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Your data matches 272 different statistics following compositions of up to 3 maps.
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Matching statistic: St000157
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Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
=> 0
[1,2] => [1,2] => [[1,2]]
=> 0
[2,1] => [2,1] => [[1],[2]]
=> 1
[2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[3,1,2] => [3,1,2] => [[1,3],[2]]
=> 1
[3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 2
[3,1,2,4] => [3,1,4,2] => [[1,3],[2,4]]
=> 2
[3,1,4,2] => [3,1,4,2] => [[1,3],[2,4]]
=> 2
[3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[4,3,1,2] => [4,3,1,2] => [[1,4],[2],[3]]
=> 2
[4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[4,2,3,1,5] => [4,2,5,1,3] => [[1,3],[2,5],[4]]
=> 2
[4,2,5,1,3] => [4,2,5,1,3] => [[1,3],[2,5],[4]]
=> 2
[4,3,2,1,5] => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 3
[5,4,3,1,2] => [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> 3
[5,4,3,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 4
[5,3,1,2,4,6] => [5,3,1,6,4,2] => [[1,4],[2,5],[3,6]]
=> 4
[5,3,1,2,6,4] => [5,3,1,6,4,2] => [[1,4],[2,5],[3,6]]
=> 4
[5,3,1,4,2,6] => [5,3,1,6,4,2] => [[1,4],[2,5],[3,6]]
=> 4
[5,3,1,4,6,2] => [5,3,1,6,4,2] => [[1,4],[2,5],[3,6]]
=> 4
[5,3,1,6,2,4] => [5,3,1,6,4,2] => [[1,4],[2,5],[3,6]]
=> 4
[5,3,1,6,4,2] => [5,3,1,6,4,2] => [[1,4],[2,5],[3,6]]
=> 4
[5,4,3,2,1,6] => [5,4,3,2,1,6] => [[1,6],[2],[3],[4],[5]]
=> 4
[6,5,4,3,1,2] => [6,5,4,3,1,2] => [[1,6],[2],[3],[4],[5]]
=> 4
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6]]
=> 5
[6,4,2,3,5,1,7] => [6,4,2,7,5,1,3] => [[1,4],[2,5],[3,7],[6]]
=> 4
[6,4,2,3,7,1,5] => [6,4,2,7,5,1,3] => [[1,4],[2,5],[3,7],[6]]
=> 4
[6,4,2,5,3,1,7] => [6,4,2,7,5,1,3] => [[1,4],[2,5],[3,7],[6]]
=> 4
[6,4,2,5,7,1,3] => [6,4,2,7,5,1,3] => [[1,4],[2,5],[3,7],[6]]
=> 4
[6,4,2,7,3,1,5] => [6,4,2,7,5,1,3] => [[1,4],[2,5],[3,7],[6]]
=> 4
[6,4,2,7,5,1,3] => [6,4,2,7,5,1,3] => [[1,4],[2,5],[3,7],[6]]
=> 4
[6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [[1,7],[2],[3],[4],[5],[6]]
=> 5
[7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [[1,7],[2],[3],[4],[5],[6]]
=> 5
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7]]
=> 6
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000024
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(load all 4 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[2,1] => [2,1] => [2,1] => [1,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[3,1,2] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 2
[3,1,2,4] => [3,1,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[3,1,4,2] => [3,1,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[4,3,1,2] => [4,3,1,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[4,2,3,1,5] => [4,2,5,1,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2
[4,2,5,1,3] => [4,2,5,1,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 3
[5,4,3,1,2] => [5,4,3,1,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 4
[5,3,1,2,4,6] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[5,3,1,2,6,4] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[5,3,1,4,2,6] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[5,3,1,4,6,2] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[5,3,1,6,2,4] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[5,3,1,6,4,2] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[5,4,3,2,1,6] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 4
[6,5,4,3,1,2] => [6,5,4,3,1,2] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[6,4,2,3,5,1,7] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[6,4,2,3,7,1,5] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[6,4,2,5,3,1,7] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[6,4,2,5,7,1,3] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[6,4,2,7,3,1,5] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[6,4,2,7,5,1,3] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 5
[7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 6
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000053
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000053: 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] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[2,1] => [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 2
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 3
[4,2,3,1,5] => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[4,2,5,1,3] => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[4,3,2,1,5] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[5,4,3,1,2] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[5,4,3,2,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[5,3,1,2,4,6] => [5,3,1,6,4,2] => [[[.,.],.],[[[.,.],.],.]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 4
[5,3,1,2,6,4] => [5,3,1,6,4,2] => [[[.,.],.],[[[.,.],.],.]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 4
[5,3,1,4,2,6] => [5,3,1,6,4,2] => [[[.,.],.],[[[.,.],.],.]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 4
[5,3,1,4,6,2] => [5,3,1,6,4,2] => [[[.,.],.],[[[.,.],.],.]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 4
[5,3,1,6,2,4] => [5,3,1,6,4,2] => [[[.,.],.],[[[.,.],.],.]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 4
[5,3,1,6,4,2] => [5,3,1,6,4,2] => [[[.,.],.],[[[.,.],.],.]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 4
[5,4,3,2,1,6] => [5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[6,5,4,3,1,2] => [6,5,4,3,1,2] => [[[[[.,.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5
[6,4,2,3,5,1,7] => [6,4,2,7,5,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> 4
[6,4,2,3,7,1,5] => [6,4,2,7,5,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> 4
[6,4,2,5,3,1,7] => [6,4,2,7,5,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> 4
[6,4,2,5,7,1,3] => [6,4,2,7,5,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> 4
[6,4,2,7,3,1,5] => [6,4,2,7,5,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> 4
[6,4,2,7,5,1,3] => [6,4,2,7,5,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> 4
[6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [[[[[[.,.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [[[[[[.,.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
Description
The number of valleys of the Dyck path.
Matching statistic: St000171
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [1,2] => ([],2)
=> 0
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1
[3,1,2] => [3,1,2] => [1,3,2] => ([(1,2)],3)
=> 1
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1,2,4] => [3,1,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,1,4,2] => [3,1,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1,5] => [4,2,5,1,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[4,2,5,1,3] => [4,2,5,1,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,4,3,1,2] => [5,4,3,1,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[5,3,1,2,4,6] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[5,3,1,2,6,4] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[5,3,1,4,2,6] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[5,3,1,4,6,2] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[5,3,1,6,2,4] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[5,3,1,6,4,2] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[5,4,3,2,1,6] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[6,5,4,3,1,2] => [6,5,4,3,1,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(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)
=> 5
[6,4,2,3,5,1,7] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
[6,4,2,3,7,1,5] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
[6,4,2,5,3,1,7] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
[6,4,2,5,7,1,3] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
[6,4,2,7,3,1,5] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
[6,4,2,7,5,1,3] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
[6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
Description
The degree of the graph.
This is the maximal vertex degree of a graph.
Matching statistic: St000211
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => {{1}}
=> 0
[1,2] => [1,2] => [1,2] => {{1},{2}}
=> 0
[2,1] => [2,1] => [2,1] => {{1,2}}
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 1
[3,1,2] => [3,1,2] => [3,1,2] => {{1,3},{2}}
=> 1
[3,2,1] => [3,2,1] => [2,3,1] => {{1,2,3}}
=> 2
[3,1,2,4] => [3,1,4,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 2
[3,1,4,2] => [3,1,4,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 2
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 2
[4,3,1,2] => [4,3,1,2] => [3,1,4,2] => {{1,3,4},{2}}
=> 2
[4,3,2,1] => [4,3,2,1] => [2,3,4,1] => {{1,2,3,4}}
=> 3
[4,2,3,1,5] => [4,2,5,1,3] => [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 2
[4,2,5,1,3] => [4,2,5,1,3] => [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 2
[4,3,2,1,5] => [4,3,2,1,5] => [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 3
[5,4,3,1,2] => [5,4,3,1,2] => [3,1,4,5,2] => {{1,3,4,5},{2}}
=> 3
[5,4,3,2,1] => [5,4,3,2,1] => [2,3,4,5,1] => {{1,2,3,4,5}}
=> 4
[5,3,1,2,4,6] => [5,3,1,6,4,2] => [3,5,4,6,2,1] => {{1,3,4,6},{2,5}}
=> 4
[5,3,1,2,6,4] => [5,3,1,6,4,2] => [3,5,4,6,2,1] => {{1,3,4,6},{2,5}}
=> 4
[5,3,1,4,2,6] => [5,3,1,6,4,2] => [3,5,4,6,2,1] => {{1,3,4,6},{2,5}}
=> 4
[5,3,1,4,6,2] => [5,3,1,6,4,2] => [3,5,4,6,2,1] => {{1,3,4,6},{2,5}}
=> 4
[5,3,1,6,2,4] => [5,3,1,6,4,2] => [3,5,4,6,2,1] => {{1,3,4,6},{2,5}}
=> 4
[5,3,1,6,4,2] => [5,3,1,6,4,2] => [3,5,4,6,2,1] => {{1,3,4,6},{2,5}}
=> 4
[5,4,3,2,1,6] => [5,4,3,2,1,6] => [2,3,4,5,1,6] => {{1,2,3,4,5},{6}}
=> 4
[6,5,4,3,1,2] => [6,5,4,3,1,2] => [3,1,4,5,6,2] => {{1,3,4,5,6},{2}}
=> 4
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [2,3,4,5,6,1] => {{1,2,3,4,5,6}}
=> 5
[6,4,2,3,5,1,7] => [6,4,2,7,5,1,3] => [5,4,2,7,6,1,3] => {{1,5,6},{2,4,7},{3}}
=> 4
[6,4,2,3,7,1,5] => [6,4,2,7,5,1,3] => [5,4,2,7,6,1,3] => {{1,5,6},{2,4,7},{3}}
=> 4
[6,4,2,5,3,1,7] => [6,4,2,7,5,1,3] => [5,4,2,7,6,1,3] => {{1,5,6},{2,4,7},{3}}
=> 4
[6,4,2,5,7,1,3] => [6,4,2,7,5,1,3] => [5,4,2,7,6,1,3] => {{1,5,6},{2,4,7},{3}}
=> 4
[6,4,2,7,3,1,5] => [6,4,2,7,5,1,3] => [5,4,2,7,6,1,3] => {{1,5,6},{2,4,7},{3}}
=> 4
[6,4,2,7,5,1,3] => [6,4,2,7,5,1,3] => [5,4,2,7,6,1,3] => {{1,5,6},{2,4,7},{3}}
=> 4
[6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [2,3,4,5,6,1,7] => {{1,2,3,4,5,6},{7}}
=> 5
[7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [3,1,4,5,6,7,2] => {{1,3,4,5,6,7},{2}}
=> 5
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => {{1,2,3,4,5,6,7}}
=> 6
Description
The rank of the set partition.
This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.
Matching statistic: St000245
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00329: Permutations —Tanimoto⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [2,1] => [1,2] => 1
[2,1,3] => [2,1,3] => [1,3,2] => [3,1,2] => 1
[3,1,2] => [3,1,2] => [2,3,1] => [2,1,3] => 1
[3,2,1] => [3,2,1] => [3,2,1] => [1,2,3] => 2
[3,1,2,4] => [3,1,4,2] => [4,2,1,3] => [1,3,4,2] => 2
[3,1,4,2] => [3,1,4,2] => [4,2,1,3] => [1,3,4,2] => 2
[3,2,1,4] => [3,2,1,4] => [1,4,3,2] => [4,1,2,3] => 2
[4,3,1,2] => [4,3,1,2] => [4,2,3,1] => [1,3,2,4] => 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 3
[4,2,3,1,5] => [4,2,5,1,3] => [5,3,1,2,4] => [1,3,5,4,2] => 2
[4,2,5,1,3] => [4,2,5,1,3] => [5,3,1,2,4] => [1,3,5,4,2] => 2
[4,3,2,1,5] => [4,3,2,1,5] => [1,5,4,3,2] => [5,1,2,3,4] => 3
[5,4,3,1,2] => [5,4,3,1,2] => [5,4,2,3,1] => [1,2,4,3,5] => 3
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 4
[5,3,1,2,4,6] => [5,3,1,6,4,2] => [6,4,2,1,5,3] => [1,3,5,6,2,4] => 4
[5,3,1,2,6,4] => [5,3,1,6,4,2] => [6,4,2,1,5,3] => [1,3,5,6,2,4] => 4
[5,3,1,4,2,6] => [5,3,1,6,4,2] => [6,4,2,1,5,3] => [1,3,5,6,2,4] => 4
[5,3,1,4,6,2] => [5,3,1,6,4,2] => [6,4,2,1,5,3] => [1,3,5,6,2,4] => 4
[5,3,1,6,2,4] => [5,3,1,6,4,2] => [6,4,2,1,5,3] => [1,3,5,6,2,4] => 4
[5,3,1,6,4,2] => [5,3,1,6,4,2] => [6,4,2,1,5,3] => [1,3,5,6,2,4] => 4
[5,4,3,2,1,6] => [5,4,3,2,1,6] => [1,6,5,4,3,2] => [6,1,2,3,4,5] => 4
[6,5,4,3,1,2] => [6,5,4,3,1,2] => [6,5,4,2,3,1] => [1,2,3,5,4,6] => 4
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 5
[6,4,2,3,5,1,7] => [6,4,2,7,5,1,3] => [7,5,3,1,6,2,4] => [1,3,5,7,2,6,4] => 4
[6,4,2,3,7,1,5] => [6,4,2,7,5,1,3] => [7,5,3,1,6,2,4] => [1,3,5,7,2,6,4] => 4
[6,4,2,5,3,1,7] => [6,4,2,7,5,1,3] => [7,5,3,1,6,2,4] => [1,3,5,7,2,6,4] => 4
[6,4,2,5,7,1,3] => [6,4,2,7,5,1,3] => [7,5,3,1,6,2,4] => [1,3,5,7,2,6,4] => 4
[6,4,2,7,3,1,5] => [6,4,2,7,5,1,3] => [7,5,3,1,6,2,4] => [1,3,5,7,2,6,4] => 4
[6,4,2,7,5,1,3] => [6,4,2,7,5,1,3] => [7,5,3,1,6,2,4] => [1,3,5,7,2,6,4] => 4
[6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [1,7,6,5,4,3,2] => [7,1,2,3,4,5,6] => 5
[7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [7,6,5,4,2,3,1] => [1,2,3,4,6,5,7] => 5
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 6
Description
The number of ascents of a permutation.
Matching statistic: St000272
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [1,2] => ([],2)
=> 0
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1
[3,1,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 1
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1,2,4] => [1,3,2,4] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1,5] => [2,4,3,1,5] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[4,2,5,1,3] => [2,4,1,5,3] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[5,3,1,2,4,6] => [1,3,2,5,4,6] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[5,3,1,2,6,4] => [1,3,2,5,6,4] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[5,3,1,4,2,6] => [1,3,5,4,2,6] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[5,3,1,4,6,2] => [1,3,5,4,6,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[5,3,1,6,2,4] => [1,3,5,2,6,4] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[5,3,1,6,4,2] => [1,3,5,6,4,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[5,4,3,2,1,6] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[6,5,4,3,1,2] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(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)
=> 5
[6,4,2,3,5,1,7] => [2,4,3,6,5,1,7] => [2,7,6,5,4,1,3] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[6,4,2,3,7,1,5] => [2,4,3,6,1,7,5] => [2,7,6,5,1,4,3] => ([(0,3),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[6,4,2,5,3,1,7] => [2,4,6,5,3,1,7] => [2,7,6,5,4,1,3] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[6,4,2,5,7,1,3] => [2,4,6,5,1,7,3] => [2,7,6,5,1,4,3] => ([(0,3),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[6,4,2,7,3,1,5] => [2,4,6,3,1,7,5] => [2,7,6,5,1,4,3] => ([(0,3),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[6,4,2,7,5,1,3] => [2,4,6,1,7,5,3] => [2,7,6,1,5,4,3] => ([(0,1),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
Description
The treewidth of a graph.
A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.
Matching statistic: St000319
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 0
[1,2] => [1,2] => [1,2] => [1,1]
=> 0
[2,1] => [2,1] => [2,1] => [2]
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,1]
=> 1
[3,1,2] => [1,3,2] => [1,3,2] => [2,1]
=> 1
[3,2,1] => [3,2,1] => [3,2,1] => [3]
=> 2
[3,1,2,4] => [1,3,2,4] => [1,4,3,2] => [3,1]
=> 2
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => [3,1]
=> 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,1]
=> 2
[4,3,1,2] => [1,4,3,2] => [1,4,3,2] => [3,1]
=> 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4]
=> 3
[4,2,3,1,5] => [2,4,3,1,5] => [2,5,4,1,3] => [3,1,1]
=> 2
[4,2,5,1,3] => [2,4,1,5,3] => [2,5,1,4,3] => [3,1,1]
=> 2
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => [4,1]
=> 3
[5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => [4,1]
=> 3
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [5]
=> 4
[5,3,1,2,4,6] => [1,3,2,5,4,6] => [1,6,5,4,3,2] => [5,1]
=> 4
[5,3,1,2,6,4] => [1,3,2,5,6,4] => [1,6,5,4,3,2] => [5,1]
=> 4
[5,3,1,4,2,6] => [1,3,5,4,2,6] => [1,6,5,4,3,2] => [5,1]
=> 4
[5,3,1,4,6,2] => [1,3,5,4,6,2] => [1,6,5,4,3,2] => [5,1]
=> 4
[5,3,1,6,2,4] => [1,3,5,2,6,4] => [1,6,5,4,3,2] => [5,1]
=> 4
[5,3,1,6,4,2] => [1,3,5,6,4,2] => [1,6,5,4,3,2] => [5,1]
=> 4
[5,4,3,2,1,6] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => [5,1]
=> 4
[6,5,4,3,1,2] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [5,1]
=> 4
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [6]
=> 5
[6,4,2,3,5,1,7] => [2,4,3,6,5,1,7] => [2,7,6,5,4,1,3] => [5,1,1]
=> 4
[6,4,2,3,7,1,5] => [2,4,3,6,1,7,5] => [2,7,6,5,1,4,3] => [5,1,1]
=> 4
[6,4,2,5,3,1,7] => [2,4,6,5,3,1,7] => [2,7,6,5,4,1,3] => [5,1,1]
=> 4
[6,4,2,5,7,1,3] => [2,4,6,5,1,7,3] => [2,7,6,5,1,4,3] => [5,1,1]
=> 4
[6,4,2,7,3,1,5] => [2,4,6,3,1,7,5] => [2,7,6,5,1,4,3] => [5,1,1]
=> 4
[6,4,2,7,5,1,3] => [2,4,6,1,7,5,3] => [2,7,6,1,5,4,3] => [5,1,1]
=> 4
[6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [6,1]
=> 5
[7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [6,1]
=> 5
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7]
=> 6
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 0
[1,2] => [1,2] => [1,2] => [1,1]
=> 0
[2,1] => [2,1] => [2,1] => [2]
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,1]
=> 1
[3,1,2] => [1,3,2] => [1,3,2] => [2,1]
=> 1
[3,2,1] => [3,2,1] => [3,2,1] => [3]
=> 2
[3,1,2,4] => [1,3,2,4] => [1,4,3,2] => [3,1]
=> 2
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => [3,1]
=> 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,1]
=> 2
[4,3,1,2] => [1,4,3,2] => [1,4,3,2] => [3,1]
=> 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4]
=> 3
[4,2,3,1,5] => [2,4,3,1,5] => [2,5,4,1,3] => [3,1,1]
=> 2
[4,2,5,1,3] => [2,4,1,5,3] => [2,5,1,4,3] => [3,1,1]
=> 2
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => [4,1]
=> 3
[5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => [4,1]
=> 3
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [5]
=> 4
[5,3,1,2,4,6] => [1,3,2,5,4,6] => [1,6,5,4,3,2] => [5,1]
=> 4
[5,3,1,2,6,4] => [1,3,2,5,6,4] => [1,6,5,4,3,2] => [5,1]
=> 4
[5,3,1,4,2,6] => [1,3,5,4,2,6] => [1,6,5,4,3,2] => [5,1]
=> 4
[5,3,1,4,6,2] => [1,3,5,4,6,2] => [1,6,5,4,3,2] => [5,1]
=> 4
[5,3,1,6,2,4] => [1,3,5,2,6,4] => [1,6,5,4,3,2] => [5,1]
=> 4
[5,3,1,6,4,2] => [1,3,5,6,4,2] => [1,6,5,4,3,2] => [5,1]
=> 4
[5,4,3,2,1,6] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => [5,1]
=> 4
[6,5,4,3,1,2] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => [5,1]
=> 4
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [6]
=> 5
[6,4,2,3,5,1,7] => [2,4,3,6,5,1,7] => [2,7,6,5,4,1,3] => [5,1,1]
=> 4
[6,4,2,3,7,1,5] => [2,4,3,6,1,7,5] => [2,7,6,5,1,4,3] => [5,1,1]
=> 4
[6,4,2,5,3,1,7] => [2,4,6,5,3,1,7] => [2,7,6,5,4,1,3] => [5,1,1]
=> 4
[6,4,2,5,7,1,3] => [2,4,6,5,1,7,3] => [2,7,6,5,1,4,3] => [5,1,1]
=> 4
[6,4,2,7,3,1,5] => [2,4,6,3,1,7,5] => [2,7,6,5,1,4,3] => [5,1,1]
=> 4
[6,4,2,7,5,1,3] => [2,4,6,1,7,5,3] => [2,7,6,1,5,4,3] => [5,1,1]
=> 4
[6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [6,1]
=> 5
[7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [6,1]
=> 5
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7]
=> 6
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St000362
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [1,2] => ([],2)
=> 0
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1
[3,1,2] => [3,1,2] => [1,3,2] => ([(1,2)],3)
=> 1
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1,2,4] => [3,1,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,1,4,2] => [3,1,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1,5] => [4,2,5,1,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[4,2,5,1,3] => [4,2,5,1,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,4,3,1,2] => [5,4,3,1,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[5,3,1,2,4,6] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[5,3,1,2,6,4] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[5,3,1,4,2,6] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[5,3,1,4,6,2] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[5,3,1,6,2,4] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[5,3,1,6,4,2] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[5,4,3,2,1,6] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[6,5,4,3,1,2] => [6,5,4,3,1,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(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)
=> 5
[6,4,2,3,5,1,7] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
[6,4,2,3,7,1,5] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
[6,4,2,5,3,1,7] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
[6,4,2,5,7,1,3] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
[6,4,2,7,3,1,5] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
[6,4,2,7,5,1,3] => [6,4,2,7,5,1,3] => [1,6,7,4,5,2,3] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
[6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
Description
The size of a minimal vertex cover of a graph.
A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. Finding a minimal vertex cover is an NP-hard optimization problem.
The following 262 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000377The dinv defect of an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000662The staircase size of the code of a permutation. St001176The size of a partition minus its first part. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000058The order of a permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000507The number of ascents of a standard tableau. St000676The number of odd rises of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001250The number of parts of a partition that are not congruent 0 modulo 3. St001389The number of partitions of the same length below the given integer partition. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001883The mutual visibility number of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000392The length of the longest run of ones in a binary word. St001372The length of a longest cyclic run of ones of a binary word. St000306The bounce count of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000503The maximal difference between two elements in a common block. St000730The maximal arc length of a set partition. St000877The depth of the binary word interpreted as a path. St000326The position of the first one in a binary word after appending a 1 at the end. St000444The length of the maximal rise of a Dyck path. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000806The semiperimeter of the associated bargraph. St000702The number of weak deficiencies of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000956The maximal displacement of a permutation. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St000632The jump number of the poset. St000809The reduced reflection length of the permutation. St001644The dimension of a graph. St000308The height of the tree associated to a permutation. St000485The length of the longest cycle of a permutation. St000527The width of the poset. St000141The maximum drop size of a permutation. St000703The number of deficiencies of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000167The number of leaves of an ordered tree. St000451The length of the longest pattern of the form k 1 2. St000883The number of longest increasing subsequences of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000007The number of saliances of the permutation. St001497The position of the largest weak excedence of a permutation. St000209Maximum difference of elements in cycles. St001330The hat guessing number of a graph. St000021The number of descents of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000155The number of exceedances (also excedences) of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000168The number of internal nodes of an ordered tree. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000864The number of circled entries of the shifted recording tableau of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001427The number of descents of a signed permutation. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St000822The Hadwiger number of the graph. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001180Number of indecomposable injective modules with projective dimension at most 1. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000653The last descent of a permutation. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001812The biclique partition number of a graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000567The sum of the products of all pairs of parts. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001820The size of the image of the pop stack sorting operator. St001863The number of weak excedances of a signed permutation. St000144The pyramid weight of the Dyck path. St000863The length of the first row of the shifted shape of a permutation. St000477The weight of a partition according to Alladi. St000928The sum of the coefficients of the character polynomial of an integer partition. St001668The number of points of the poset minus the width of the poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000264The girth of a graph, which is not a tree. St001430The number of positive entries in a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001896The number of right descents of a signed permutations. St001875The number of simple modules with projective dimension at most 1. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001946The number of descents in a parking function. St001060The distinguishing index of a graph. St001769The reflection length of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001864The number of excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001626The number of maximal proper sublattices of a lattice. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000225Difference between largest and smallest parts in a partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001645The pebbling number of a connected graph. St001432The order dimension of the partition. St000455The second largest eigenvalue of a graph if it is integral. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000761The number of ascents in an integer composition. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000025The number of initial rises of a Dyck path. St000061The number of nodes on the left branch of a binary tree. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000815The number of semistandard Young tableaux of partition weight of given shape. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001498The normalised height of a Nakayama algebra with magnitude 1. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000929The constant term of the character polynomial of an integer partition. 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. St001877Number of indecomposable injective modules with projective dimension 2. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000456The monochromatic index of a connected graph. St000619The number of cyclic descents of a permutation. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001926Sparre Andersen's position of the maximum of a signed permutation. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000741The Colin de Verdière graph invariant. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001545The second Elser number of a connected graph. St000707The product of the factorials of the parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000509The diagonal index (content) of a partition. St000997The even-odd crank of an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000736The last entry in the first row of a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001948The number of augmented double ascents of a permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St000075The orbit size of a standard tableau under promotion.
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