Identifier
- St001365: Binary words ⟶ ℤ
Values
0 => 2
1 => 1
00 => 4
01 => 3
10 => 2
11 => 1
000 => 8
001 => 7
010 => 6
011 => 4
100 => 4
101 => 3
110 => 2
111 => 1
0000 => 16
0001 => 15
0010 => 14
0011 => 11
0100 => 12
0101 => 10
0110 => 8
0111 => 5
1000 => 8
1001 => 7
1010 => 6
1011 => 4
1100 => 4
1101 => 3
1110 => 2
1111 => 1
00000 => 32
00001 => 31
00010 => 30
00011 => 26
00100 => 28
00101 => 25
00110 => 22
00111 => 16
01000 => 24
01001 => 22
01010 => 20
01011 => 15
01100 => 16
01101 => 13
01110 => 10
01111 => 6
10000 => 16
10001 => 15
10010 => 14
10011 => 11
10100 => 12
10101 => 10
10110 => 8
10111 => 5
11000 => 8
11001 => 7
11010 => 6
11011 => 4
11100 => 4
11101 => 3
11110 => 2
11111 => 1
000000 => 64
000001 => 63
000010 => 62
000011 => 57
000100 => 60
000101 => 56
000110 => 52
000111 => 42
001000 => 56
001001 => 53
001010 => 50
001011 => 41
001100 => 44
001101 => 38
001110 => 32
001111 => 22
010000 => 48
010001 => 46
010010 => 44
010011 => 37
010100 => 40
010101 => 35
010110 => 30
010111 => 21
011000 => 32
011001 => 29
011010 => 26
011011 => 19
011100 => 20
011101 => 16
011110 => 12
011111 => 7
100000 => 32
100001 => 31
100010 => 30
100011 => 26
100100 => 28
100101 => 25
100110 => 22
>>> Load all 1022 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The number of lattice paths of the same length weakly above the path given by a binary word.
In particular, there are $2^n$ lattice paths weakly above the the length $n$ binary word $0\dots 0$, there is a unique path weakly above $1\dots 1$, and there are $\binom{2n}{n}$ paths weakly above the length $2n$ binary word $10\dots 10$.
In particular, there are $2^n$ lattice paths weakly above the the length $n$ binary word $0\dots 0$, there is a unique path weakly above $1\dots 1$, and there are $\binom{2n}{n}$ paths weakly above the length $2n$ binary word $10\dots 10$.
Code
@cached_function
def all_paths_above(L, i=0):
if i < 0:
return []
elif len(L) == 0:
return [tuple()]
else:
steps = []
if L[0] == 1:
steps.append((1,i))
steps.append((0,i-1))
if L[0] == 0:
steps.append((1,i+1))
steps.append((0,i))
return [ tuple([a]) + path for a,i in steps for path in all_paths_above(L[1:],i)]
def statistic(D):
return len(all_paths_above(tuple(D)))
Created
Mar 16, 2019 at 14:35 by Martin Rubey
Updated
Mar 16, 2019 at 14:35 by Martin Rubey
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!