Identifier
-
Mp00095:
Integer partitions
—to binary word⟶
Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ
Values
[1] => 10 => [1,1] => [1,0,1,0] => 1
[2] => 100 => [1,2] => [1,0,1,1,0,0] => 2
[1,1] => 110 => [2,1] => [1,1,0,0,1,0] => 1
[3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0] => 3
[1,1,1] => 1110 => [3,1] => [1,1,1,0,0,0,1,0] => 1
[4] => 10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 4
[2,2] => 1100 => [2,2] => [1,1,0,0,1,1,0,0] => 2
[1,1,1,1] => 11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
[5] => 100000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[3,1,1] => 100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
[1,1,1,1,1] => 111110 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[6] => 1000000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[4,1,1] => 1000110 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 6
[3,3] => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 3
[3,1,1,1] => 1001110 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 6
[2,2,2] => 11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
[1,1,1,1,1,1] => 1111110 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[4,4] => 110000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
[4,2,2] => 1001100 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6
[3,3,1,1] => 1100110 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 5
[2,2,2,2] => 111100 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
[3,3,3] => 111000 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
[5,5] => 1100000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 5
[2,2,2,2,2] => 1111100 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 2
[4,4,4] => 1110000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 4
[3,3,3,3] => 1111000 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 3
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Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to binary word
Description
Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.
Map
delta morphism
Description
Applies the delta morphism to a binary word.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
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