- FindStat

Identifier

Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000531: Integer partitions ⟶ ℤ

Values

[1] => [[1],[]] => ([],1) => [1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[5,2] => [[5,2],[]] => ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7) => [5,2] => 8

[5,1,1] => [[5,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7) => [5,2] => 8

[4,3] => [[4,3],[]] => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7) => [4,3] => 9

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

[4,1,1,1] => [[4,1,1,1],[]] => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7) => [4,3] => 9

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

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

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

[3,1,1,1,1] => [[3,1,1,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7) => [5,2] => 8

[2,2,2,1] => [[2,2,2,1],[]] => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7) => [4,3] => 9

[2,2,1,1,1] => [[2,2,1,1,1],[]] => ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7) => [5,2] => 8

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

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

[8] => [[8],[]] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => [8] => 8

[7,1] => [[7,1],[]] => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8) => [7,1] => 6

[6,2] => [[6,2],[]] => ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8) => [6,2] => 10

[6,1,1] => [[6,1,1],[]] => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8) => [6,2] => 10

[5,3] => [[5,3],[]] => ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8) => [5,3] => 12

[5,2,1] => [[5,2,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8) => [5,2,1] => 3

[5,1,1,1] => [[5,1,1,1],[]] => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8) => [5,3] => 12

[4,4] => [[4,4],[]] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => [5,3] => 12

[4,3,1] => [[4,3,1],[]] => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8) => [4,3,1] => 4

[4,2,2] => [[4,2,2],[]] => ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8) => [4,3,1] => 4

[4,2,1,1] => [[4,2,1,1],[]] => ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8) => [4,3,1] => 4

[4,1,1,1,1] => [[4,1,1,1,1],[]] => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8) => [5,3] => 12

[3,3,2] => [[3,3,2],[]] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8) => [4,3,1] => 4

[3,3,1,1] => [[3,3,1,1],[]] => ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8) => [4,3,1] => 4

[3,2,2,1] => [[3,2,2,1],[]] => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8) => [4,3,1] => 4

[3,2,1,1,1] => [[3,2,1,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8) => [5,2,1] => 3

[3,1,1,1,1,1] => [[3,1,1,1,1,1],[]] => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8) => [6,2] => 10

[2,2,2,2] => [[2,2,2,2],[]] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => [5,3] => 12

[2,2,2,1,1] => [[2,2,2,1,1],[]] => ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8) => [5,3] => 12

[2,2,1,1,1,1] => [[2,2,1,1,1,1],[]] => ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8) => [6,2] => 10

[2,1,1,1,1,1,1] => [[2,1,1,1,1,1,1],[]] => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8) => [7,1] => 6

[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => [8] => 8

[9] => [[9],[]] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) => [9] => 9

[8,1] => [[8,1],[]] => ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9) => [8,1] => 7

[7,2] => [[7,2],[]] => ([(0,2),(0,7),(2,8),(3,4),(4,6),(5,3),(6,1),(7,5),(7,8)],9) => [7,2] => 12

[7,1,1] => [[7,1,1],[]] => ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9) => [7,2] => 12

[6,3] => [[6,3],[]] => ([(0,2),(0,6),(2,7),(3,5),(4,3),(4,8),(5,1),(6,4),(6,7),(7,8)],9) => [6,3] => 15

[6,2,1] => [[6,2,1],[]] => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(6,8),(7,1),(7,8)],9) => [6,2,1] => 4

[6,1,1,1] => [[6,1,1,1],[]] => ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9) => [6,3] => 15

[5,4] => [[5,4],[]] => ([(0,2),(0,5),(2,6),(3,4),(3,8),(4,1),(4,7),(5,3),(5,6),(6,8),(8,7)],9) => [5,4] => 16

[5,3,1] => [[5,3,1],[]] => ([(0,5),(0,6),(3,4),(3,8),(4,2),(5,3),(5,7),(6,1),(6,7),(7,8)],9) => [5,3,1] => 6

[5,2,2] => [[5,2,2],[]] => ([(0,5),(0,6),(2,8),(3,4),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8)],9) => [5,3,1] => 6

[5,2,1,1] => [[5,2,1,1],[]] => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(6,8),(7,3),(7,8)],9) => [5,3,1] => 6

[5,1,1,1,1] => [[5,1,1,1,1],[]] => ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(8,4)],9) => [5,4] => 16

[4,4,1] => [[4,4,1],[]] => ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9) => [5,3,1] => 6

[4,3,2] => [[4,3,2],[]] => ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9) => [4,3,2] => 8

[4,3,1,1] => [[4,3,1,1],[]] => ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9) => [4,3,2] => 8

[4,2,2,1] => [[4,2,2,1],[]] => ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9) => [4,3,2] => 8

[4,2,1,1,1] => [[4,2,1,1,1],[]] => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(6,8),(7,3),(7,8)],9) => [5,3,1] => 6

[4,1,1,1,1,1] => [[4,1,1,1,1,1],[]] => ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9) => [6,3] => 15

[3,3,3] => [[3,3,3],[]] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => [5,3,1] => 6

[3,3,2,1] => [[3,3,2,1],[]] => ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9) => [4,3,2] => 8

[3,3,1,1,1] => [[3,3,1,1,1],[]] => ([(0,5),(0,6),(2,8),(3,4),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8)],9) => [5,3,1] => 6

[3,2,2,2] => [[3,2,2,2],[]] => ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9) => [5,3,1] => 6

[3,2,2,1,1] => [[3,2,2,1,1],[]] => ([(0,5),(0,6),(3,4),(3,8),(4,2),(5,3),(5,7),(6,1),(6,7),(7,8)],9) => [5,3,1] => 6

[3,2,1,1,1,1] => [[3,2,1,1,1,1],[]] => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(6,8),(7,1),(7,8)],9) => [6,2,1] => 4

[3,1,1,1,1,1,1] => [[3,1,1,1,1,1,1],[]] => ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9) => [7,2] => 12

[2,2,2,2,1] => [[2,2,2,2,1],[]] => ([(0,2),(0,5),(2,6),(3,4),(3,8),(4,1),(4,7),(5,3),(5,6),(6,8),(8,7)],9) => [5,4] => 16

[2,2,2,1,1,1] => [[2,2,2,1,1,1],[]] => ([(0,2),(0,6),(2,7),(3,5),(4,3),(4,8),(5,1),(6,4),(6,7),(7,8)],9) => [6,3] => 15

[2,2,1,1,1,1,1] => [[2,2,1,1,1,1,1],[]] => ([(0,2),(0,7),(2,8),(3,4),(4,6),(5,3),(6,1),(7,5),(7,8)],9) => [7,2] => 12

[2,1,1,1,1,1,1,1] => [[2,1,1,1,1,1,1,1],[]] => ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9) => [8,1] => 7

[1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1],[]] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) => [9] => 9

[10] => [[10],[]] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => [10] => 10

[9,1] => [[9,1],[]] => ([(0,2),(0,9),(3,4),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => [9,1] => 8

[8,2] => [[8,2],[]] => ([(0,2),(0,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6),(8,9)],10) => [8,2] => 14

[8,1,1] => [[8,1,1],[]] => ([(0,8),(0,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6),(9,1)],10) => [8,2] => 14

[7,3] => [[7,3],[]] => ([(0,2),(0,7),(2,8),(3,4),(4,6),(5,3),(5,9),(6,1),(7,5),(7,8),(8,9)],10) => [7,3] => 18

>>> Load all 144 entries. <<<

[7,2,1] => [[7,2,1],[]] => ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(7,9),(8,1),(8,9)],10) => [7,2,1] => 5

[7,1,1,1] => [[7,1,1,1],[]] => ([(0,8),(0,9),(3,4),(4,6),(5,3),(6,2),(7,1),(8,7),(9,5)],10) => [7,3] => 18

[6,4] => [[6,4],[]] => ([(0,2),(0,6),(2,7),(3,1),(4,3),(4,8),(5,4),(5,9),(6,5),(6,7),(7,9),(9,8)],10) => [6,4] => 20

[6,3,1] => [[6,3,1],[]] => ([(0,6),(0,7),(3,5),(4,3),(4,9),(5,2),(6,4),(6,8),(7,1),(7,8),(8,9)],10) => [6,3,1] => 8

[6,2,2] => [[6,2,2],[]] => ([(0,6),(0,7),(2,9),(3,5),(4,3),(5,1),(6,4),(6,8),(7,2),(7,8),(8,9)],10) => [6,3,1] => 8

[6,2,1,1] => [[6,2,1,1],[]] => ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(7,9),(8,4),(8,9)],10) => [6,3,1] => 8

[6,1,1,1,1] => [[6,1,1,1,1],[]] => ([(0,8),(0,9),(3,7),(4,3),(5,6),(6,1),(7,2),(8,4),(9,5)],10) => [6,4] => 20

[5,5] => [[5,5],[]] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => [6,4] => 20

[5,4,1] => [[5,4,1],[]] => ([(0,5),(0,6),(3,4),(3,9),(4,2),(4,8),(5,3),(5,7),(6,1),(6,7),(7,9),(9,8)],10) => [5,4,1] => 9

[5,3,2] => [[5,3,2],[]] => ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9)],10) => [5,3,2] => 12

[5,3,1,1] => [[5,3,1,1],[]] => ([(0,6),(0,7),(3,4),(3,9),(4,1),(5,2),(6,5),(6,8),(7,3),(7,8),(8,9)],10) => [5,3,2] => 12

[5,2,2,1] => [[5,2,2,1],[]] => ([(0,6),(0,7),(3,4),(4,1),(5,2),(5,9),(6,5),(6,8),(7,3),(7,8),(8,9)],10) => [5,3,2] => 12

[5,2,1,1,1] => [[5,2,1,1,1],[]] => ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(7,9),(8,4),(8,9)],10) => [5,4,1] => 9

[5,1,1,1,1,1] => [[5,1,1,1,1,1],[]] => ([(0,8),(0,9),(3,7),(4,3),(5,6),(6,1),(7,2),(8,4),(9,5)],10) => [6,4] => 20

[4,4,2] => [[4,4,2],[]] => ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(9,7)],10) => [5,3,2] => 12

[4,4,1,1] => [[4,4,1,1],[]] => ([(0,5),(0,6),(2,8),(3,1),(4,2),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(9,8)],10) => [5,3,2] => 12

[4,3,3] => [[4,3,3],[]] => ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8),(7,9),(8,9)],10) => [5,3,2] => 12

[4,3,2,1] => [[4,3,2,1],[]] => ([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10) => [4,3,2,1] => 1

[4,3,1,1,1] => [[4,3,1,1,1],[]] => ([(0,6),(0,7),(3,4),(4,1),(5,2),(5,9),(6,5),(6,8),(7,3),(7,8),(8,9)],10) => [5,3,2] => 12

[4,2,2,2] => [[4,2,2,2],[]] => ([(0,5),(0,6),(2,8),(3,1),(4,2),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(9,8)],10) => [5,3,2] => 12

[4,2,2,1,1] => [[4,2,2,1,1],[]] => ([(0,6),(0,7),(3,4),(3,9),(4,1),(5,2),(6,5),(6,8),(7,3),(7,8),(8,9)],10) => [5,3,2] => 12

[4,2,1,1,1,1] => [[4,2,1,1,1,1],[]] => ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(7,9),(8,4),(8,9)],10) => [6,3,1] => 8

[4,1,1,1,1,1,1] => [[4,1,1,1,1,1,1],[]] => ([(0,8),(0,9),(3,4),(4,6),(5,3),(6,2),(7,1),(8,7),(9,5)],10) => [7,3] => 18

[3,3,3,1] => [[3,3,3,1],[]] => ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8),(7,9),(8,9)],10) => [5,3,2] => 12

[3,3,2,2] => [[3,3,2,2],[]] => ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(9,7)],10) => [5,3,2] => 12

[3,3,2,1,1] => [[3,3,2,1,1],[]] => ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9)],10) => [5,3,2] => 12

[3,3,1,1,1,1] => [[3,3,1,1,1,1],[]] => ([(0,6),(0,7),(2,9),(3,5),(4,3),(5,1),(6,4),(6,8),(7,2),(7,8),(8,9)],10) => [6,3,1] => 8

[3,2,2,2,1] => [[3,2,2,2,1],[]] => ([(0,5),(0,6),(3,4),(3,9),(4,2),(4,8),(5,3),(5,7),(6,1),(6,7),(7,9),(9,8)],10) => [5,4,1] => 9

[3,2,2,1,1,1] => [[3,2,2,1,1,1],[]] => ([(0,6),(0,7),(3,5),(4,3),(4,9),(5,2),(6,4),(6,8),(7,1),(7,8),(8,9)],10) => [6,3,1] => 8

[3,2,1,1,1,1,1] => [[3,2,1,1,1,1,1],[]] => ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(7,9),(8,1),(8,9)],10) => [7,2,1] => 5

[3,1,1,1,1,1,1,1] => [[3,1,1,1,1,1,1,1],[]] => ([(0,8),(0,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6),(9,1)],10) => [8,2] => 14

[2,2,2,2,2] => [[2,2,2,2,2],[]] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => [6,4] => 20

[2,2,2,2,1,1] => [[2,2,2,2,1,1],[]] => ([(0,2),(0,6),(2,7),(3,1),(4,3),(4,8),(5,4),(5,9),(6,5),(6,7),(7,9),(9,8)],10) => [6,4] => 20

[2,2,2,1,1,1,1] => [[2,2,2,1,1,1,1],[]] => ([(0,2),(0,7),(2,8),(3,4),(4,6),(5,3),(5,9),(6,1),(7,5),(7,8),(8,9)],10) => [7,3] => 18

[2,2,1,1,1,1,1,1] => [[2,2,1,1,1,1,1,1],[]] => ([(0,2),(0,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6),(8,9)],10) => [8,2] => 14

[2,1,1,1,1,1,1,1,1] => [[2,1,1,1,1,1,1,1,1],[]] => ([(0,2),(0,9),(3,4),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => [9,1] => 8

[1,1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1,1],[]] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => [10] => 10

[4,4,3] => [[4,4,3],[]] => ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(8,10),(9,7),(9,10)],11) => [5,4,2] => 18

[3,3,3,2] => [[3,3,3,2],[]] => ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(8,10),(9,7),(9,10)],11) => [5,4,2] => 18

[6,6] => [[6,6],[]] => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12) => [7,5] => 30

[4,4,4] => [[4,4,4],[]] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => [6,4,2] => 24

[3,3,3,3] => [[3,3,3,3],[]] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => [6,4,2] => 24

[2,2,2,2,2,2] => [[2,2,2,2,2,2],[]] => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12) => [7,5] => 30

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$F_{1} = q$

$F_{2} = 2\ q^{2}$

$F_{3} = q + 2\ q^{3}$

$F_{4} = 3\ q^{2} + 2\ q^{4}$

$F_{5} = 2\ q^{3} + 3\ q^{4} + 2\ q^{5}$

$F_{6} = q + 2\ q^{4} + 8\ q^{6}$

$F_{7} = 4\ q^{2} + 2\ q^{5} + 2\ q^{7} + 4\ q^{8} + 3\ q^{9}$

$F_{8} = 2\ q^{3} + 6\ q^{4} + 2\ q^{6} + 2\ q^{8} + 4\ q^{10} + 6\ q^{12}$

$F_{9} = 2\ q^{4} + 9\ q^{6} + 2\ q^{7} + 4\ q^{8} + 2\ q^{9} + 4\ q^{12} + 4\ q^{15} + 3\ q^{16}$

$F_{10} = q + 2\ q^{5} + 8\ q^{8} + 3\ q^{9} + 2\ q^{10} + 12\ q^{12} + 4\ q^{14} + 4\ q^{18} + 6\ q^{20}$

Description

The leading coefficient of the rook polynomial of an integer partition.
Let

$m$ be the minimum of the number of parts and the size of the first part of an integer partition

$\lambda$ . Then this statistic yields the number of ways to place

$m$ non-attacking rooks on the Ferrers board of

$\lambda$ .

Map

Greene-Kleitman invariant

Description

The Greene-Kleitman invariant of a poset.
This is the partition

$(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)$ , where

$c_k$ is the maximum cardinality of a union of

$k$ chains of the poset. Equivalently, this is the conjugate of the partition

$(a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)$ , where

$a_k$ is the maximum cardinality of a union of

$k$ antichains of the poset.

Map

to skew partition

Description

The partition regarded as a skew partition.

Map

cell poset

Description

The Young diagram of a skew partition regarded as a poset.
This is the poset on the cells of the Young diagram, such that a cell

$d$ is greater than a cell

$c$ if the entry in

$d$ must be larger than the entry of

$c$ in any standard Young tableau on the skew partition.