- FindStat

Identifier

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: 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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 2

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

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

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

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

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

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

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

>>> Load all 5555 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 1 + 11\ q + 2\ q^{2}$

$F_{5} = 1 + 26\ q + 15\ q^{2}$

$F_{6} = 1 + 57\ q + 69\ q^{2} + 5\ q^{3}$

$F_{7} = 1 + 120\ q + 252\ q^{2} + 56\ q^{3}$

$F_{8} = 1 + 247\ q + 804\ q^{2} + 364\ q^{3} + 14\ q^{4}$

Description

The number of parts of an integer partition that are at least two.

Map

to partition

Description

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

Map

rise composition

Description

Send a Dyck path to the composition of sizes of its rises.