Identifier
-
Mp00043:
Integer partitions
—to Dyck path⟶
Dyck paths
St000335: Dyck paths ⟶ ℤ
Values
[1] => [1,0,1,0] => 1
[2] => [1,1,0,0,1,0] => 2
[1,1] => [1,0,1,1,0,0] => 2
[3] => [1,1,1,0,0,0,1,0] => 3
[2,1] => [1,0,1,0,1,0] => 1
[1,1,1] => [1,0,1,1,1,0,0,0] => 3
[4] => [1,1,1,1,0,0,0,0,1,0] => 4
[3,1] => [1,1,0,1,0,0,1,0] => 1
[2,2] => [1,1,0,0,1,1,0,0] => 2
[2,1,1] => [1,0,1,1,0,1,0,0] => 1
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => 4
[5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
[4,1] => [1,1,1,0,1,0,0,0,1,0] => 1
[3,2] => [1,1,0,0,1,0,1,0] => 2
[3,1,1] => [1,0,1,1,0,0,1,0] => 2
[2,2,1] => [1,0,1,0,1,1,0,0] => 2
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => 1
[1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => 1
[4,2] => [1,1,1,0,0,1,0,0,1,0] => 2
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 2
[3,3] => [1,1,1,0,0,0,1,1,0,0] => 3
[3,2,1] => [1,0,1,0,1,0,1,0] => 1
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => 2
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => 3
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => 2
[2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => 1
[5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => 2
[5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 2
[4,3] => [1,1,1,0,0,0,1,0,1,0] => 3
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => 1
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => 3
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => 1
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => 1
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => 1
[3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => 2
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => 3
[2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2
[5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => 3
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => 1
[5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0] => 3
[4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => 1
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => 2
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => 1
[4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => 3
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => 3
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => 2
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => 1
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => 1
[2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
[2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => 3
[5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 4
[5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => 1
[5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0] => 2
[5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => 1
[5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 4
[4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => 1
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => 2
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => 2
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => 2
[4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 1
[3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => 2
[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => 2
[3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0] => 1
[3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => 1
[2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => 4
[5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0] => 1
[5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => 2
[5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => 2
[5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0] => 2
[5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0] => 1
[4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0] => 2
[4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0] => 2
[4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0] => 2
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 1
[4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 2
[4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => 2
[4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => 2
[3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => 2
[3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => 2
[3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,0] => 2
[3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,0] => 1
[5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0] => 2
[5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,0] => 2
[5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
[5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0] => 1
[5,3,1,1,1] => [1,0,1,1,1,0,0,1,0,0,1,0] => 2
[5,2,2,2] => [1,1,0,0,1,1,1,0,0,0,1,0] => 3
[5,2,2,1,1] => [1,0,1,1,0,1,1,0,0,0,1,0] => 2
[4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => 4
[4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0] => 1
[4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0] => 3
[4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0] => 0
[4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,0] => 1
[4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 1
[4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 2
[3,3,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0] => 4
[3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3
[3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,0] => 2
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Description
The difference of lower and upper interactions.
An upper interaction in a Dyck path is the occurrence of a factor $0^k 1^k$ with $k \geq 1$ (see St000331The number of upper interactions of a Dyck path.), and a lower interaction is the occurrence of a factor $1^k 0^k$ with $k \geq 1$. In both cases, $1$ denotes an up-step $0$ denotes a a down-step.
An upper interaction in a Dyck path is the occurrence of a factor $0^k 1^k$ with $k \geq 1$ (see St000331The number of upper interactions of a Dyck path.), and a lower interaction is the occurrence of a factor $1^k 0^k$ with $k \geq 1$. In both cases, $1$ denotes an up-step $0$ denotes a a down-step.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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