Identifier
- St000006: Dyck paths ⟶ ℤ
Values
=>
Cc0005;cc-rep
[1,0]=>0
[1,0,1,0]=>1
[1,1,0,0]=>0
[1,0,1,0,1,0]=>3
[1,0,1,1,0,0]=>1
[1,1,0,0,1,0]=>2
[1,1,0,1,0,0]=>1
[1,1,1,0,0,0]=>0
[1,0,1,0,1,0,1,0]=>6
[1,0,1,0,1,1,0,0]=>3
[1,0,1,1,0,0,1,0]=>4
[1,0,1,1,0,1,0,0]=>2
[1,0,1,1,1,0,0,0]=>1
[1,1,0,0,1,0,1,0]=>5
[1,1,0,0,1,1,0,0]=>3
[1,1,0,1,0,0,1,0]=>4
[1,1,0,1,0,1,0,0]=>3
[1,1,0,1,1,0,0,0]=>1
[1,1,1,0,0,0,1,0]=>2
[1,1,1,0,0,1,0,0]=>2
[1,1,1,0,1,0,0,0]=>1
[1,1,1,1,0,0,0,0]=>0
[1,0,1,0,1,0,1,0,1,0]=>10
[1,0,1,0,1,0,1,1,0,0]=>6
[1,0,1,0,1,1,0,0,1,0]=>7
[1,0,1,0,1,1,0,1,0,0]=>4
[1,0,1,0,1,1,1,0,0,0]=>3
[1,0,1,1,0,0,1,0,1,0]=>8
[1,0,1,1,0,0,1,1,0,0]=>5
[1,0,1,1,0,1,0,0,1,0]=>6
[1,0,1,1,0,1,0,1,0,0]=>4
[1,0,1,1,0,1,1,0,0,0]=>2
[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]=>2
[1,0,1,1,1,1,0,0,0,0]=>1
[1,1,0,0,1,0,1,0,1,0]=>9
[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]=>5
[1,1,0,0,1,1,1,0,0,0]=>3
[1,1,0,1,0,0,1,0,1,0]=>8
[1,1,0,1,0,0,1,1,0,0]=>6
[1,1,0,1,0,1,0,0,1,0]=>7
[1,1,0,1,0,1,0,1,0,0]=>6
[1,1,0,1,0,1,1,0,0,0]=>3
[1,1,0,1,1,0,0,0,1,0]=>4
[1,1,0,1,1,0,0,1,0,0]=>4
[1,1,0,1,1,0,1,0,0,0]=>2
[1,1,0,1,1,1,0,0,0,0]=>1
[1,1,1,0,0,0,1,0,1,0]=>5
[1,1,1,0,0,0,1,1,0,0]=>4
[1,1,1,0,0,1,0,0,1,0]=>5
[1,1,1,0,0,1,0,1,0,0]=>5
[1,1,1,0,0,1,1,0,0,0]=>3
[1,1,1,0,1,0,0,0,1,0]=>3
[1,1,1,0,1,0,0,1,0,0]=>4
[1,1,1,0,1,0,1,0,0,0]=>3
[1,1,1,0,1,1,0,0,0,0]=>1
[1,1,1,1,0,0,0,0,1,0]=>2
[1,1,1,1,0,0,0,1,0,0]=>2
[1,1,1,1,0,0,1,0,0,0]=>2
[1,1,1,1,0,1,0,0,0,0]=>1
[1,1,1,1,1,0,0,0,0,0]=>0
[1,0,1,0,1,0,1,0,1,0,1,0]=>15
[1,0,1,0,1,0,1,0,1,1,0,0]=>10
[1,0,1,0,1,0,1,1,0,0,1,0]=>11
[1,0,1,0,1,0,1,1,0,1,0,0]=>7
[1,0,1,0,1,0,1,1,1,0,0,0]=>6
[1,0,1,0,1,1,0,0,1,0,1,0]=>12
[1,0,1,0,1,1,0,0,1,1,0,0]=>8
[1,0,1,0,1,1,0,1,0,0,1,0]=>9
[1,0,1,0,1,1,0,1,0,1,0,0]=>6
[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]=>7
[1,0,1,0,1,1,1,0,0,1,0,0]=>5
[1,0,1,0,1,1,1,0,1,0,0,0]=>4
[1,0,1,0,1,1,1,1,0,0,0,0]=>3
[1,0,1,1,0,0,1,0,1,0,1,0]=>13
[1,0,1,1,0,0,1,0,1,1,0,0]=>9
[1,0,1,1,0,0,1,1,0,0,1,0]=>10
[1,0,1,1,0,0,1,1,0,1,0,0]=>7
[1,0,1,1,0,0,1,1,1,0,0,0]=>5
[1,0,1,1,0,1,0,0,1,0,1,0]=>11
[1,0,1,1,0,1,0,0,1,1,0,0]=>8
[1,0,1,1,0,1,0,1,0,0,1,0]=>9
[1,0,1,1,0,1,0,1,0,1,0,0]=>7
[1,0,1,1,0,1,0,1,1,0,0,0]=>4
[1,0,1,1,0,1,1,0,0,0,1,0]=>6
[1,0,1,1,0,1,1,0,0,1,0,0]=>5
[1,0,1,1,0,1,1,0,1,0,0,0]=>3
[1,0,1,1,0,1,1,1,0,0,0,0]=>2
[1,0,1,1,1,0,0,0,1,0,1,0]=>8
[1,0,1,1,1,0,0,0,1,1,0,0]=>6
[1,0,1,1,1,0,0,1,0,0,1,0]=>7
[1,0,1,1,1,0,0,1,0,1,0,0]=>6
[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]=>5
[1,0,1,1,1,0,1,0,1,0,0,0]=>4
[1,0,1,1,1,0,1,1,0,0,0,0]=>2
[1,0,1,1,1,1,0,0,0,0,1,0]=>4
[1,0,1,1,1,1,0,0,0,1,0,0]=>3
[1,0,1,1,1,1,0,0,1,0,0,0]=>3
[1,0,1,1,1,1,0,1,0,0,0,0]=>2
[1,0,1,1,1,1,1,0,0,0,0,0]=>1
[1,1,0,0,1,0,1,0,1,0,1,0]=>14
[1,1,0,0,1,0,1,0,1,1,0,0]=>10
[1,1,0,0,1,0,1,1,0,0,1,0]=>11
[1,1,0,0,1,0,1,1,0,1,0,0]=>8
[1,1,0,0,1,0,1,1,1,0,0,0]=>6
[1,1,0,0,1,1,0,0,1,0,1,0]=>12
[1,1,0,0,1,1,0,0,1,1,0,0]=>9
[1,1,0,0,1,1,0,1,0,0,1,0]=>10
[1,1,0,0,1,1,0,1,0,1,0,0]=>8
[1,1,0,0,1,1,0,1,1,0,0,0]=>5
[1,1,0,0,1,1,1,0,0,0,1,0]=>7
[1,1,0,0,1,1,1,0,0,1,0,0]=>6
[1,1,0,0,1,1,1,0,1,0,0,0]=>4
[1,1,0,0,1,1,1,1,0,0,0,0]=>3
[1,1,0,1,0,0,1,0,1,0,1,0]=>13
[1,1,0,1,0,0,1,0,1,1,0,0]=>10
[1,1,0,1,0,0,1,1,0,0,1,0]=>11
[1,1,0,1,0,0,1,1,0,1,0,0]=>9
[1,1,0,1,0,0,1,1,1,0,0,0]=>6
[1,1,0,1,0,1,0,0,1,0,1,0]=>12
[1,1,0,1,0,1,0,0,1,1,0,0]=>10
[1,1,0,1,0,1,0,1,0,0,1,0]=>11
[1,1,0,1,0,1,0,1,0,1,0,0]=>10
[1,1,0,1,0,1,0,1,1,0,0,0]=>6
[1,1,0,1,0,1,1,0,0,0,1,0]=>7
[1,1,0,1,0,1,1,0,0,1,0,0]=>7
[1,1,0,1,0,1,1,0,1,0,0,0]=>4
[1,1,0,1,0,1,1,1,0,0,0,0]=>3
[1,1,0,1,1,0,0,0,1,0,1,0]=>8
[1,1,0,1,1,0,0,0,1,1,0,0]=>7
[1,1,0,1,1,0,0,1,0,0,1,0]=>8
[1,1,0,1,1,0,0,1,0,1,0,0]=>8
[1,1,0,1,1,0,0,1,1,0,0,0]=>5
[1,1,0,1,1,0,1,0,0,0,1,0]=>5
[1,1,0,1,1,0,1,0,0,1,0,0]=>6
[1,1,0,1,1,0,1,0,1,0,0,0]=>4
[1,1,0,1,1,0,1,1,0,0,0,0]=>2
[1,1,0,1,1,1,0,0,0,0,1,0]=>4
[1,1,0,1,1,1,0,0,0,1,0,0]=>4
[1,1,0,1,1,1,0,0,1,0,0,0]=>3
[1,1,0,1,1,1,0,1,0,0,0,0]=>2
[1,1,0,1,1,1,1,0,0,0,0,0]=>1
[1,1,1,0,0,0,1,0,1,0,1,0]=>9
[1,1,1,0,0,0,1,0,1,1,0,0]=>7
[1,1,1,0,0,0,1,1,0,0,1,0]=>8
[1,1,1,0,0,0,1,1,0,1,0,0]=>7
[1,1,1,0,0,0,1,1,1,0,0,0]=>5
[1,1,1,0,0,1,0,0,1,0,1,0]=>9
[1,1,1,0,0,1,0,0,1,1,0,0]=>8
[1,1,1,0,0,1,0,1,0,0,1,0]=>9
[1,1,1,0,0,1,0,1,0,1,0,0]=>9
[1,1,1,0,0,1,0,1,1,0,0,0]=>6
[1,1,1,0,0,1,1,0,0,0,1,0]=>6
[1,1,1,0,0,1,1,0,0,1,0,0]=>7
[1,1,1,0,0,1,1,0,1,0,0,0]=>5
[1,1,1,0,0,1,1,1,0,0,0,0]=>3
[1,1,1,0,1,0,0,0,1,0,1,0]=>6
[1,1,1,0,1,0,0,0,1,1,0,0]=>6
[1,1,1,0,1,0,0,1,0,0,1,0]=>7
[1,1,1,0,1,0,0,1,0,1,0,0]=>8
[1,1,1,0,1,0,0,1,1,0,0,0]=>6
[1,1,1,0,1,0,1,0,0,0,1,0]=>5
[1,1,1,0,1,0,1,0,0,1,0,0]=>7
[1,1,1,0,1,0,1,0,1,0,0,0]=>6
[1,1,1,0,1,0,1,1,0,0,0,0]=>3
[1,1,1,0,1,1,0,0,0,0,1,0]=>3
[1,1,1,0,1,1,0,0,0,1,0,0]=>4
[1,1,1,0,1,1,0,0,1,0,0,0]=>4
[1,1,1,0,1,1,0,1,0,0,0,0]=>2
[1,1,1,0,1,1,1,0,0,0,0,0]=>1
[1,1,1,1,0,0,0,0,1,0,1,0]=>5
[1,1,1,1,0,0,0,0,1,1,0,0]=>4
[1,1,1,1,0,0,0,1,0,0,1,0]=>5
[1,1,1,1,0,0,0,1,0,1,0,0]=>5
[1,1,1,1,0,0,0,1,1,0,0,0]=>4
[1,1,1,1,0,0,1,0,0,0,1,0]=>4
[1,1,1,1,0,0,1,0,0,1,0,0]=>5
[1,1,1,1,0,0,1,0,1,0,0,0]=>5
[1,1,1,1,0,0,1,1,0,0,0,0]=>3
[1,1,1,1,0,1,0,0,0,0,1,0]=>3
[1,1,1,1,0,1,0,0,0,1,0,0]=>3
[1,1,1,1,0,1,0,0,1,0,0,0]=>4
[1,1,1,1,0,1,0,1,0,0,0,0]=>3
[1,1,1,1,0,1,1,0,0,0,0,0]=>1
[1,1,1,1,1,0,0,0,0,0,1,0]=>2
[1,1,1,1,1,0,0,0,0,1,0,0]=>2
[1,1,1,1,1,0,0,0,1,0,0,0]=>2
[1,1,1,1,1,0,0,1,0,0,0,0]=>2
[1,1,1,1,1,0,1,0,0,0,0,0]=>1
[1,1,1,1,1,1,0,0,0,0,0,0]=>0
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Description
The dinv of a Dyck path.
Let $a=(a_1,\ldots,a_n)$ be the area sequence of a Dyck path $D$ (see St000012The area of a Dyck path.).
The dinv statistic of $D$ is
$$ \operatorname{dinv}(D) = \# \big\{ i < j : a_i-a_j \in \{ 0,1 \} \big\}.$$
Equivalently, $\operatorname{dinv}(D)$ is also equal to the number of boxes in the partition above $D$ whose arm length is one larger or equal to its leg length.
There is a recursive definition of the $(\operatorname{area},\operatorname{dinv})$ pair of statistics, see [2].
Let $a=(0,a_2,\ldots,a_r,0,a_{r+2},\ldots,a_n)$ be the area sequence of the Dyck path $D$ with $a_i > 0$ for $2\leq i\leq r$ (so that the path touches the diagonal for the first time after $r$ steps). Assume that $D$ has $v$ entries where $a_i=0$. Let $D'$ be the path with the area sequence $(0,a_{r+2},\ldots,a_n,a_2-1,a_3-1,\ldots,a_r-1)$, then the statistics are related by
$$(\operatorname{area}(D),\operatorname{dinv}(D)) = (\operatorname{area}(D')+r-1,\operatorname{dinv}(D')+v-1).$$
Let $a=(a_1,\ldots,a_n)$ be the area sequence of a Dyck path $D$ (see St000012The area of a Dyck path.).
The dinv statistic of $D$ is
$$ \operatorname{dinv}(D) = \# \big\{ i < j : a_i-a_j \in \{ 0,1 \} \big\}.$$
Equivalently, $\operatorname{dinv}(D)$ is also equal to the number of boxes in the partition above $D$ whose arm length is one larger or equal to its leg length.
There is a recursive definition of the $(\operatorname{area},\operatorname{dinv})$ pair of statistics, see [2].
Let $a=(0,a_2,\ldots,a_r,0,a_{r+2},\ldots,a_n)$ be the area sequence of the Dyck path $D$ with $a_i > 0$ for $2\leq i\leq r$ (so that the path touches the diagonal for the first time after $r$ steps). Assume that $D$ has $v$ entries where $a_i=0$. Let $D'$ be the path with the area sequence $(0,a_{r+2},\ldots,a_n,a_2-1,a_3-1,\ldots,a_r-1)$, then the statistics are related by
$$(\operatorname{area}(D),\operatorname{dinv}(D)) = (\operatorname{area}(D')+r-1,\operatorname{dinv}(D')+v-1).$$
References
[1] Haglund, J. The $q$,$t$-Catalan numbers and the space of diagonal harmonics MathSciNet:2371044
[2] Garsia, A. M., Xin, G., Zabrocki, M. Hall-Littlewood operators in the theory of parking functions and diagonal harmonics MathSciNet:2899952
[2] Garsia, A. M., Xin, G., Zabrocki, M. Hall-Littlewood operators in the theory of parking functions and diagonal harmonics MathSciNet:2899952
Code
def statistic(x):
return x.dinv()
Created
Sep 21, 2011 at 03:34 by Chris Berg
Updated
Jun 17, 2019 at 17:06 by Christian Stump
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