Identifier
-
Mp00097:
Binary words
—delta morphism⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001238: Dyck paths ⟶ ℤ
Values
0 => [1] => [1,0] => 1
1 => [1] => [1,0] => 1
00 => [2] => [1,1,0,0] => 1
01 => [1,1] => [1,0,1,0] => 1
10 => [1,1] => [1,0,1,0] => 1
11 => [2] => [1,1,0,0] => 1
000 => [3] => [1,1,1,0,0,0] => 1
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] => 1
100 => [1,2] => [1,0,1,1,0,0] => 1
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] => 1
0000 => [4] => [1,1,1,1,0,0,0,0] => 1
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] => 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0] => 2
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] => 1
0111 => [1,3] => [1,0,1,1,1,0,0,0] => 1
1000 => [1,3] => [1,0,1,1,1,0,0,0] => 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => 1
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] => 2
1100 => [2,2] => [1,1,0,0,1,1,0,0] => 1
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] => 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0] => 1
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] => 1
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 2
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] => 1
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 1
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 2
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 2
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] => 3
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 1
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 2
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 1
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 1
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 2
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 1
10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 3
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] => 2
10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 2
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 1
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 1
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] => 2
11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 1
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] => 1
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
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] => 1
000100 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 2
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] => 1
000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 1
001000 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 2
001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 2
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] => 3
001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 1
001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 2
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 1
001111 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 1
010000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2
010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 2
010010 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 3
010011 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 2
010100 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 4
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] => 3
010111 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3
011000 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 1
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 1
011010 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 3
011011 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 2
011100 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 1
011101 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 2
011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 1
011111 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
100000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
100001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 1
100010 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 2
100011 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 1
100100 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 2
100101 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 3
100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 1
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Description
The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S.
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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