2,2,1],[]]=>[[3,3,3,3,3],[2,1,1]]
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]=>[[5,5,2],[]]=>[[3,3,2,2,2],[]]=>[[3,3,3,3,3],[1,1,1]]
[1,1,1,0,1,1,1,0,0,0,0,1,0,0]=>[[4,3,3,2],[]]=>[[4,4,3,1],[]]=>[[4,4,4,4],[3,1]]
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]=>[[4,4,3,2],[]]=>[[4,4,3,2],[]]=>[[4,4,4,4],[2,1]]
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]=>[[3,3,3,3,2],[]]=>[[5,5,4],[]]=>[[5,5,5],[1]]
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]=>[[4,4,4,2],[]]=>[[4,4,3,3],[]]=>[[4,4,4,4],[1,1]]
[1,1,1,1,0,0,0,1,0,1,0,1,0,0]=>[[6,3],[]]=>[[2,2,2,1,1,1],[]]=>[[2,2,2,2,2,2],[1,1,1]]
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]=>[[6,4],[]]=>[[2,2,2,2,1,1],[]]=>[[2,2,2,2,2,2],[1,1]]
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]=>[[6,5],[]]=>[[2,2,2,2,2,1],[]]=>[[2,2,2,2,2,2],[1]]
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]=>[[6,6],[]]=>[[2,2,2,2,2,2],[]]=>[[2,2,2,2,2,2],[]]
[1,1,1,1,1,0,0,0,0,1,0,1,0,0]=>[[5,3,3],[]]=>[[3,3,3,1,1],[]]=>[[3,3,3,3,3],[2,2]]
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]=>[[5,4,3],[]]=>[[3,3,3,2,1],[]]=>[[3,3,3,3,3],[2,1]]
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]=>[[5,5,3],[]]=>[[3,3,3,2,2],[]]=>[[3,3,3,3,3],[1,1]]
[1,1,1,1,1,0,1,0,0,0,0,1,0,0]=>[[4,3,3,3],[]]=>[[4,4,4,1],[]]=>[[4,4,4,4],[3]]
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]=>[[4,4,3,3],[]]=>[[4,4,4,2],[]]=>[[4,4,4,4],[2]]
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]=>[[3,3,3,3,3],[]]=>[[5,5,5],[]]=>[[5,5,5],[]]
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]=>[[4,4,4,3],[]]=>[[4,4,4,3],[]]=>[[4,4,4,4],[1]]
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]=>[[5,4,4],[]]=>[[3,3,3,3,1],[]]=>[[3,3,3,3,3],[2]]
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]=>[[5,5,4],[]]=>[[3,3,3,3,2],[]]=>[[3,3,3,3,3],[1]]
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]=>[[5,5,5],[]]=>[[3,3,3,3,3],[]]=>[[3,3,3,3,3],[]]
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]=>[[4,4,4,4],[]]=>[[4,4,4,4],[]]=>[[4,4,4,4],[]]
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,1,1,1,1,1,1,1],[]]=>[[8],[]]=>[[8],[]]
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[2,1,1,1,1,1,1],[]]=>[[7,1],[]]=>[[7,7],[6]]
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[[3,1,1,1,1,1],[]]=>[[6,1,1],[]]=>[[6,6,6],[5,5]]
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[2,2,1,1,1,1,1],[]]=>[[7,2],[]]=>[[7,7],[5]]
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[[4,1,1,1,1],[]]=>[[5,1,1,1],[]]=>[[5,5,5,5],[4,4,4]]
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[[3,2,1,1,1,1],[]]=>[[6,2,1],[]]=>[[6,6,6],[5,4]]
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]=>[[2,2,2,1,1,1,1],[]]=>[[7,3],[]]=>[[7,7],[4]]
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]=>[[3,3,1,1,1,1],[]]=>[[6,2,2],[]]=>[[6,6,6],[4,4]]
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[[5,1,1,1],[]]=>[[4,1,1,1,1],[]]=>[[4,4,4,4,4],[3,3,3,3]]
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[[4,2,1,1,1],[]]=>[[5,2,1,1],[]]=>[[5,5,5,5],[4,4,3]]
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]=>[[3,2,2,1,1,1],[]]=>[[6,3,1],[]]=>[[6,6,6],[5,3]]
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]=>[[4,3,1,1,1],[]]=>[[5,2,2,1],[]]=>[[5,5,5,5],[4,3,3]]
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[[6,1,1],[]]=>[[3,1,1,1,1,1],[]]=>[[3,3,3,3,3,3],[2,2,2,2,2]]
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[[5,2,1,1],[]]=>[[4,2,1,1,1],[]]=>[[4,4,4,4,4],[3,3,3,2]]
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]=>[[4,2,2,1,1],[]]=>[[5,3,1,1],[]]=>[[5,5,5,5],[4,4,2]]
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]=>[[5,3,1,1],[]]=>[[4,2,2,1,1],[]]=>[[4,4,4,4,4],[3,3,2,2]]
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]=>[[4,4,3,1,1],[]]=>[[5,3,3,2],[]]=>[[5,5,5,5],[3,2,2]]
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7,1],[]]=>[[2,1,1,1,1,1,1],[]]=>[[2,2,2,2,2,2,2],[1,1,1,1,1,1]]
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[[6,2,1],[]]=>[[3,2,1,1,1,1],[]]=>[[3,3,3,3,3,3],[2,2,2,2,1]]
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]=>[[5,2,2,1],[]]=>[[4,3,1,1,1],[]]=>[[4,4,4,4,4],[3,3,3,1]]
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]=>[[3,2,2,2,2,1],[]]=>[[6,5,1],[]]=>[[6,6,6],[5,1]]
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2,2,1],[]]=>[[7,6],[]]=>[[7,7],[1]]
[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]=>[[6,3,1],[]]=>[[3,2,2,1,1,1],[]]=>[[3,3,3,3,3,3],[2,2,2,1,1]]
[1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]=>[[6,5,1],[]]=>[[3,2,2,2,2,1],[]]=>[[3,3,3,3,3,3],[2,1,1,1,1]]
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]=>[[6,6,1],[]]=>[[3,2,2,2,2,2],[]]=>[[3,3,3,3,3,3],[1,1,1,1,1]]
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[8],[]]=>[[1,1,1,1,1,1,1,1],[]]=>[[1,1,1,1,1,1,1,1],[]]
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7,2],[]]=>[[2,2,1,1,1,1,1],[]]=>[[2,2,2,2,2,2,2],[1,1,1,1,1]]
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]=>[[6,2,2],[]]=>[[3,3,1,1,1,1],[]]=>[[3,3,3,3,3,3],[2,2,2,2]]
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]=>[[5,2,2,2],[]]=>[[4,4,1,1,1],[]]=>[[4,4,4,4,4],[3,3,3]]
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]=>[[3,2,2,2,2,2],[]]=>[[6,6,1],[]]=>[[6,6,6],[5]]
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2,2,2],[]]=>[[7,7],[]]=>[[7,7],[]]
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]=>[[3,3,2,2,2,2],[]]=>[[6,6,2],[]]=>[[6,6,6],[4]]
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]=>[[6,6,2],[]]=>[[3,3,2,2,2,2],[]]=>[[3,3,3,3,3,3],[1,1,1,1]]
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]=>[[7,3],[]]=>[[2,2,2,1,1,1,1],[]]=>[[2,2,2,2,2,2,2],[1,1,1,1]]
[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]=>[[7,6],[]]=>[[2,2,2,2,2,2,1],[]]=>[[2,2,2,2,2,2,2],[1]]
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[[7,7],[]]=>[[2,2,2,2,2,2,2],[]]=>[[2,2,2,2,2,2,2],[]]
[1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]=>[[6,6,3],[]]=>[[3,3,3,2,2,2],[]]=>[[3,3,3,3,3,3],[1,1,1]]
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]=>[[3,3,3,3,3,3],[]]=>[[6,6,6],[]]=>[[6,6,6],[]]
[1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]=>[[6,5,4],[]]=>[[3,3,3,3,2,1],[]]=>[[3,3,3,3,3,3],[2,1]]
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]=>[[6,6,6],[]]=>[[3,3,3,3,3,3],[]]=>[[3,3,3,3,3,3],[]]
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]=>[[4,4,4,4,4],[]]=>[[5,5,5,5],[]]=>[[5,5,5,5],[]]
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]=>[[5,5,5,5],[]]=>[[4,4,4,4,4],[]]=>[[4,4,4,4,4],[]]
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[9],[]]=>[[1,1,1,1,1,1,1,1,1],[]]=>[[1,1,1,1,1,1,1,1,1],[]]
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[8,2],[]]=>[[2,2,1,1,1,1,1,1],[]]=>[[2,2,2,2,2,2,2,2],[1,1,1,1,1,1]]
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7,2,2],[]]=>[[3,3,1,1,1,1,1],[]]=>[[3,3,3,3,3,3,3],[2,2,2,2,2]]
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]=>[[2,2,2,2,2,2,2,2],[]]=>[[8,8],[]]=>[[8,8],[]]
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[[8,8],[]]=>[[2,2,2,2,2,2,2,2],[]]=>[[2,2,2,2,2,2,2,2],[]]
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]=>[[4,4,4,4,4,4],[]]=>[[6,6,6,6],[]]=>[[6,6,6,6],[]]
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]=>[[6,6,6,6],[]]=>[[4,4,4,4,4,4],[]]=>[[4,4,4,4,4,4],[]]
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]=>[[5,5,5,5,5],[]]=>[[5,5,5,5,5],[]]=>[[5,5,5,5,5],[]]
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,1,1,1,1,1,1,1,1],[]]=>[[9],[]]=>[[9],[]]
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[2,1,1,1,1,1,1,1],[]]=>[[8,1],[]]=>[[8,8],[7]]
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[[3,1,1,1,1,1,1],[]]=>[[7,1,1],[]]=>[[7,7,7],[6,6]]
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[[4,1,1,1,1,1],[]]=>[[6,1,1,1],[]]=>[[6,6,6,6],[5,5,5]]
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[[5,1,1,1,1],[]]=>[[5,1,1,1,1],[]]=>[[5,5,5,5,5],[4,4,4,4]]
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[[6,1,1,1],[]]=>[[4,1,1,1,1,1],[]]=>[[4,4,4,4,4,4],[3,3,3,3,3]]
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7,1,1],[]]=>[[3,1,1,1,1,1,1],[]]=>[[3,3,3,3,3,3,3],[2,2,2,2,2,2]]
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[8,1],[]]=>[[2,1,1,1,1,1,1,1],[]]=>[[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]]
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,1,1,1,1,1,1,1,1,1],[]]=>[[10],[]]=>[[10],[]]
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[2,1,1,1,1,1,1,1,1],[]]=>[[9,1],[]]=>[[9,9],[8]]
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]=>[[2,2,1,1,1,1,1,1],[]]=>[[8,2],[]]=>[[8,8],[6]]
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]=>[[3,1,1,1,1,1,1,1],[]]=>[[8,1,1],[]]=>[[8,8,8],[7,7]]
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]=>[[3,2,1,1,1,1,1],[]]=>[[7,2,1],[]]=>[[7,7,7],[6,5]]
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]=>[[4,1,1,1,1,1,1],[]]=>[[7,1,1,1],[]]=>[[7,7,7,7],[6,6,6]]
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]=>[[4,2,1,1,1,1],[]]=>[[6,2,1,1],[]]=>[[6,6,6,6],[5,5,4]]
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]=>[[5,1,1,1,1,1],[]]=>[[6,1,1,1,1],[]]=>[[6,6,6,6,6],[5,5,5,5]]
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]=>[[5,2,1,1,1],[]]=>[[5,2,1,1,1],[]]=>[[5,5,5,5,5],[4,4,4,3]]
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]=>[[6,1,1,1,1],[]]=>[[5,1,1,1,1,1],[]]=>[[5,5,5,5,5,5],[4,4,4,4,4]]
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]=>[[6,2,1,1],[]]=>[[4,2,1,1,1,1],[]]=>[[4,4,4,4,4,4],[3,3,3,3,2]]
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7,1,1,1],[]]=>[[4,1,1,1,1,1,1],[]]=>[[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]=>[[7,2,1],[]]=>[[3,2,1,1,1,1,1],[]]=>[[3,3,3,3,3,3,3],[2,2,2,2,2,1]]
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[8,1,1],[]]=>[[3,1,1,1,1,1,1,1],[]]=>[[3,3,3,3,3,3,3,3],[2,2,2,2,2,2,2]]
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[9,1],[]]=>[[2,1,1,1,1,1,1,1,1],[]]=>[[2,2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1,1]]
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[10],[]]=>[[1,1,1,1,1,1,1,1,1,1],[]]=>[[1,1,1,1,1,1,1,1,1,1],[]]
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]=>[[2,1,1,1,1,1,1,1,1,1],[]]=>[[10,1],[]]=>[[10,10],[9]]
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]=>[[11],[]]=>[[1,1,1,1,1,1,1,1,1,1,1],[]]=>[[1,1,1,1,1,1,1,1,1,1,1],[]]
[1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]=>[[6,2,2,2,2],[]]=>[[5,5,1,1,1,1],[]]=>[[5,5,5,5,5,5],[4,4,4,4]]
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]=>[[3,2,2,2,2,2,2,2],[]]=>[[8,8,1],[]]=>[[8,8,8],[7]]
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[[8,8,1],[]]=>[[3,2,2,2,2,2,2,2],[]]=>[[3,3,3,3,3,3,3,3],[1,1,1,1,1,1,1]]
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]=>[[6,6,6,6,6],[]]=>[[5,5,5,5,5,5],[]]=>[[5,5,5,5,5,5],[]]
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]=>[[5,5,5,5,5,5],[]]=>[[6,6,6,6,6],[]]=>[[6,6,6,6,6],[]]
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]=>[[9,9],[]]=>[[2,2,2,2,2,2,2,2,2],[]]=>[[2,2,2,2,2,2,2,2,2],[]]
Map
skew partition
Description
The parallelogram polyomino corresponding to a Dyck path, interpreted as a skew partition.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
This map returns the skew partition definded by the diagram of $\gamma(D)$.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
This map returns the skew partition definded by the diagram of $\gamma(D)$.
Map
conjugate
Description
The conjugate of the skew partition.
The conjugate of a skew partition $\lambda$ is the skew partition $\lambda^*$ whose Ferrers diagram is obtained from the Ferrers diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
The conjugate of a skew partition $\lambda$ is the skew partition $\lambda^*$ whose Ferrers diagram is obtained from the Ferrers diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
Map
rotate
Description
The rotation of a skew partition.
This is the skew partition obtained by rotating the diagram by 180 degrees. Equivalently, given a skew partition $\lambda/\mu$, its rotation $(\lambda/\mu)^\natural$ is the skew partition with cells $\{(a-i, b-j)| (i, j) \in \lambda/\mu\}$, where $b$ and $a$ are the first part and the number of parts of $\lambda$ respectively.
This is the skew partition obtained by rotating the diagram by 180 degrees. Equivalently, given a skew partition $\lambda/\mu$, its rotation $(\lambda/\mu)^\natural$ is the skew partition with cells $\{(a-i, b-j)| (i, j) \in \lambda/\mu\}$, where $b$ and $a$ are the first part and the number of parts of $\lambda$ respectively.
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