Identifier
Values
0 => [1] => [1,0] => [1,0] => 2
1 => [1] => [1,0] => [1,0] => 2
00 => [2] => [1,1,0,0] => [1,0,1,0] => 3
01 => [1,1] => [1,0,1,0] => [1,1,0,0] => 2
10 => [1,1] => [1,0,1,0] => [1,1,0,0] => 2
11 => [2] => [1,1,0,0] => [1,0,1,0] => 3
000 => [3] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 4
001 => [2,1] => [1,1,0,0,1,0] => [1,1,1,0,0,0] => 2
010 => [1,1,1] => [1,0,1,0,1,0] => [1,1,0,0,1,0] => 3
011 => [1,2] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
100 => [1,2] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
101 => [1,1,1] => [1,0,1,0,1,0] => [1,1,0,0,1,0] => 3
110 => [2,1] => [1,1,0,0,1,0] => [1,1,1,0,0,0] => 2
111 => [3] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 4
0000 => [4] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 5
0001 => [3,1] => [1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,0] => 2
0010 => [2,1,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,0] => 2
0011 => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,1,0,0,0,0] => 2
0100 => [1,1,2] => [1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 2
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0] => 3
0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,0,1,0] => 4
0111 => [1,3] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 3
1000 => [1,3] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 3
1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,0,1,0] => 4
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0] => 3
1011 => [1,1,2] => [1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 2
1100 => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,1,0,0,0,0] => 2
1101 => [2,1,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,0] => 2
1110 => [3,1] => [1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,0] => 2
1111 => [4] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 5
00000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 6
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => 2
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 2
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0] => 2
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 3
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] => 3
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 3
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 2
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] => 3
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] => 4
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] => 3
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 3
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] => 3
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 5
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 3
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 3
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 5
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] => 3
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 3
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] => 3
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] => 4
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] => 3
10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 2
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 3
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] => 3
11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 3
11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0] => 2
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 2
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => 2
11111 => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 6
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 7
000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 3
000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 2
000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
000100 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => 3
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => 2
000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 2
001000 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => 2
001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => 3
001010 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,0,1,1,0,0] => 3
001011 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => 3
001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => 4
001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,0,1,0,1,0] => 4
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 3
001111 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => 2
010000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => 3
010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,1,0,0] => 4
010010 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 5
010011 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => 4
010100 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => 3
010101 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
010110 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,1,1,0,0,0] => 3
010111 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 3
011000 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 2
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 3
011010 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,1,0,0] => 3
011011 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => 3
011100 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 3
011101 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,1,0,0,0] => 3
011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 6
011111 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 4
100000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 4
100001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 6
100010 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,1,0,0,0] => 3
100011 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 3
100100 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => 3
100101 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,1,0,0] => 3
100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 3
>>> Load all 126 entries. <<<
100111 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 2
101000 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 3
101001 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,1,1,0,0,0] => 3
101010 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
101011 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => 3
101100 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => 4
101101 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 5
101110 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,1,0,0] => 4
101111 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => 3
110000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => 2
110001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 3
110010 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,0,1,0,1,0] => 4
110011 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => 4
110100 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => 3
110101 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,0,1,1,0,0] => 3
110110 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => 3
110111 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => 2
111000 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 2
111001 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => 2
111010 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => 3
111011 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
111100 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
111101 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 2
111110 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 3
111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 7
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Description
The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A.
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
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.