- FindStat

Identifier

Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[-1,-2,-3,-4] => 1111 => [4] => [1,1,1,1,0,0,0,0] => 0

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

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

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

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

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

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

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

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

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

[-1,-2,-4,-3] => 1111 => [4] => [1,1,1,1,0,0,0,0] => 0

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

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

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

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

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

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

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

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

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

[-1,-3,-2,-4] => 1111 => [4] => [1,1,1,1,0,0,0,0] => 0

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

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

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

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

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

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

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

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

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

[-1,-3,-4,-2] => 1111 => [4] => [1,1,1,1,0,0,0,0] => 0

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

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

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

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

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

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

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

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

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

[-1,-4,-2,-3] => 1111 => [4] => [1,1,1,1,0,0,0,0] => 0

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

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

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

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

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

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

bounce path

Description

The bounce path determined by an integer composition.

Map

signs

Description

The binary word recording the signs of a signed permutation.
This map sends a signed permutation $\pi\in\mathfrak H_n$ to the binary word $w$ of length $n$ such that $w_i = 0$ if $\pi(i) > 0$.