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Your data matches 29 different statistics following compositions of up to 3 maps.
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Matching statistic: St000008
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 0
[[1,2]]
=> [2] => 0
[[1],[2]]
=> [1,1] => 1
[[1,2,3]]
=> [3] => 0
[[1,3],[2]]
=> [1,2] => 1
[[1,2],[3]]
=> [2,1] => 2
[[1],[2],[3]]
=> [1,1,1] => 3
[[1,2,3,4]]
=> [4] => 0
[[1,3,4],[2]]
=> [1,3] => 1
[[1,2,4],[3]]
=> [2,2] => 2
[[1,2,3],[4]]
=> [3,1] => 3
[[1,3],[2,4]]
=> [1,2,1] => 4
[[1,2],[3,4]]
=> [2,2] => 2
[[1,4],[2],[3]]
=> [1,1,2] => 3
[[1,3],[2],[4]]
=> [1,2,1] => 4
[[1,2],[3],[4]]
=> [2,1,1] => 5
[[1],[2],[3],[4]]
=> [1,1,1,1] => 6
[[1,2,3,4,5]]
=> [5] => 0
[[1,3,4,5],[2]]
=> [1,4] => 1
[[1,2,4,5],[3]]
=> [2,3] => 2
[[1,2,3,5],[4]]
=> [3,2] => 3
[[1,2,3,4],[5]]
=> [4,1] => 4
[[1,3,5],[2,4]]
=> [1,2,2] => 4
[[1,2,5],[3,4]]
=> [2,3] => 2
[[1,3,4],[2,5]]
=> [1,3,1] => 5
[[1,2,4],[3,5]]
=> [2,2,1] => 6
[[1,2,3],[4,5]]
=> [3,2] => 3
[[1,4,5],[2],[3]]
=> [1,1,3] => 3
[[1,3,5],[2],[4]]
=> [1,2,2] => 4
[[1,2,5],[3],[4]]
=> [2,1,2] => 5
[[1,3,4],[2],[5]]
=> [1,3,1] => 5
[[1,2,4],[3],[5]]
=> [2,2,1] => 6
[[1,2,3],[4],[5]]
=> [3,1,1] => 7
[[1,4],[2,5],[3]]
=> [1,1,2,1] => 7
[[1,3],[2,5],[4]]
=> [1,2,2] => 4
[[1,2],[3,5],[4]]
=> [2,1,2] => 5
[[1,3],[2,4],[5]]
=> [1,2,1,1] => 8
[[1,2],[3,4],[5]]
=> [2,2,1] => 6
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => 6
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => 7
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => 8
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => 9
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 10
[[1,2,3,4,5,6]]
=> [6] => 0
[[1,3,4,5,6],[2]]
=> [1,5] => 1
[[1,2,4,5,6],[3]]
=> [2,4] => 2
[[1,2,3,5,6],[4]]
=> [3,3] => 3
[[1,2,3,4,6],[5]]
=> [4,2] => 4
[[1,2,3,4,5],[6]]
=> [5,1] => 5
[[1,3,5,6],[2,4]]
=> [1,2,3] => 4
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
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
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%
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%
Values
[[1]]
=> [1] => [1,0]
=> 0
[[1,2]]
=> [2] => [1,1,0,0]
=> 0
[[1],[2]]
=> [1,1] => [1,0,1,0]
=> 1
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> 0
[[1,3],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> 1
[[1,2],[3]]
=> [2,1] => [1,1,0,0,1,0]
=> 2
[[1],[2],[3]]
=> [1,1,1] => [1,0,1,0,1,0]
=> 3
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
[[1,3,4],[2]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[[1,2,4],[3]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[1,2,3],[4]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[[1,3],[2,4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[[1,2],[3,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[1,4],[2],[3]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[[1,3],[2],[4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[[1,2],[3],[4]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 6
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[1,3,4,5],[2]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[[1,2,4,5],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[[1,2,3,5],[4]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[[1,2,3,4],[5]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[1,3,5],[2,4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,5],[3,4]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[[1,3,4],[2,5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 5
[[1,2,4],[3,5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[[1,2,3],[4,5]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[[1,4,5],[2],[3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
[[1,3,4],[2],[5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 5
[[1,2,4],[3],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 7
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 7
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 8
[[1,2],[3,4],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 7
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 8
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 9
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 10
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[[1,3,4,5,6],[2]]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2,4,5,6],[3]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[[1,2,3,5,6],[4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[[1,2,3,4,6],[5]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[[1,2,3,4,5],[6]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 4
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: St000081
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => ([],1)
=> 0
[[1,2]]
=> [2] => ([],2)
=> 0
[[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 1
[[1,2,3]]
=> [3] => ([],3)
=> 0
[[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> 1
[[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,3,4]]
=> [4] => ([],4)
=> 0
[[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> 1
[[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,3],[2,4]]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[1,3],[2],[4]]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[1,2],[3],[4]]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[[1,2,3,4,5]]
=> [5] => ([],5)
=> 0
[[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> 1
[[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[[1,3,5],[2,4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[[1,3,4],[2,5]]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[[1,3,5],[2],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[1,2,5],[3],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[[1,3,4],[2],[5]]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[[1,2,3],[4],[5]]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[[1,4],[2,5],[3]]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[1,2],[3,5],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[[1,3],[2,4],[5]]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 9
[[1],[2],[3],[4],[5]]
=> [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
[[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 0
[[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> 1
[[1,2,4,5,6],[3]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 2
[[1,2,3,5,6],[4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
[[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,2,3,4,5],[6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[[1,3,5,6],[2,4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 4
Description
The number of edges of a graph.
Matching statistic: St001161
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
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%
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%
Values
[[1]]
=> [1] => [1,0]
=> 0
[[1,2]]
=> [2] => [1,1,0,0]
=> 0
[[1],[2]]
=> [1,1] => [1,0,1,0]
=> 1
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> 0
[[1,3],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> 1
[[1,2],[3]]
=> [2,1] => [1,1,0,0,1,0]
=> 2
[[1],[2],[3]]
=> [1,1,1] => [1,0,1,0,1,0]
=> 3
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
[[1,3,4],[2]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[[1,2,4],[3]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[1,2,3],[4]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[[1,3],[2,4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[[1,2],[3,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[1,4],[2],[3]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[[1,3],[2],[4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[[1,2],[3],[4]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 6
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[1,3,4,5],[2]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[[1,2,4,5],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[[1,2,3,5],[4]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[[1,2,3,4],[5]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[1,3,5],[2,4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,5],[3,4]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[[1,3,4],[2,5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 5
[[1,2,4],[3,5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[[1,2,3],[4,5]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[[1,4,5],[2],[3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
[[1,3,4],[2],[5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 5
[[1,2,4],[3],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 7
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 7
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 8
[[1,2],[3,4],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 7
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 8
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 9
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 10
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[[1,3,4,5,6],[2]]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2,4,5,6],[3]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[[1,2,3,5,6],[4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[[1,2,3,4,6],[5]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[[1,2,3,4,5],[6]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 4
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: St000012
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1,0]
=> [1,0]
=> 0
[[1,2]]
=> [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[[1],[2]]
=> [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[[1,3],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[1,2],[3]]
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[1],[2],[3]]
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[1,3,4],[2]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[1,2,4],[3]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,2,3],[4]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[1,3],[2,4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[[1,2],[3,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,4],[2],[3]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[1,3],[2],[4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[[1,2],[3],[4]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 5
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 6
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[1,3,4,5],[2]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,2,4,5],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[[1,2,3,5],[4]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[[1,2,3,4],[5]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[[1,3,5],[2,4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[[1,2,5],[3,4]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[[1,3,4],[2,5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[[1,2,4],[3,5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 6
[[1,2,3],[4,5]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[[1,4,5],[2],[3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 5
[[1,3,4],[2],[5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[[1,2,4],[3],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 6
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 7
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 7
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 5
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 8
[[1,2],[3,4],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 6
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 7
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 8
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 9
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 10
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[[1,3,4,5,6],[2]]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,2,4,5,6],[3]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
[[1,2,3,5,6],[4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[[1,2,3,4,6],[5]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 4
[[1,2,3,4,5],[6]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 4
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: St000156
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000156: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000156: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1,0]
=> [1] => 0
[[1,2]]
=> [2] => [1,1,0,0]
=> [1,2] => 0
[[1],[2]]
=> [1,1] => [1,0,1,0]
=> [2,1] => 1
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[[1,3],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[[1,2],[3]]
=> [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 2
[[1],[2],[3]]
=> [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 3
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[[1,3,4],[2]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[[1,2,4],[3]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 2
[[1,2,3],[4]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 3
[[1,3],[2,4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 4
[[1,2],[3,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 2
[[1,4],[2],[3]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 3
[[1,3],[2],[4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 4
[[1,2],[3],[4]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 5
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 6
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[1,3,4,5],[2]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[[1,2,4,5],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 2
[[1,2,3,5],[4]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 3
[[1,2,3,4],[5]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 4
[[1,3,5],[2,4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 4
[[1,2,5],[3,4]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 2
[[1,3,4],[2,5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 5
[[1,2,4],[3,5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 6
[[1,2,3],[4,5]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 3
[[1,4,5],[2],[3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 3
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 4
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 5
[[1,3,4],[2],[5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 5
[[1,2,4],[3],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 6
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 7
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 7
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 4
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 5
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 8
[[1,2],[3,4],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 6
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 6
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 7
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 8
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 9
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 10
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[[1,3,4,5,6],[2]]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 1
[[1,2,4,5,6],[3]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => 2
[[1,2,3,5,6],[4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => 3
[[1,2,3,4,6],[5]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => 4
[[1,2,3,4,5],[6]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => 5
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => 4
Description
The Denert index of a permutation.
It is defined as
$$
\begin{align*}
den(\sigma) &= \#\{ 1\leq l < k \leq n : \sigma(k) < \sigma(l) \leq k \} \\
&+ \#\{ 1\leq l < k \leq n : \sigma(l) \leq k < \sigma(k) \} \\
&+ \#\{ 1\leq l < k \leq n : k < \sigma(k) < \sigma(l) \}
\end{align*}
$$
where $n$ is the size of $\sigma$. It was studied by Denert in [1], and it was shown by Foata and Zeilberger in [2] that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $exc$ is the number of weak exceedences, see [[St000155]].
Matching statistic: St000305
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000305: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000305: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1,0]
=> [1] => 0
[[1,2]]
=> [2] => [1,1,0,0]
=> [1,2] => 0
[[1],[2]]
=> [1,1] => [1,0,1,0]
=> [2,1] => 1
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[[1,3],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 1
[[1,2],[3]]
=> [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 2
[[1],[2],[3]]
=> [1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[[1,3,4],[2]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[[1,2,4],[3]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[[1,2,3],[4]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[[1,3],[2,4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 4
[[1,2],[3,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[[1,4],[2],[3]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 3
[[1,3],[2],[4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 4
[[1,2],[3],[4]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 6
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[1,3,4,5],[2]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[[1,2,4,5],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2
[[1,2,3,5],[4]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 3
[[1,2,3,4],[5]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[[1,3,5],[2,4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 4
[[1,2,5],[3,4]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2
[[1,3,4],[2,5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 5
[[1,2,4],[3,5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 6
[[1,2,3],[4,5]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 3
[[1,4,5],[2],[3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 3
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 4
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 5
[[1,3,4],[2],[5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 5
[[1,2,4],[3],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 6
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 7
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 7
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 4
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 5
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 8
[[1,2],[3,4],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 6
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 6
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 7
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 8
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 9
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 10
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[[1,3,4,5,6],[2]]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1
[[1,2,4,5,6],[3]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => 2
[[1,2,3,5,6],[4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => 3
[[1,2,3,4,6],[5]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => 4
[[1,2,3,4,5],[6]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => 4
Description
The inverse major index of a permutation.
This is the major index [[St000004]] of the inverse permutation [[Mp00066]].
Matching statistic: St000947
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1,0]
=> [1,1,0,0]
=> 0
[[1,2]]
=> [2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[[1],[2]]
=> [1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,3],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[[1,2],[3]]
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[1],[2],[3]]
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[[1,3,4],[2]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[[1,2,4],[3]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[[1,2,3],[4]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[[1,3],[2,4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[[1,2],[3,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[[1,4],[2],[3]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[[1,3],[2],[4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[[1,2],[3],[4]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 6
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[[1,3,4,5],[2]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2,4,5],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2
[[1,2,3,5],[4]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3
[[1,2,3,4],[5]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 4
[[1,3,5],[2,4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 4
[[1,2,5],[3,4]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2
[[1,3,4],[2,5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 5
[[1,2,4],[3,5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 6
[[1,2,3],[4,5]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3
[[1,4,5],[2],[3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 4
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 5
[[1,3,4],[2],[5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 5
[[1,2,4],[3],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 6
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 7
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 7
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 4
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 5
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 8
[[1,2],[3,4],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 6
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 6
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 7
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 8
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 9
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 10
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
[[1,3,4,5,6],[2]]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[[1,2,4,5,6],[3]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1,2,3,5,6],[4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> 3
[[1,2,3,4,6],[5]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> 4
[[1,2,3,4,5],[6]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 5
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> 4
Description
The major index east 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 east count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$.
Matching statistic: St001295
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001295: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001295: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1,0]
=> [1,0]
=> 0
[[1,2]]
=> [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[[1],[2]]
=> [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[[1,3],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[1,2],[3]]
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[1],[2],[3]]
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[1,3,4],[2]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[1,2,4],[3]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,2,3],[4]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[1,3],[2,4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[[1,2],[3,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,4],[2],[3]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[1,3],[2],[4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[[1,2],[3],[4]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 5
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 6
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[1,3,4,5],[2]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,2,4,5],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[[1,2,3,5],[4]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[[1,2,3,4],[5]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[[1,3,5],[2,4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[[1,2,5],[3,4]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[[1,3,4],[2,5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[[1,2,4],[3,5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 6
[[1,2,3],[4,5]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[[1,4,5],[2],[3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 5
[[1,3,4],[2],[5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[[1,2,4],[3],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 6
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 7
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 7
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 5
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 8
[[1,2],[3,4],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 6
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 7
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 8
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 9
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 10
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[[1,3,4,5,6],[2]]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[[1,2,4,5,6],[3]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
[[1,2,3,5,6],[4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[[1,2,3,4,6],[5]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 4
[[1,2,3,4,5],[6]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 4
Description
Gives the vector space dimension of the homomorphism space between J^2 and J^2.
Matching statistic: St001671
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001671: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001671: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1,0]
=> [1] => 0
[[1,2]]
=> [2] => [1,1,0,0]
=> [1,2] => 0
[[1],[2]]
=> [1,1] => [1,0,1,0]
=> [2,1] => 1
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[[1,3],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[[1,2],[3]]
=> [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 2
[[1],[2],[3]]
=> [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 3
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[[1,3,4],[2]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[[1,2,4],[3]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 2
[[1,2,3],[4]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 3
[[1,3],[2,4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 4
[[1,2],[3,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 2
[[1,4],[2],[3]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 3
[[1,3],[2],[4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 4
[[1,2],[3],[4]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 5
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 6
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[1,3,4,5],[2]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[[1,2,4,5],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 2
[[1,2,3,5],[4]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 3
[[1,2,3,4],[5]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 4
[[1,3,5],[2,4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 4
[[1,2,5],[3,4]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 2
[[1,3,4],[2,5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 5
[[1,2,4],[3,5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 6
[[1,2,3],[4,5]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 3
[[1,4,5],[2],[3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 3
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 4
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 5
[[1,3,4],[2],[5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 5
[[1,2,4],[3],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 6
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 7
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 7
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 4
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 5
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 8
[[1,2],[3,4],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 6
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 6
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 7
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 8
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 9
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 10
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[[1,3,4,5,6],[2]]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 1
[[1,2,4,5,6],[3]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => 2
[[1,2,3,5,6],[4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => 3
[[1,2,3,4,6],[5]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => 4
[[1,2,3,4,5],[6]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => 5
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => 4
Description
Haglund's hag of a permutation.
Let $edif$ be the sum of the differences of exceedence tops and bottoms, let $\pi_E$ the subsequence of exceedence tops and let $\pi_N$ be the subsequence of non-exceedence tops. Finally, let $L$ be the number of pairs of indices $k < i$ such that $\pi_k \leq i < \pi_i$.
Then $hag(\pi) = edif + inv(\pi_E) - inv(\pi_N) + L$, where $inv$ denotes the number of inversions of a word.
The following 19 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000492The rob 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. St000796The stat' of a permutation. St000798The makl of a permutation. St000833The comajor index of a permutation. St001311The cyclomatic number of a graph. St000456The monochromatic index of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000101The cocharge of a semistandard tableau. St000102The charge of a semistandard tableau. St000450The number of edges minus the number of vertices plus 2 of a graph. St000454The largest eigenvalue of a graph if it is integral. St001645The pebbling number of a connected graph. St001209The pmaj statistic of a parking function. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001877Number of indecomposable injective modules with projective dimension 2. St001330The hat guessing number of a graph.
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