- FindStat

Identifier

Mp00130: Permutations —descent tops⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ

Values

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

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

[1,2,3] => 00 => [2] => [1,1,0,0] => 0

[1,3,2] => 01 => [1,1] => [1,0,1,0] => 1

[2,1,3] => 10 => [1,1] => [1,0,1,0] => 1

[2,3,1] => 01 => [1,1] => [1,0,1,0] => 1

[3,1,2] => 01 => [1,1] => [1,0,1,0] => 1

[3,2,1] => 11 => [2] => [1,1,0,0] => 0

[1,2,3,4] => 000 => [3] => [1,1,1,0,0,0] => 0

[1,2,4,3] => 001 => [2,1] => [1,1,0,0,1,0] => 1

[1,3,4,2] => 001 => [2,1] => [1,1,0,0,1,0] => 1

[1,4,2,3] => 001 => [2,1] => [1,1,0,0,1,0] => 1

[1,4,3,2] => 011 => [1,2] => [1,0,1,1,0,0] => 2

[2,1,3,4] => 100 => [1,2] => [1,0,1,1,0,0] => 2

[2,3,4,1] => 001 => [2,1] => [1,1,0,0,1,0] => 1

[2,4,1,3] => 001 => [2,1] => [1,1,0,0,1,0] => 1

[2,4,3,1] => 011 => [1,2] => [1,0,1,1,0,0] => 2

[3,1,4,2] => 011 => [1,2] => [1,0,1,1,0,0] => 2

[3,2,1,4] => 110 => [2,1] => [1,1,0,0,1,0] => 1

[3,2,4,1] => 011 => [1,2] => [1,0,1,1,0,0] => 2

[3,4,1,2] => 001 => [2,1] => [1,1,0,0,1,0] => 1

[4,1,2,3] => 001 => [2,1] => [1,1,0,0,1,0] => 1

[4,1,3,2] => 011 => [1,2] => [1,0,1,1,0,0] => 2

[4,2,3,1] => 011 => [1,2] => [1,0,1,1,0,0] => 2

[4,3,1,2] => 011 => [1,2] => [1,0,1,1,0,0] => 2

[4,3,2,1] => 111 => [3] => [1,1,1,0,0,0] => 0

[1,2,3,4,5] => 0000 => [4] => [1,1,1,1,0,0,0,0] => 0

[1,2,3,5,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 1

[1,2,4,5,3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 1

[1,2,5,3,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 1

[1,2,5,4,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[1,3,4,5,2] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 1

[1,3,5,2,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 1

[1,3,5,4,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[1,4,2,5,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[1,4,3,2,5] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

[1,4,3,5,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[1,4,5,2,3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 1

[1,5,2,3,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 1

[1,5,2,4,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[1,5,3,4,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[1,5,4,2,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[1,5,4,3,2] => 0111 => [1,3] => [1,0,1,1,1,0,0,0] => 3

[2,1,3,4,5] => 1000 => [1,3] => [1,0,1,1,1,0,0,0] => 3

[2,1,3,5,4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

[2,1,4,5,3] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

[2,1,5,3,4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

[2,3,4,5,1] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 1

[2,3,5,1,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 1

[2,3,5,4,1] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[2,4,1,5,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[2,4,3,1,5] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

[2,4,3,5,1] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[2,4,5,1,3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 1

[2,5,1,3,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 1

[2,5,1,4,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[2,5,3,4,1] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[2,5,4,1,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[2,5,4,3,1] => 0111 => [1,3] => [1,0,1,1,1,0,0,0] => 3

[3,1,4,2,5] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

[3,1,5,4,2] => 0111 => [1,3] => [1,0,1,1,1,0,0,0] => 3

[3,2,1,4,5] => 1100 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[3,2,4,1,5] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

[3,2,5,4,1] => 0111 => [1,3] => [1,0,1,1,1,0,0,0] => 3

[3,4,1,5,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[3,4,2,5,1] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[3,4,5,1,2] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 1

[3,4,5,2,1] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

[3,5,1,2,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 1

[3,5,1,4,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[3,5,2,1,4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

[3,5,2,4,1] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[3,5,4,1,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[4,1,2,5,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[4,1,3,2,5] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

[4,1,3,5,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[4,1,5,2,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[4,1,5,3,2] => 0111 => [1,3] => [1,0,1,1,1,0,0,0] => 3

[4,2,3,1,5] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

[4,2,3,5,1] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[4,2,5,1,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[4,2,5,3,1] => 0111 => [1,3] => [1,0,1,1,1,0,0,0] => 3

[4,3,1,2,5] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

[4,3,1,5,2] => 0111 => [1,3] => [1,0,1,1,1,0,0,0] => 3

[4,3,2,1,5] => 1110 => [3,1] => [1,1,1,0,0,0,1,0] => 1

[4,3,2,5,1] => 0111 => [1,3] => [1,0,1,1,1,0,0,0] => 3

[4,3,5,1,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[4,5,1,2,3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 1

[4,5,2,1,3] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

[5,1,2,3,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0] => 1

[5,1,2,4,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[5,1,3,4,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[5,1,4,2,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[5,1,4,3,2] => 0111 => [1,3] => [1,0,1,1,1,0,0,0] => 3

[5,2,1,3,4] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

[5,2,3,4,1] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[5,2,4,1,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[5,2,4,3,1] => 0111 => [1,3] => [1,0,1,1,1,0,0,0] => 3

[5,3,1,4,2] => 0111 => [1,3] => [1,0,1,1,1,0,0,0] => 3

[5,3,2,4,1] => 0111 => [1,3] => [1,0,1,1,1,0,0,0] => 3

[5,3,4,1,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

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$F_{2} = 2$

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

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

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

$w$ .

Map

descent tops

Description

The descent tops of a permutation as a binary word.
Since 1 is never a descent top, it is omitted and the first letter of the word corresponds to the element 2.