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Identifier

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤ

Values

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

[2] => [] => [] => [] => 0

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

[3] => [] => [] => [] => 0

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

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

[4] => [] => [] => [] => 0

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

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

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

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

[5] => [] => [] => [] => 0

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

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

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

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

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

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

[6] => [] => [] => [] => 0

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

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

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

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

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

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

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

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

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

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

[7] => [] => [] => [] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[8] => [] => [] => [] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[9] => [] => [] => [] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[10] => [] => [] => [] => 0

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + q + q^{2}$

$F_{4} = 1 + 2\ q + q^{2} + q^{3}$

$F_{5} = 1 + 2\ q + 2\ q^{2} + q^{3} + q^{4}$

$F_{6} = 1 + 3\ q + 3\ q^{2} + 2\ q^{3} + q^{4} + q^{5}$

$F_{7} = 1 + 3\ q + 4\ q^{2} + 3\ q^{3} + 2\ q^{4} + q^{5} + q^{6}$

$F_{8} = 1 + 4\ q + 5\ q^{2} + 5\ q^{3} + 3\ q^{4} + 2\ q^{5} + q^{6} + q^{7}$

$F_{9} = 1 + 4\ q + 7\ q^{2} + 6\ q^{3} + 5\ q^{4} + 3\ q^{5} + 2\ q^{6} + q^{7} + q^{8}$

Description

The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength

$n$ with

$k$ up steps in odd positions and

$k$ returns to the main diagonal are counted by the binomial coefficient

$\binom{n-1}{k-1}$ [3,4].

Map

first row removal

Description

Removes the first entry of an integer partition

Map

conjugate

Description

Return the conjugate partition of the partition.
The conjugate partition of the partition

$\lambda$ of

$n$ is the partition

$\lambda^*$ whose Ferrers diagram is obtained from the diagram of

$\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.

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.