Identifier
-
Mp00097:
Binary words
—delta morphism⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤ
Values
0 => [1] => [1,0] => 0
1 => [1] => [1,0] => 0
00 => [2] => [1,1,0,0] => 0
01 => [1,1] => [1,0,1,0] => 1
10 => [1,1] => [1,0,1,0] => 1
11 => [2] => [1,1,0,0] => 0
000 => [3] => [1,1,1,0,0,0] => 0
001 => [2,1] => [1,1,0,0,1,0] => 1
010 => [1,1,1] => [1,0,1,0,1,0] => 2
011 => [1,2] => [1,0,1,1,0,0] => 2
100 => [1,2] => [1,0,1,1,0,0] => 2
101 => [1,1,1] => [1,0,1,0,1,0] => 2
110 => [2,1] => [1,1,0,0,1,0] => 1
111 => [3] => [1,1,1,0,0,0] => 0
0000 => [4] => [1,1,1,1,0,0,0,0] => 0
0001 => [3,1] => [1,1,1,0,0,0,1,0] => 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0] => 2
0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2
0100 => [1,1,2] => [1,0,1,0,1,1,0,0] => 3
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 3
0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3
0111 => [1,3] => [1,0,1,1,1,0,0,0] => 3
1000 => [1,3] => [1,0,1,1,1,0,0,0] => 3
1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 3
1011 => [1,1,2] => [1,0,1,0,1,1,0,0] => 3
1100 => [2,2] => [1,1,0,0,1,1,0,0] => 2
1101 => [2,1,1] => [1,1,0,0,1,0,1,0] => 2
1110 => [3,1] => [1,1,1,0,0,0,1,0] => 1
1111 => [4] => [1,1,1,1,0,0,0,0] => 0
00000 => [5] => [1,1,1,1,1,0,0,0,0,0] => 0
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 2
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 3
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 3
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 3
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 4
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 4
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 4
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 4
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 4
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 4
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 4
10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4
10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 4
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 3
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 3
11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 3
11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 2
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0] => 0
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
000100 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 3
000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 3
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
001000 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 4
001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 4
001010 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 4
001011 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 4
001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 4
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
001111 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
010000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 5
010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 5
010010 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 5
010011 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 5
010100 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
010101 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
010110 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 5
010111 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 5
011000 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
011010 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 5
011011 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 5
011100 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
011101 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 5
011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
011111 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
100000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
100001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
100010 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 5
100011 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
100100 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 5
100101 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 5
100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
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Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
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 $w$.
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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