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Identifier

St001956: Dyck paths ⟶ ℤ

Values

[1,0] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,1,0,1,0,1,0,0] => 7

[1,0,1,1,0,1,1,0,0,0] => 7

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

[1,0,1,1,1,0,0,1,0,0] => 7

[1,0,1,1,1,0,1,0,0,0] => 7

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

[1,1,0,0,1,0,1,0,1,0] => 7

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

[1,1,0,0,1,1,0,0,1,0] => 7

[1,1,0,0,1,1,0,1,0,0] => 8

[1,1,0,0,1,1,1,0,0,0] => 8

[1,1,0,1,0,0,1,0,1,0] => 8

[1,1,0,1,0,0,1,1,0,0] => 9

[1,1,0,1,0,1,0,0,1,0] => 9

[1,1,0,1,0,1,0,1,0,0] => 10

[1,1,0,1,0,1,1,0,0,0] => 7

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 4

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0] => 10

[1,0,1,0,1,0,1,0,1,1,0,0] => 9

[1,0,1,0,1,0,1,1,0,0,1,0] => 7

[1,0,1,0,1,0,1,1,0,1,0,0] => 8

[1,0,1,0,1,0,1,1,1,0,0,0] => 8

[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] => 6

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

[1,0,1,0,1,1,0,1,0,1,0,0] => 7

[1,0,1,0,1,1,0,1,1,0,0,0] => 7

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

[1,0,1,0,1,1,1,0,0,1,0,0] => 7

[1,0,1,0,1,1,1,0,1,0,0,0] => 7

[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] => 8

[1,0,1,1,0,0,1,0,1,1,0,0] => 7

[1,0,1,1,0,0,1,1,0,0,1,0] => 8

[1,0,1,1,0,0,1,1,0,1,0,0] => 9

[1,0,1,1,0,0,1,1,1,0,0,0] => 9

[1,0,1,1,0,1,0,0,1,0,1,0] => 9

[1,0,1,1,0,1,0,0,1,1,0,0] => 10

[1,0,1,1,0,1,0,1,0,0,1,0] => 10

[1,0,1,1,0,1,0,1,0,1,0,0] => 11

[1,0,1,1,0,1,0,1,1,0,0,0] => 11

[1,0,1,1,0,1,1,0,0,0,1,0] => 10

[1,0,1,1,0,1,1,0,0,1,0,0] => 11

[1,0,1,1,0,1,1,0,1,0,0,0] => 11

[1,0,1,1,0,1,1,1,0,0,0,0] => 8

[1,0,1,1,1,0,0,0,1,0,1,0] => 9

[1,0,1,1,1,0,0,0,1,1,0,0] => 10

[1,0,1,1,1,0,0,1,0,0,1,0] => 10

[1,0,1,1,1,0,0,1,0,1,0,0] => 11

[1,0,1,1,1,0,0,1,1,0,0,0] => 11

[1,0,1,1,1,0,1,0,0,0,1,0] => 10

[1,0,1,1,1,0,1,0,0,1,0,0] => 11

[1,0,1,1,1,0,1,0,1,0,0,0] => 11

[1,0,1,1,1,0,1,1,0,0,0,0] => 8

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Description

The comajor index for set-valued two-row standard Young tableaux.
The comajorindex is the sum $\sum_k (n+1-k)$ over all natural descents $k$.
Bijections via bicolored Motzkin paths (with two restrictions, see [1]) give the following for Dyck paths.
Let $j$ be smallest integer such that $2j$ is a down step.
Then $k$ is a natural descent if

$k-2\ge j$ and positions $2(k-1)-1,2(k-1)$ are a valley i.e. [0,1], or
$k-2\ge j$ and positions $2(k-1)-1,2(k-1)$ are a peak i.e. [1,0], or
$k-1\ge j$ and positions $2(k-1),2k-1,2k$ form [0,1,1], or
$k=j$ and positions $2k-1,2k$ are double down i.e. [0,0], or
$k < j$ and positions $2k-1,2k$ are a valley i.e. [0,1].

References

[1] Alexander Lazar and Svante Linusson, "Two-row Set-Valued Tableaux: Catalan$^+k$ Combinatorics", Extended abstract at FPSAC 2024.

Code

def natural_descents(D):
    for j in range(1, D.semilength()+1):
        if D[2*j-1] == 0:
            break
    else:
        raise ValueError
    des = []
    for k in range(1, len(D)+2):
        if k-2 >= j and D[2*(k-1)-2:2*(k-1)] == [0,1]:
            des.append(k)
        elif k-2 >= j and D[2*(k-1)-2:2*(k-1)] == [1,0]:
            des.append(k)
        elif k-1 >= j and D[2*(k-1)-1:2*k] == [0,1,1]:
            des.append(k)
        elif k == j and D[2*k-2:2*k] == [0,0]:
            des.append(k)
        elif k < j and D[2*k-2:2*k] == [0,1]:
            des.append(k)
    return des

def statistic(D):
    n = D.semilength()
    return sum(n+1-k for k in natural_descents(D))

Created

Jul 25, 2024 at 23:26 by Martin Rubey

Updated

Jul 25, 2024 at 23:26 by Martin Rubey