- FindStat

Identifier

Mp00022: Cores —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤ

Values

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

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$F_{(2, 3)} = 2\ q^{2}$

$F_{(3, 3)} = 2\ q^{4}$

$F_{(4, 3)} = q^{5} + 2\ q^{6}$

$F_{(5, 3)} = q^{8} + 2\ q^{9}$

$F_{(6, 3)} = 2\ q^{10} + 2\ q^{12}$

$F_{(2, 4)} = 2\ q^{2}$

$F_{(3, 4)} = 3\ q^{3}$

$F_{(4, 4)} = q^{4} + 3\ q^{5}$

$F_{(5, 4)} = 3\ q^{6} + 2\ q^{7}$

$F_{(6, 4)} = q^{7} + 4\ q^{8} + 2\ q^{9}$

$F_{(2, 5)} = 2\ q^{2}$

$F_{(3, 5)} = 3\ q^{3}$

$F_{(4, 5)} = 5\ q^{4}$

$F_{(5, 5)} = 2\ q^{5} + 4\ q^{6}$

$F_{(6, 5)} = 2\ q^{6} + 5\ q^{7} + 2\ q^{8}$

$F_{(2, 6)} = 2\ q^{2}$

$F_{(3, 6)} = 3\ q^{3}$

$F_{(4, 6)} = 5\ q^{4}$

$F_{(5, 6)} = 7\ q^{5}$

$F_{(6, 6)} = 5\ q^{6} + 5\ q^{7}$

$F_{(7, 6)} = 4\ q^{7} + 7\ q^{8} + 2\ q^{9}$

Description

The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].

Map

parallelogram polyomino

Description

Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.

Map

to partition

Description

Considers a core as a partition.
This embedding is graded and injective but not surjective on

$k$ -cores for a given parameter

$k$ , while it is surjective and neither graded nor injective on the collection of all cores.