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Your data matches 129 different statistics following compositions of up to 3 maps.
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Matching statistic: St000008
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 1
[2] => 0
[1,1,1] => 3
[1,2] => 1
[2,1] => 2
[3] => 0
[1,1,1,1] => 6
[1,1,2] => 3
[1,2,1] => 4
[1,3] => 1
[2,1,1] => 5
[2,2] => 2
[3,1] => 3
[4] => 0
[1,1,1,1,1] => 10
[1,1,1,2] => 6
[1,1,2,1] => 7
[1,1,3] => 3
[1,2,1,1] => 8
[1,2,2] => 4
[1,3,1] => 5
[1,4] => 1
[2,1,1,1] => 9
[2,1,2] => 5
[2,2,1] => 6
[2,3] => 2
[3,1,1] => 7
[3,2] => 3
[4,1] => 4
[5] => 0
[1,1,1,1,1,1] => 15
[1,1,1,1,2] => 10
[1,1,1,2,1] => 11
[1,1,1,3] => 6
[1,1,2,1,1] => 12
[1,1,2,2] => 7
[1,1,3,1] => 8
[1,1,4] => 3
[1,2,1,1,1] => 13
[1,2,1,2] => 8
[1,2,2,1] => 9
[1,2,3] => 4
[1,3,1,1] => 10
[1,3,2] => 5
[1,4,1] => 6
[1,5] => 1
[2,1,1,1,1] => 14
[2,1,1,2] => 9
[2,1,2,1] => 10
Description
The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000005
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000005: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000005: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 0
[1,1] => [1,0,1,0]
=> 1
[2] => [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2] => [1,0,1,1,0,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> 2
[3] => [1,1,1,0,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 6
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 10
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 7
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 8
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 9
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 7
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 15
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 10
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 11
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 6
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 12
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 7
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 8
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 13
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 8
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 9
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 4
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 10
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 14
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 9
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 10
Description
The bounce statistic of a Dyck path.
The '''bounce path''' $D'$ of a Dyck path $D$ is the Dyck path obtained from $D$ by starting at the end point $(2n,0)$, traveling north-west until hitting $D$, then bouncing back south-west to the $x$-axis, and repeating this procedure until finally reaching the point $(0,0)$.
The points where $D'$ touches the $x$-axis are called '''bounce points''', and a bounce path is uniquely determined by its bounce points.
This statistic is given by the sum of all $i$ for which the bounce path $D'$ of $D$ touches the $x$-axis at $(2i,0)$.
In particular, the bounce statistics of $D$ and $D'$ coincide.
Matching statistic: St000006
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000006: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000006: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 0
[1,1] => [1,0,1,0]
=> 1
[2] => [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2] => [1,0,1,1,0,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> 2
[3] => [1,1,1,0,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 6
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[2,2] => [1,1,0,0,1,1,0,0]
=> 3
[3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 10
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 7
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 8
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 5
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 9
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 6
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 7
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 5
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 15
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 10
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 11
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 6
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 12
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 8
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 7
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 13
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 9
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 10
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 8
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 6
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 4
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 14
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 10
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 11
Description
The dinv of a Dyck path.
Let $a=(a_1,\ldots,a_n)$ be the area sequence of a Dyck path $D$ (see [[St000012]]).
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
$$(\operatorname{area}(D),\operatorname{dinv}(D)) = (\operatorname{area}(D')+r-1,\operatorname{dinv}(D')+v-1).$$
Matching statistic: St000081
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 0
[1,1] => ([(0,1)],2)
=> 1
[2] => ([],2)
=> 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2] => ([(1,2)],3)
=> 1
[2,1] => ([(0,2),(1,2)],3)
=> 2
[3] => ([],3)
=> 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,3] => ([(2,3)],4)
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4] => ([],4)
=> 0
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4] => ([(3,4)],5)
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 9
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[2,3] => ([(2,4),(3,4)],5)
=> 2
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[5] => ([],5)
=> 0
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 15
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 10
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 11
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 12
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 13
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 9
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 10
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,5] => ([(4,5)],6)
=> 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 14
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 9
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 10
Description
The number of edges of a graph.
Matching statistic: St001161
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001161: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 0
[1,1] => [1,0,1,0]
=> 1
[2] => [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2] => [1,0,1,1,0,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> 2
[3] => [1,1,1,0,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 6
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 10
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 7
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 8
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 9
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 7
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 15
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 10
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 11
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 6
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 12
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 7
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 8
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 13
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 8
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 9
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 4
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 10
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 14
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 9
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 10
Description
The major index north count of a Dyck path.
The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$.
The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]].
The '''major index north count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = N\}$.
Matching statistic: St000004
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0
[1,1] => [1,0,1,0]
=> [1,2] => 0
[2] => [1,1,0,0]
=> [2,1] => 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2
[2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[3] => [1,1,1,0,0,0]
=> [3,2,1] => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 5
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 6
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 7
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 6
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 9
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 8
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 7
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 6
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 10
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => 9
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => 8
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => 7
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => 12
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 7
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => 6
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => 11
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 5
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 10
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 9
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 14
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 6
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 5
Description
The major index of a permutation.
This is the sum of the positions of its descents,
$$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$$
Its generating function is $[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for $[k]_q = 1 + q + q^2 + \dots q^{k-1}$.
A statistic equidistributed with the major index is called '''Mahonian statistic'''.
Matching statistic: St000012
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 0
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 5
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 6
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 7
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 6
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 9
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 8
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 7
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 10
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 9
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 8
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 7
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 12
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 7
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 6
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 11
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 5
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 10
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 9
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 14
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 6
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 5
Description
The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
Matching statistic: St000133
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0
[1,1] => [1,0,1,0]
=> [1,2] => 1
[2] => [1,1,0,0]
=> [2,1] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2
[2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[3] => [1,1,1,0,0,0]
=> [3,2,1] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 6
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 5
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 4
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 10
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 9
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 8
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 7
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 7
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 6
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 4
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 6
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 15
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 14
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => 13
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => 12
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 12
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => 11
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => 10
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => 9
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => 11
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 10
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => 9
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => 8
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 8
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 7
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 6
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 5
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 10
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 9
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 8
Description
The "bounce" of a permutation.
Matching statistic: St000147
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> []
=> 0
[1,1] => ([(0,1)],2)
=> [1]
=> 1
[2] => ([],2)
=> []
=> 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
[1,2] => ([(1,2)],3)
=> [1]
=> 1
[2,1] => ([(0,2),(1,2)],3)
=> [2]
=> 2
[3] => ([],3)
=> []
=> 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 6
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[1,3] => ([(2,3)],4)
=> [1]
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 5
[2,2] => ([(1,3),(2,3)],4)
=> [2]
=> 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> [3]
=> 3
[4] => ([],4)
=> []
=> 0
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> 10
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 6
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 7
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 3
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> 8
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 4
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 5
[1,4] => ([(3,4)],5)
=> [1]
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> 9
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 5
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 6
[2,3] => ([(2,4),(3,4)],5)
=> [2]
=> 2
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 7
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 4
[5] => ([],5)
=> []
=> 0
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [15]
=> 15
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> 10
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> 11
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 6
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [12]
=> 12
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> 7
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> 8
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> [3]
=> 3
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [13]
=> 13
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> 8
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> 9
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> 4
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> 10
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> 5
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 6
[1,5] => ([(4,5)],6)
=> [1]
=> 1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [14]
=> 14
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> 9
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> 10
Description
The largest part of an integer partition.
Matching statistic: St000154
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000154: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000154: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0
[1,1] => [1,0,1,0]
=> [1,2] => 0
[2] => [1,1,0,0]
=> [2,1] => 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2
[2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[3] => [1,1,1,0,0,0]
=> [3,2,1] => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 5
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 6
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 7
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 6
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 9
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 8
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 7
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 6
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 10
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => 9
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => 8
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => 7
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => 12
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 7
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => 6
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => 11
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 5
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 10
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 9
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 14
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 6
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 5
Description
The sum of the descent bottoms of a permutation.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_{i+1}.$$
For the descent tops, see [[St000111]].
The following 119 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000156The Denert index of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000304The load of a permutation. St000305The inverse major index of a permutation. St000378The diagonal inversion number of an integer partition. St000446The disorder of a permutation. St000459The hook length of the base cell of a partition. St000947The major index east count of a Dyck path. St000992The alternating sum of the parts of an integer partition. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001671Haglund's hag of a permutation. St001759The Rajchgot index of a permutation. St001400The total number of Littlewood-Richardson tableaux of given shape. St001814The number of partitions interlacing the given partition. St000009The charge of a standard tableau. St000010The length of the partition. St000018The number of inversions of a permutation. St000041The number of nestings of a perfect matching. St000057The Shynar inversion number of a standard tableau. St000067The inversion number of the alternating sign matrix. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000148The number of odd parts of a partition. St000169The cocharge of a standard tableau. St000246The number of non-inversions of a permutation. St000330The (standard) major index of a standard tableau. St000391The sum of the positions of the ones in a binary word. St000493The los statistic of a set partition. St000548The number of different non-empty partial sums of an integer partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000692Babson and Steingrímsson's statistic of a permutation. St000867The sum of the hook lengths in the first row of an integer partition. St000921The number of internal inversions of a binary word. St001094The depth index of a set partition. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001956The comajor index for set-valued two-row standard Young tableaux. St000946The sum of the skew hook positions in a Dyck path. St000448The number of pairs of vertices of a graph with distance 2. St000472The sum of the ascent bottoms of a permutation. St000492The rob statistic of a set partition. St000498The lcs statistic of a set partition. St000499The rcb statistic of a set partition. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000794The mak of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000833The comajor index of a permutation. St001341The number of edges in the center of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St000490The intertwining number of a set partition. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000795The mad of a permutation. St000384The maximal part of the shifted composition of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St001055The Grundy value for the game of removing cells of a row in an integer partition. St000063The number of linear extensions of a certain poset defined for an integer partition. St000532The total number of rook placements on a Ferrers board. St000160The multiplicity of the smallest part of a partition. St000475The number of parts equal to 1 in a partition. St001127The sum of the squares of the parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001311The cyclomatic number of a graph. St001360The number of covering relations in Young's lattice below a partition. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St000357The number of occurrences of the pattern 12-3. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001933The largest multiplicity of a part in an integer partition. St000228The size of a partition. St000108The number of partitions contained in the given partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000667The greatest common divisor of the parts of the partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001571The Cartan determinant of the integer partition. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000993The multiplicity of the largest part of an integer partition. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000719The number of alignments in a perfect matching. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001746The coalition number of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001725The harmonious chromatic number of a graph. St001645The pebbling number of a connected graph. St000456The monochromatic index of a connected graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000360The number of occurrences of the pattern 32-1. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001622The number of join-irreducible elements of a lattice. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001621The number of atoms of a lattice. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000565The major index of a set partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001415The length of the longest palindromic prefix of a binary word. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph.
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