Identifier
- St000877: Binary words ⟶ ℤ
Values
=>
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11=>0
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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].
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].
References
[1] Lothaire, M. Applied combinatorics on words MathSciNet:2165687
[2] Square table read by antidiagonals: a(n,k) = binomial(n+k, floor(k/2)). OEIS:A061554
[2] Square table read by antidiagonals: a(n,k) = binomial(n+k, floor(k/2)). OEIS:A061554
Code
def statistic(w): h = m = 0 for i, s in enumerate(w): if s == 1: h += 1 else: h -= 1 m = min(h, m) return -m
Created
Jul 03, 2017 at 08:07 by Martin Rubey
Updated
Dec 19, 2017 at 18:24 by Martin Rubey
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