Identifier
Values
0 => [1] => [1,0] => [2,1] => 0
1 => [1] => [1,0] => [2,1] => 0
00 => [2] => [1,1,0,0] => [2,3,1] => 1
01 => [1,1] => [1,0,1,0] => [3,1,2] => 1
10 => [1,1] => [1,0,1,0] => [3,1,2] => 1
11 => [2] => [1,1,0,0] => [2,3,1] => 1
000 => [3] => [1,1,1,0,0,0] => [2,3,4,1] => 1
001 => [2,1] => [1,1,0,0,1,0] => [2,4,1,3] => 2
010 => [1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => 1
011 => [1,2] => [1,0,1,1,0,0] => [3,1,4,2] => 2
100 => [1,2] => [1,0,1,1,0,0] => [3,1,4,2] => 2
101 => [1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => 1
110 => [2,1] => [1,1,0,0,1,0] => [2,4,1,3] => 2
111 => [3] => [1,1,1,0,0,0] => [2,3,4,1] => 1
0000 => [4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
0001 => [3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 2
0010 => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 2
0011 => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 3
0100 => [1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 2
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 3
0111 => [1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 2
1000 => [1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 3
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 2
1100 => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 3
1101 => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 2
1110 => [3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 2
1111 => [4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 1
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 2
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 2
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 3
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 3
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 2
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 4
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 3
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 2
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 3
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 1
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 2
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 4
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 3
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 3
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 2
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 2
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 3
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 3
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 4
10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 2
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 1
10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 3
10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 2
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 3
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 4
11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 2
11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 3
11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 3
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 2
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 2
11111 => [5] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 1
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 1
000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => 2
000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => 2
000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [2,3,4,6,1,7,5] => 3
001010 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => 2
010100 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => 2
010101 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => 1
010110 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => 3
010111 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => 2
101000 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => 2
101001 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => 3
101010 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => 1
101011 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => 2
110101 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => 2
111100 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [2,3,4,6,1,7,5] => 3
111101 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => 2
111110 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => 2
111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 1
0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => 1
0000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [2,3,4,5,6,8,1,7] => 2
0010101 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,8,1,3,4,5,6,7] => 2
0101010 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => 1
0101011 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [7,1,2,3,4,5,8,6] => 2
0110011 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [3,1,5,2,7,4,8,6] => 6
1001100 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [3,1,5,2,7,4,8,6] => 6
1010100 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [7,1,2,3,4,5,8,6] => 2
1010101 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => 1
1101010 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,8,1,3,4,5,6,7] => 2
1111110 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [2,3,4,5,6,8,1,7] => 2
1111111 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => 1
00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 1
00000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [2,3,4,5,6,7,9,1,8] => 2
00101010 => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,9,1,3,4,5,6,7,8] => 2
01010100 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [8,1,2,3,4,5,6,9,7] => 2
01010101 => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,1,2,3,4,5,6,7,8] => 1
10101010 => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,1,2,3,4,5,6,7,8] => 1
10101011 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [8,1,2,3,4,5,6,9,7] => 2
11010101 => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,9,1,3,4,5,6,7,8] => 2
11111110 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [2,3,4,5,6,7,9,1,8] => 2
>>> Load all 114 entries. <<<
11111111 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 1
000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,1] => 1
001010101 => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,10,1,3,4,5,6,7,8,9] => 2
010101010 => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,1,2,3,4,5,6,7,8,9] => 1
010101011 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [9,1,2,3,4,5,6,7,10,8] => 2
101010100 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [9,1,2,3,4,5,6,7,10,8] => 2
101010101 => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,1,2,3,4,5,6,7,8,9] => 1
110101010 => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,10,1,3,4,5,6,7,8,9] => 2
111111111 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,1] => 1
1010101010 => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [11,1,2,3,4,5,6,7,8,9,10] => 1
1111111111 => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
0101010101 => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [11,1,2,3,4,5,6,7,8,9,10] => 1
0000000000 => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => 1
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Description
The number of times a permutation switches from increasing to decreasing or decreasing to increasing.
This is the same as the number of inner peaks plus the number of inner valleys and called alternating runs in [2]
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
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.