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Identifier

St000676: Dyck paths ⟶ ℤ

Values

[] => 0

[1,0] => 1

[1,0,1,0] => 2

[1,1,0,0] => 1

[1,0,1,0,1,0] => 3

[1,0,1,1,0,0] => 2

[1,1,0,0,1,0] => 2

[1,1,0,1,0,0] => 1

[1,1,1,0,0,0] => 2

[1,0,1,0,1,0,1,0] => 4

[1,0,1,0,1,1,0,0] => 3

[1,0,1,1,0,0,1,0] => 3

[1,0,1,1,0,1,0,0] => 2

[1,0,1,1,1,0,0,0] => 3

[1,1,0,0,1,0,1,0] => 3

[1,1,0,0,1,1,0,0] => 2

[1,1,0,1,0,0,1,0] => 2

[1,1,0,1,0,1,0,0] => 1

[1,1,0,1,1,0,0,0] => 2

[1,1,1,0,0,0,1,0] => 3

[1,1,1,0,0,1,0,0] => 2

[1,1,1,0,1,0,0,0] => 3

[1,1,1,1,0,0,0,0] => 2

[1,0,1,0,1,0,1,0,1,0] => 5

[1,0,1,0,1,0,1,1,0,0] => 4

[1,0,1,0,1,1,0,0,1,0] => 4

[1,0,1,0,1,1,0,1,0,0] => 3

[1,0,1,0,1,1,1,0,0,0] => 4

[1,0,1,1,0,0,1,0,1,0] => 4

[1,0,1,1,0,0,1,1,0,0] => 3

[1,0,1,1,0,1,0,0,1,0] => 3

[1,0,1,1,0,1,0,1,0,0] => 2

[1,0,1,1,0,1,1,0,0,0] => 3

[1,0,1,1,1,0,0,0,1,0] => 4

[1,0,1,1,1,0,0,1,0,0] => 3

[1,0,1,1,1,0,1,0,0,0] => 4

[1,0,1,1,1,1,0,0,0,0] => 3

[1,1,0,0,1,0,1,0,1,0] => 4

[1,1,0,0,1,0,1,1,0,0] => 3

[1,1,0,0,1,1,0,0,1,0] => 3

[1,1,0,0,1,1,0,1,0,0] => 2

[1,1,0,0,1,1,1,0,0,0] => 3

[1,1,0,1,0,0,1,0,1,0] => 3

[1,1,0,1,0,0,1,1,0,0] => 2

[1,1,0,1,0,1,0,0,1,0] => 2

[1,1,0,1,0,1,0,1,0,0] => 1

[1,1,0,1,0,1,1,0,0,0] => 2

[1,1,0,1,1,0,0,0,1,0] => 3

[1,1,0,1,1,0,0,1,0,0] => 2

[1,1,0,1,1,0,1,0,0,0] => 3

[1,1,0,1,1,1,0,0,0,0] => 2

[1,1,1,0,0,0,1,0,1,0] => 4

[1,1,1,0,0,0,1,1,0,0] => 3

[1,1,1,0,0,1,0,0,1,0] => 3

[1,1,1,0,0,1,0,1,0,0] => 2

[1,1,1,0,0,1,1,0,0,0] => 3

[1,1,1,0,1,0,0,0,1,0] => 4

[1,1,1,0,1,0,0,1,0,0] => 3

[1,1,1,0,1,0,1,0,0,0] => 4

[1,1,1,0,1,1,0,0,0,0] => 3

[1,1,1,1,0,0,0,0,1,0] => 3

[1,1,1,1,0,0,0,1,0,0] => 2

[1,1,1,1,0,0,1,0,0,0] => 3

[1,1,1,1,0,1,0,0,0,0] => 2

[1,1,1,1,1,0,0,0,0,0] => 3

[1,0,1,0,1,0,1,0,1,0,1,0] => 6

[1,0,1,0,1,0,1,0,1,1,0,0] => 5

[1,0,1,0,1,0,1,1,0,0,1,0] => 5

[1,0,1,0,1,0,1,1,0,1,0,0] => 4

[1,0,1,0,1,0,1,1,1,0,0,0] => 5

[1,0,1,0,1,1,0,0,1,0,1,0] => 5

[1,0,1,0,1,1,0,0,1,1,0,0] => 4

[1,0,1,0,1,1,0,1,0,0,1,0] => 4

[1,0,1,0,1,1,0,1,0,1,0,0] => 3

[1,0,1,0,1,1,0,1,1,0,0,0] => 4

[1,0,1,0,1,1,1,0,0,0,1,0] => 5

[1,0,1,0,1,1,1,0,0,1,0,0] => 4

[1,0,1,0,1,1,1,0,1,0,0,0] => 5

[1,0,1,0,1,1,1,1,0,0,0,0] => 4

[1,0,1,1,0,0,1,0,1,0,1,0] => 5

[1,0,1,1,0,0,1,0,1,1,0,0] => 4

[1,0,1,1,0,0,1,1,0,0,1,0] => 4

[1,0,1,1,0,0,1,1,0,1,0,0] => 3

[1,0,1,1,0,0,1,1,1,0,0,0] => 4

[1,0,1,1,0,1,0,0,1,0,1,0] => 4

[1,0,1,1,0,1,0,0,1,1,0,0] => 3

[1,0,1,1,0,1,0,1,0,0,1,0] => 3

[1,0,1,1,0,1,0,1,0,1,0,0] => 2

[1,0,1,1,0,1,0,1,1,0,0,0] => 3

[1,0,1,1,0,1,1,0,0,0,1,0] => 4

[1,0,1,1,0,1,1,0,0,1,0,0] => 3

[1,0,1,1,0,1,1,0,1,0,0,0] => 4

[1,0,1,1,0,1,1,1,0,0,0,0] => 3

[1,0,1,1,1,0,0,0,1,0,1,0] => 5

[1,0,1,1,1,0,0,0,1,1,0,0] => 4

[1,0,1,1,1,0,0,1,0,0,1,0] => 4

[1,0,1,1,1,0,0,1,0,1,0,0] => 3

[1,0,1,1,1,0,0,1,1,0,0,0] => 4

[1,0,1,1,1,0,1,0,0,0,1,0] => 5

[1,0,1,1,1,0,1,0,0,1,0,0] => 4

[1,0,1,1,1,0,1,0,1,0,0,0] => 5

>>> Load all 1200 entries. <<<

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = q + 3\ q^{2} + q^{3}$

$F_{4} = q + 6\ q^{2} + 6\ q^{3} + q^{4}$

$F_{5} = q + 10\ q^{2} + 20\ q^{3} + 10\ q^{4} + q^{5}$

$F_{6} = q + 15\ q^{2} + 50\ q^{3} + 50\ q^{4} + 15\ q^{5} + q^{6}$

$F_{7} = q + 21\ q^{2} + 105\ q^{3} + 175\ q^{4} + 105\ q^{5} + 21\ q^{6} + q^{7}$

Description

The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength

$n$ with

$k$ up steps in odd positions and

$k$ returns to the main diagonal are counted by the binomial coefficient

$\binom{n-1}{k-1}$ [3,4].

References

[1] Elizalde, S., Deutsch, E. A simple and unusual bijection for Dyck paths and its consequences MathSciNet:2028229
[2] The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000329
[3] Pascal's triangle read by rows: C(n,k) = binomial(n,k) = n!/(k!*(n-k)!), 0 <= k <= n. OEIS:A007318
[4] Houston, R. Counting particular Dyck paths MathOverflow:285079

Code

def statistic(D):
    return D[::2].count(1)

Created

Dec 27, 2016 at 00:06 by Martin Rubey

Updated

Nov 03, 2017 at 20:52 by Martin Rubey