- FindStat

Identifier

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ

Values

[1,1] => [1] => [1,0,1,0] => [1,1,0,0] => 0

[2,1] => [1] => [1,0,1,0] => [1,1,0,0] => 0

[1,1,1] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2

[3,1] => [1] => [1,0,1,0] => [1,1,0,0] => 0

[2,2] => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1

[2,1,1] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2

[4,1] => [1] => [1,0,1,0] => [1,1,0,0] => 0

[3,2] => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1

[3,1,1] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2

[2,2,1] => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0

[5,1] => [1] => [1,0,1,0] => [1,1,0,0] => 0

[4,2] => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1

[4,1,1] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2

[3,2,1] => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0

[2,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 4

[6,1] => [1] => [1,0,1,0] => [1,1,0,0] => 0

[5,2] => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1

[5,1,1] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2

[4,2,1] => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0

[3,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2

[3,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 4

[2,2,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 3

[7,1] => [1] => [1,0,1,0] => [1,1,0,0] => 0

[6,2] => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1

[6,1,1] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2

[5,2,1] => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0

[4,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2

[4,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 4

[3,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1

[3,3,1,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2

[3,2,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 3

[8,1] => [1] => [1,0,1,0] => [1,1,0,0] => 0

[7,2] => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1

[7,1,1] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2

[6,2,1] => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0

[5,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2

[5,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 4

[4,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1

[4,3,1,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2

[4,2,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 3

[3,3,2,1] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 0

[9,1] => [1] => [1,0,1,0] => [1,1,0,0] => 0

[8,2] => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1

[8,1,1] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2

[7,2,1] => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0

[6,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2

[6,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 4

[5,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1

[5,3,1,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2

[5,2,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 3

[4,4,2] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3

[4,3,2,1] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 0

[3,3,2,2] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 5

[3,3,2,1,1] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => 6

[10,1] => [1] => [1,0,1,0] => [1,1,0,0] => 0

[9,2] => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1

[9,1,1] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2

[8,2,1] => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0

[7,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2

[7,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 4

[6,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1

[6,3,1,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2

[6,2,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 3

[5,4,2] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3

[5,3,2,1] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 0

[4,4,2,1] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 3

[4,3,2,2] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 5

[4,3,2,1,1] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => 6

[3,3,2,2,1] => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => 6

[11,1] => [1] => [1,0,1,0] => [1,1,0,0] => 0

[10,2] => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1

[10,1,1] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2

[9,2,1] => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0

[8,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2

[8,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 4

[7,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1

[7,3,1,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2

[7,2,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 3

[6,4,2] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3

[6,3,2,1] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 0

[5,4,2,1] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 3

[5,3,2,2] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 5

[5,3,2,1,1] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => 6

[4,4,3,1] => [4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 2

[4,4,2,1,1] => [4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 4

[4,3,2,2,1] => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => 6

[3,3,3,2,1] => [3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4

[12,1] => [1] => [1,0,1,0] => [1,1,0,0] => 0

[11,2] => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1

[11,1,1] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2

[10,2,1] => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0

[9,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2

[9,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 4

[8,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1

[8,3,1,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2

[8,2,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 3

[7,4,2] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3

[7,3,2,1] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 0

[6,4,2,1] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 3

[6,3,2,2] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 5

[6,3,2,1,1] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => 6

>>> Load all 312 entries. <<<

[5,4,3,1] => [4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 2

[5,4,2,1,1] => [4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 4

[5,3,2,2,1] => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => 6

[4,4,3,2] => [4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 1

[4,4,3,1,1] => [4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 2

[4,4,2,2,1] => [4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3

[4,3,3,2,1] => [3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4

[13,1] => [1] => [1,0,1,0] => [1,1,0,0] => 0

[12,2] => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1

[12,1,1] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2

[11,2,1] => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0

[10,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2

[10,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 4

[9,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1

[9,3,1,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2

[9,2,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 3

[8,4,2] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3

[8,3,2,1] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 0

[7,4,2,1] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 3

[7,3,2,2] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 5

[7,3,2,1,1] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => 6

[6,4,3,1] => [4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 2

[6,4,2,1,1] => [4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 4

[6,3,2,2,1] => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => 6

[5,5,3,1] => [5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4

[5,4,3,2] => [4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 1

[5,4,3,1,1] => [4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 2

[5,4,2,2,1] => [4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3

[5,3,3,2,1] => [3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4

[4,4,3,2,1] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 0

[4,4,3,1,1,1] => [4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8

[14,1] => [1] => [1,0,1,0] => [1,1,0,0] => 0

[13,2] => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1

[13,1,1] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2

[12,2,1] => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0

[11,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2

[11,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 4

[10,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1

[10,3,1,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2

[10,2,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 3

[9,4,2] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3

[9,3,2,1] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 0

[8,4,2,1] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 3

[8,3,2,2] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 5

[8,3,2,1,1] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => 6

[7,4,3,1] => [4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 2

[7,4,2,1,1] => [4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 4

[7,3,2,2,1] => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => 6

[6,5,3,1] => [5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4

[6,4,3,2] => [4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 1

[6,4,3,1,1] => [4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 2

[6,4,2,2,1] => [4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3

[6,3,3,2,1] => [3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4

[5,5,3,2] => [5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4

[5,5,3,1,1] => [5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,1,0,0] => 5

[5,4,3,2,1] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 0

[5,4,3,1,1,1] => [4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8

[4,4,3,3,1] => [4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => 6

[4,4,3,2,2] => [4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => 7

[4,4,3,2,1,1] => [4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 8

[15,1] => [1] => [1,0,1,0] => [1,1,0,0] => 0

[14,2] => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1

[14,1,1] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2

[13,2,1] => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0

[12,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2

[12,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 4

[11,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1

[11,3,1,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2

[11,2,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 3

[10,4,2] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3

[10,3,2,1] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 0

[9,4,2,1] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 3

[9,3,2,2] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 5

[9,3,2,1,1] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => 6

[8,4,3,1] => [4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 2

[8,4,2,1,1] => [4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 4

[8,3,2,2,1] => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => 6

[7,5,3,1] => [5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4

[7,4,3,2] => [4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 1

[7,4,3,1,1] => [4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 2

[7,4,2,2,1] => [4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3

[7,3,3,2,1] => [3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4

[6,5,3,2] => [5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4

[6,5,3,1,1] => [5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,1,0,0] => 5

[6,4,3,2,1] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 0

[6,4,3,1,1,1] => [4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8

[5,5,4,2] => [5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3

[5,5,3,2,1] => [5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4

[5,4,3,3,1] => [4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => 6

[5,4,3,2,2] => [4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => 7

[5,4,3,2,1,1] => [4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 8

[4,4,3,3,2] => [4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7

[4,4,3,2,2,1] => [4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 9

[16,1] => [1] => [1,0,1,0] => [1,1,0,0] => 0

[15,2] => [2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1

[15,1,1] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2

[14,2,1] => [2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0

[13,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 2

[13,2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 4

[12,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 1

[12,3,1,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 2

[12,2,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 3

[11,4,2] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3

[11,3,2,1] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 0

[10,4,2,1] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 3

[10,3,2,2] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 5

[10,3,2,1,1] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => 6

[9,4,3,1] => [4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 2

[9,4,2,1,1] => [4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 4

[9,3,2,2,1] => [3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => 6

[8,5,3,1] => [5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4

[8,4,3,2] => [4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 1

[8,4,3,1,1] => [4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 2

[8,4,2,2,1] => [4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3

[8,3,3,2,1] => [3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4

[7,5,3,2] => [5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4

[7,5,3,1,1] => [5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,1,0,0] => 5

[7,4,3,2,1] => [4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 0

[7,4,3,1,1,1] => [4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8

[6,5,4,2] => [5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3

[6,5,3,2,1] => [5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4

[6,4,3,3,1] => [4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => 6

[6,4,3,2,2] => [4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => 7

[6,4,3,2,1,1] => [4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 8

[5,5,4,2,1] => [5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3

[5,5,3,2,2] => [5,3,2,2] => [1,1,0,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0,1,0] => 5

[5,5,3,2,1,1] => [5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 6

[5,4,3,3,2] => [4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7

[5,4,3,2,2,1] => [4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 9

[4,4,3,3,2,1] => [4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 8

[6,5,4,3,2,1] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[5,5,4,3,2,1] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[6,4,4,3,2,1] => [4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 5

[5,4,4,3,2,1] => [4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 5

[4,4,4,3,2,1] => [4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 5

[6,5,3,3,2,1] => [5,3,3,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 4

[5,5,3,3,2,1] => [5,3,3,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 4

[6,4,3,3,2,1] => [4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 8

[5,4,3,3,2,1] => [4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 8

[6,5,4,2,2,1] => [5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 3

[5,5,4,2,2,1] => [5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 3

[6,5,3,2,2,1] => [5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 6

[5,5,3,2,2,1] => [5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 6

[6,4,3,2,2,1] => [4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 9

[6,5,4,3,1,1] => [5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2

[5,5,4,3,1,1] => [5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2

[6,5,4,2,1,1] => [5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4

[5,5,4,2,1,1] => [5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4

[6,5,3,2,1,1] => [5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 6

[6,5,4,3,2] => [5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1

[5,5,4,3,2] => [5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1

[6,4,3,3,2] => [4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7

[6,5,3,2,2] => [5,3,2,2] => [1,1,0,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0,1,0] => 5

[6,5,4,3,1] => [5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2

[5,5,4,3,1] => [5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2

[6,5,4,2,1] => [5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3

[7,6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0

[6,6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0

[7,5,5,4,3,2,1] => [5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 6

[6,5,5,4,3,2,1] => [5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 6

[5,5,5,4,3,2,1] => [5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 6

[7,6,4,4,3,2,1] => [6,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 5

[6,6,4,4,3,2,1] => [6,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 5

[7,5,4,4,3,2,1] => [5,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 10

[6,5,4,4,3,2,1] => [5,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 10

[5,5,4,4,3,2,1] => [5,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 10

[6,6,5,3,3,2,1] => [6,5,3,3,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 4

[6,5,4,3,3,2,1] => [5,4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0] => 12

[6,6,5,4,2,2,1] => [6,5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 3

[6,5,4,3,2,2,1] => [5,4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 12

[5,5,4,3,2,2,1] => [5,4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 12

[7,6,5,4,3,1,1] => [6,5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 2

[6,6,5,4,3,1,1] => [6,5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 2

[6,6,5,4,2,1,1] => [6,5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0] => 4

[6,5,4,3,2,1,1] => [5,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0] => 10

[5,5,4,3,2,1,1] => [5,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0] => 10

[6,5,4,3,1,1,1] => [5,4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,1,0,0,0,0,1,0,0] => 10

[7,5,4,2,1,1,1] => [5,4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,1,1,0,0,0] => 10

[6,5,4,2,1,1,1] => [5,4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,1,1,0,0,0] => 10

[7,6,5,4,3,2] => [6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1

[6,6,5,4,3,2] => [6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1

[6,5,4,4,3,2] => [5,4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => 9

[6,5,4,3,3,2] => [5,4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0] => 10

[6,5,4,3,2,2] => [5,4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0] => 9

[7,6,5,4,3,1] => [6,5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 2

[6,6,5,4,3,1] => [6,5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 2

[7,6,5,4,2,1] => [6,5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 3

[7,6,5,3,2,1] => [6,5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 4

[6,6,5,3,2,1] => [6,5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 4

[7,6,4,3,2,1] => [6,4,3,2,1] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 5

[6,6,4,3,2,1] => [6,4,3,2,1] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 5

[7,5,4,3,2,1] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[7,4,4,3,2,1] => [4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 5

[7,4,3,3,2,1] => [4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 8

[7,4,3,2,1,1] => [4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 8

[7,6,5,4,2] => [6,5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => 3

[7,6,5,3,2] => [6,5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => 4

[7,6,4,3,2] => [6,4,3,2] => [1,1,1,0,0,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 5

[7,5,4,3,2] => [5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1

[7,5,4,3,1] => [5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2

[7,6,4,2,1] => [6,4,2,1] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 5

[6,6,4,2,1] => [6,4,2,1] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 5

[7,5,4,2,1] => [5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3

[8,6,4,2] => [6,4,2] => [1,1,1,1,0,0,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 5

[8,6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0

[8,5,5,4,3,2,1] => [5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 6

[8,5,4,2] => [5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3

[8,6,5,3,1] => [6,5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => 4

[8,5,4,3,2,1] => [5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

[8,6,5,4,3,2] => [6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1

[8,5,4,3,2,1,1] => [5,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0] => 10

search for individual values

searching the database for the individual values of this statistic

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Map

first row removal

Description

Removes the first entry of an integer partition

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.

Map

decomposition reverse

Description

This map is recursively defined as follows.
The unique empty path of semilength

$0$ is sent to itself.
Let

$D$ be a Dyck path of semilength

$n > 0$ and decompose it into

$1 D_1 0 D_2$ with Dyck paths

$D_1, D_2$ of respective semilengths

$n_1$ and

$n_2$ such that

$n_1$ is minimal. One then has

$n_1+n_2 = n-1$ .
Now let

$\tilde D_1$ and

$\tilde D_2$ be the recursively defined respective images of

$D_1$ and

$D_2$ under this map. The image of

$D$ is then defined as

$1 \tilde D_2 0 \tilde D_1$ .