- FindStat

Identifier

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000256: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 4 + 10\ q$

$F_{5} = 8 + 34\ q$

$F_{6} = 24 + 98\ q + 10\ q^{2}$

$F_{7} = 45 + 356\ q + 28\ q^{2}$

$F_{8} = 97 + 1178\ q + 155\ q^{2}$

Description

The number of parts from which one can substract 2 and still get an integer partition.

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.

Map

touch composition

Description

Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.