Identifier
Values
000 => [3] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 0
001 => [2,1] => [1,1,0,0,1,0] => [1,1,1,0,0,0] => 1
010 => [1,1,1] => [1,0,1,0,1,0] => [1,1,0,0,1,0] => 0
011 => [1,2] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 1
100 => [1,2] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 1
101 => [1,1,1] => [1,0,1,0,1,0] => [1,1,0,0,1,0] => 0
110 => [2,1] => [1,1,0,0,1,0] => [1,1,1,0,0,0] => 1
111 => [3] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 0
0000 => [4] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 0
0001 => [3,1] => [1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,0] => 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,0] => 1
0011 => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,1,0,0,0,0] => 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0] => 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,0,1,0] => 0
0111 => [1,3] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 1
1000 => [1,3] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,0,1,0] => 0
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0] => 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 1
1100 => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,1,0,0,0,0] => 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,0] => 1
1110 => [3,1] => [1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,0] => 1
1111 => [4] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 0
00000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 0
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => 1
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 1
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => 0
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 1
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 1
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 1
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => 1
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 0
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 1
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 1
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 1
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 1
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 1
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 1
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 1
10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 1
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 0
10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => 1
10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 1
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 1
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 1
11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => 0
11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1
11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 1
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 0
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Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
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
Elizalde-Deutsch bijection
Description
The Elizalde-Deutsch bijection on Dyck paths.
.Let $n$ be the length of the Dyck path. Consider the steps $1,n,2,n-1,\dots$ of $D$. When considering the $i$-th step its corresponding matching step has not yet been read, let the $i$-th step of the image of $D$ be an up step, otherwise let it be a down step.