Identifier
- St001313: Binary words ⟶ ℤ
Values
0 => 1
1 => 1
00 => 1
01 => 2
10 => 1
11 => 1
000 => 1
001 => 3
010 => 2
011 => 3
100 => 1
101 => 2
110 => 1
111 => 1
0000 => 1
0001 => 4
0010 => 3
0011 => 6
0100 => 2
0101 => 5
0110 => 3
0111 => 4
1000 => 1
1001 => 3
1010 => 2
1011 => 3
1100 => 1
1101 => 2
1110 => 1
1111 => 1
00000 => 1
00001 => 5
00010 => 4
00011 => 10
00100 => 3
00101 => 9
00110 => 6
00111 => 10
01000 => 2
01001 => 7
01010 => 5
01011 => 9
01100 => 3
01101 => 7
01110 => 4
01111 => 5
10000 => 1
10001 => 4
10010 => 3
10011 => 6
10100 => 2
10101 => 5
10110 => 3
10111 => 4
11000 => 1
11001 => 3
11010 => 2
11011 => 3
11100 => 1
11101 => 2
11110 => 1
11111 => 1
000000 => 1
000001 => 6
000010 => 5
000011 => 15
000100 => 4
000101 => 14
000110 => 10
000111 => 20
001000 => 3
001001 => 12
001010 => 9
001011 => 19
001100 => 6
001101 => 16
001110 => 10
001111 => 15
010000 => 2
010001 => 9
010010 => 7
010011 => 16
010100 => 5
010101 => 14
010110 => 9
010111 => 14
011000 => 3
011001 => 10
011010 => 7
011011 => 12
011100 => 4
011101 => 9
011110 => 5
011111 => 6
100000 => 1
100001 => 5
100010 => 4
100011 => 10
100100 => 3
100101 => 9
100110 => 6
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Description
The number of Dyck paths above the lattice path given by a binary word.
One may treat a binary word as a lattice path starting at the origin and treating $1$'s as steps $(1,0)$ and $0$'s as steps $(0,1)$. Given a binary word $w$, this statistic counts the number of lattice paths from the origin to the same endpoint as $w$ that stay weakly above $w$.
See St001312Number of parabolic noncrossing partitions indexed by the composition. for this statistic on compositions treated as bounce paths.
One may treat a binary word as a lattice path starting at the origin and treating $1$'s as steps $(1,0)$ and $0$'s as steps $(0,1)$. Given a binary word $w$, this statistic counts the number of lattice paths from the origin to the same endpoint as $w$ that stay weakly above $w$.
See St001312Number of parabolic noncrossing partitions indexed by the composition. for this statistic on compositions treated as bounce paths.
Code
def lattice_paths_above_boundary(B):
B = list(B)
if sum(B) == 0:
return [B]
else:
paths = [ B[:1] + p for p in lattice_paths_above_boundary(B[1:]) ]
if B[0] == 0:
i = B.index(1)
B = B[:i] + B[i+1:]
paths.extend( [ [1] + p for p in lattice_paths_above_boundary(B) ] )
return paths
def statistic(w):
return len(lattice_paths_above_boundary(w))
Created
Dec 12, 2018 at 14:54 by Christian Stump
Updated
Dec 12, 2018 at 14:54 by Christian Stump
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