Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000877
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => 0 => 1
[2]
 => 0 => 1 => 0
[1,1]
 => 11 => 00 => 2
[3]
 => 1 => 0 => 1
[2,1]
 => 01 => 10 => 0
[1,1,1]
 => 111 => 000 => 3
[4]
 => 0 => 1 => 0
[3,1]
 => 11 => 00 => 2
[2,2]
 => 00 => 11 => 0
[2,1,1]
 => 011 => 100 => 1
[1,1,1,1]
 => 1111 => 0000 => 4
[5]
 => 1 => 0 => 1
[4,1]
 => 01 => 10 => 0
[3,2]
 => 10 => 01 => 1
[3,1,1]
 => 111 => 000 => 3
[2,2,1]
 => 001 => 110 => 0
[2,1,1,1]
 => 0111 => 1000 => 2
[1,1,1,1,1]
 => 11111 => 00000 => 5
[6]
 => 0 => 1 => 0
[5,1]
 => 11 => 00 => 2
[4,2]
 => 00 => 11 => 0
[4,1,1]
 => 011 => 100 => 1
[3,3]
 => 11 => 00 => 2
[3,2,1]
 => 101 => 010 => 1
[3,1,1,1]
 => 1111 => 0000 => 4
[2,2,2]
 => 000 => 111 => 0
[2,2,1,1]
 => 0011 => 1100 => 0
[2,1,1,1,1]
 => 01111 => 10000 => 3
[1,1,1,1,1,1]
 => 111111 => 000000 => 6
[7]
 => 1 => 0 => 1
[6,1]
 => 01 => 10 => 0
[5,2]
 => 10 => 01 => 1
[5,1,1]
 => 111 => 000 => 3
[4,3]
 => 01 => 10 => 0
[4,2,1]
 => 001 => 110 => 0
[4,1,1,1]
 => 0111 => 1000 => 2
[3,3,1]
 => 111 => 000 => 3
[3,2,2]
 => 100 => 011 => 1
[3,2,1,1]
 => 1011 => 0100 => 2
[3,1,1,1,1]
 => 11111 => 00000 => 5
[2,2,2,1]
 => 0001 => 1110 => 0
[2,2,1,1,1]
 => 00111 => 11000 => 1
[2,1,1,1,1,1]
 => 011111 => 100000 => 4
[1,1,1,1,1,1,1]
 => 1111111 => 0000000 => 7
[8]
 => 0 => 1 => 0
[7,1]
 => 11 => 00 => 2
[6,2]
 => 00 => 11 => 0
[6,1,1]
 => 011 => 100 => 1
[5,3]
 => 11 => 00 => 2
[5,2,1]
 => 101 => 010 => 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].