Identifier
Values
0 => [2] => [1,1,0,0] => 0
1 => [1,1] => [1,0,1,0] => 0
00 => [3] => [1,1,1,0,0,0] => 0
01 => [2,1] => [1,1,0,0,1,0] => 0
10 => [1,2] => [1,0,1,1,0,0] => 0
11 => [1,1,1] => [1,0,1,0,1,0] => 0
000 => [4] => [1,1,1,1,0,0,0,0] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0] => 0
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => 0
100 => [1,3] => [1,0,1,1,1,0,0,0] => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => 0
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 0
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 0
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 0
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 0
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 0
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 0
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 0
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 0
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 0
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 0
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 0
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 0
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 0
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 0
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 0
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 0
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 0
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 0
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 0
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 0
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 0
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 0
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 0
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 0
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 0
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 0
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 0
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 0
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 0
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 0
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 0
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 0
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 0
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 0
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => 0
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 0
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => 0
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => 0
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => 0
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 0
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => 0
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => 0
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => 0
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => 0
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => 0
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => 0
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => 0
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 0
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 0
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 0
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => 0
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 0
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 0
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => 0
010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => 0
011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => 0
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => 0
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => 0
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => 0
011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => 0
011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => 0
011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 0
011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 0
100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 0
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 0
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 0
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => 0
100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 0
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 0
100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => 0
>>> Load all 199 entries. <<<
100111 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => 0
101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 0
101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 0
101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 0
101011 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => 0
101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => 0
101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => 0
101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 0
101111 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => 0
110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 0
110001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => 0
110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => 0
110011 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => 0
110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => 0
110101 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => 0
110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 0
110111 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => 0
111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 0
111001 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => 0
111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 0
111011 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => 0
111100 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 0
111101 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => 0
111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
111111 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0
0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 0
0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => 0
0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => 0
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => 0
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0] => 0
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0] => 0
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0] => 0
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => 0
0100000 => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => 0
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0] => 0
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => 0
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0] => 0
0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0] => 0
0101010 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => 0
0111011 => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0] => 0
0111100 => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0] => 0
0111101 => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0] => 0
0111110 => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
0111111 => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0
1000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0
1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 0
1000011 => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0] => 0
1001000 => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 0
1001111 => [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0] => 0
1010000 => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 0
1011110 => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 0
1100000 => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 0
1100001 => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 0
1100111 => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0] => 0
1101110 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 0
1110110 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 0
1111010 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 0
1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
00000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => 0
00000001 => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => 0
00000010 => [7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0] => 0
00111111 => [3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0
01000000 => [2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0
01000001 => [2,6,1] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
01111101 => [2,1,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => 0
01111110 => [2,1,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
01111111 => [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] => 0
10000000 => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 0
10000001 => [1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => 0
10000010 => [1,6,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => 0
10000011 => [1,6,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0] => 0
10011111 => [1,3,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => 0
10100000 => [1,2,6] => [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => 0
10111110 => [1,2,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
11000000 => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0
11000001 => [1,1,6,1] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
11111010 => [1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 0
11111110 => [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] => 0
000000000 => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => 0
000000001 => [9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => 0
000000010 => [8,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0] => 0
001111111 => [3,1,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0
010000000 => [2,8] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 0
011111110 => [2,1,1,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
011111111 => [2,1,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0
100000000 => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => 0
100000001 => [1,8,1] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => 0
100000011 => [1,7,1,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0] => 0
110000000 => [1,1,8] => [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 0
110000001 => [1,1,7,1] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => 0
111111110 => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
1111111110 => [1,1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
1100000001 => [1,1,8,1] => [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => 0
1000000011 => [1,8,1,1] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0] => 0
1000000000 => [1,10] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => 0
=> [1] => [1,0] => 0
0000000001 => [10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => 0
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searching the database for statistics with the same generating function
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.