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Matching statistic: St000877
Mapping
St000877: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1
1 => 0
00 => 2
01 => 1
10 => 0
11 => 0
000 => 3
001 => 2
010 => 1
011 => 1
100 => 1
101 => 0
110 => 0
111 => 0
0000 => 4
0001 => 3
0010 => 2
0011 => 2
0100 => 2
0101 => 1
0110 => 1
0111 => 1
1000 => 2
1001 => 1
1010 => 0
1011 => 0
1100 => 0
1101 => 0
1110 => 0
1111 => 0
00000 => 5
00001 => 4
00010 => 3
00011 => 3
00100 => 3
00101 => 2
00110 => 2
00111 => 2
01000 => 3
01001 => 2
01010 => 1
01011 => 1
01100 => 1
01101 => 1
01110 => 1
01111 => 1
10000 => 3
10001 => 2
10010 => 1
10011 => 1
Description
The depth of the binary word interpreted as a path. This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2]. The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].