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Your data matches 43 different statistics following compositions of up to 3 maps.
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Matching statistic: St000707
St000707: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 2
[1,1]
=> 1
[3]
=> 6
[2,1]
=> 2
[1,1,1]
=> 1
[4]
=> 24
[3,1]
=> 6
[2,2]
=> 4
[2,1,1]
=> 2
[1,1,1,1]
=> 1
[5]
=> 120
[4,1]
=> 24
[3,2]
=> 12
[3,1,1]
=> 6
[2,2,1]
=> 4
[2,1,1,1]
=> 2
[1,1,1,1,1]
=> 1
[6]
=> 720
[5,1]
=> 120
[4,2]
=> 48
[4,1,1]
=> 24
[3,3]
=> 36
[3,2,1]
=> 12
[3,1,1,1]
=> 6
[2,2,2]
=> 8
[2,2,1,1]
=> 4
[2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 1
Description
The product of the factorials of the parts.
Matching statistic: St001346
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(load all 4 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001346: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001346: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [[1,2]]
=> [1,2] => 2
[1,1]
=> [[1],[2]]
=> [2,1] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => 6
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 24
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 6
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 4
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 120
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 24
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 12
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 6
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 4
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 720
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 120
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 48
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 24
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 36
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 12
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 6
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 8
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 4
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 1
Description
The number of parking functions that give the same permutation.
A '''parking function''' $(a_1,\dots,a_n)$ is a list of preferred parking spots of $n$ cars entering a one-way street. Once the cars have parked, the order of the cars gives a permutation of $\{1,\dots,n\}$. This statistic records the number of parking functions that yield the same permutation of cars.
Matching statistic: St000040
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000040: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000040: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [[1,2]]
=> [1,2] => [2,1] => 2
[1,1]
=> [[1],[2]]
=> [2,1] => [1,2] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [3,2,1] => 6
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [4,3,2,1] => 24
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,4,3,2] => 6
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => 4
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,2,4,3] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,4,3,2,1] => 120
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,5,4,3,2] => 24
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [2,1,5,4,3] => 12
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 6
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,2,5,4] => 4
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,3,5,4] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6,5,4,3,2,1] => 720
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => 120
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => 48
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [1,2,6,5,4,3] => 24
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => 36
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [1,3,2,6,5,4] => 12
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,2,3,6,5,4] => 6
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [2,1,4,3,6,5] => 8
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [1,2,4,3,6,5] => 4
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,2,3,4,6,5] => 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 1
Description
The number of regions of the inversion arrangement of a permutation.
The inversion arrangement $\mathcal{A}_w$ consists of the hyperplanes $x_i-x_j=0$ such that $(i,j)$ is an inversion of $w$.
Postnikov [4] conjectured that the number of regions in $\mathcal{A}_w$ equals the number of permutations in the interval $[id,w]$ in the strong Bruhat order if and only if $w$ avoids $4231$, $35142$, $42513$, $351624$. This conjecture was proved by Hultman-Linusson-Shareshian-Sjöstrand [1].
Oh-Postnikov-Yoo [3] showed that the number of regions of $\mathcal{A}_w$ is $|\chi_{G_w}(-1)|$ where $\chi_{G_w}$ is the chromatic polynomial of the inversion graph $G_w$. This is the graph with vertices ${1,2,\ldots,n}$ and edges $(i,j)$ for $i\lneq j$ $w_i\gneq w_j$.
For a permutation $w=w_1\cdots w_n$, Lewis-Morales [2] and Hultman (see appendix in [2]) showed that this number equals the number of placements of $n$ non-attacking rooks on the south-west Rothe diagram of $w$.
Matching statistic: St000109
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000109: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000109: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [[1,2]]
=> [1,2] => [2,1] => 2
[1,1]
=> [[1],[2]]
=> [2,1] => [1,2] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [3,2,1] => 6
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [4,3,2,1] => 24
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,4,3,2] => 6
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => 4
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,2,4,3] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,4,3,2,1] => 120
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,5,4,3,2] => 24
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [2,1,5,4,3] => 12
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 6
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,2,5,4] => 4
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,3,5,4] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6,5,4,3,2,1] => 720
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => 120
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => 48
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [1,2,6,5,4,3] => 24
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => 36
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [1,3,2,6,5,4] => 12
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,2,3,6,5,4] => 6
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [2,1,4,3,6,5] => 8
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [1,2,4,3,6,5] => 4
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,2,3,4,6,5] => 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 1
Description
The number of elements less than or equal to the given element in Bruhat order.
Matching statistic: St000110
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [[1,2]]
=> [1,2] => [2,1] => 2
[1,1]
=> [[1],[2]]
=> [2,1] => [1,2] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [3,2,1] => 6
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [4,3,2,1] => 24
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,4,3,2] => 6
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => 4
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,2,4,3] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,4,3,2,1] => 120
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,5,4,3,2] => 24
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [2,1,5,4,3] => 12
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 6
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,2,5,4] => 4
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,3,5,4] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6,5,4,3,2,1] => 720
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => 120
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => 48
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [1,2,6,5,4,3] => 24
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => 36
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [1,3,2,6,5,4] => 12
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,2,3,6,5,4] => 6
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [2,1,4,3,6,5] => 8
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [1,2,4,3,6,5] => 4
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,2,3,4,6,5] => 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 1
Description
The number of permutations less than or equal to a permutation in left weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St001813
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001813: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001813: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [[1,2]]
=> [1,2] => ([(0,1)],2)
=> 2
[1,1]
=> [[1],[2]]
=> [2,1] => ([],2)
=> 1
[3]
=> [[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 6
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => ([],3)
=> 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 24
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 6
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 4
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(2,3)],4)
=> 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => ([],4)
=> 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 120
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 24
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 12
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 6
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 4
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([],5)
=> 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 720
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> 120
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> 48
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6)
=> 24
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> 36
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6)
=> 12
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
=> 6
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
=> 8
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ([(2,5),(3,4)],6)
=> 4
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ([(4,5)],6)
=> 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([],6)
=> 1
Description
The product of the sizes of the principal order filters in a poset.
Matching statistic: St000033
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000033: Permutations ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 64%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000033: Permutations ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 64%
Values
[2]
=> [[1,2]]
=> [1,2] => 2
[1,1]
=> [[1],[2]]
=> [2,1] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => 6
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 24
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 6
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 4
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 120
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 24
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 12
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 6
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 4
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 720
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 120
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ? = 48
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ? = 24
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ? = 36
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ? = 12
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ? = 6
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? = 8
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ? = 4
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ? = 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ? = 1
Description
The number of permutations greater than or equal to the given permutation in (strong) Bruhat order.
Matching statistic: St000259
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 18%●distinct values known / distinct values provided: 9%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 18%●distinct values known / distinct values provided: 9%
Values
[2]
=> [[1,2]]
=> [2] => ([],2)
=> ? = 2
[1,1]
=> [[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 1
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? = 6
[2,1]
=> [[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> ? = 2
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? = 24
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> ? = 6
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 4
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? = 120
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> ? = 24
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 12
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 6
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,1,1,1,1]
=> [[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)
=> 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? = 720
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> ? = 120
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? = 48
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? = 24
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 36
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 12
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 8
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 18%●distinct values known / distinct values provided: 9%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 18%●distinct values known / distinct values provided: 9%
Values
[2]
=> [[1,2]]
=> [2] => ([],2)
=> ? = 2
[1,1]
=> [[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 1
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? = 6
[2,1]
=> [[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> ? = 2
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? = 24
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> ? = 6
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 4
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? = 120
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> ? = 24
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 12
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 6
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,1,1,1,1]
=> [[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)
=> 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? = 720
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> ? = 120
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? = 48
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? = 24
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 36
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 12
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 8
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000777
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 18%●distinct values known / distinct values provided: 9%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 18%●distinct values known / distinct values provided: 9%
Values
[2]
=> [[1,2]]
=> [2] => ([],2)
=> ? = 2 + 1
[1,1]
=> [[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? = 6 + 1
[2,1]
=> [[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> ? = 2 + 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? = 24 + 1
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> ? = 6 + 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 4 + 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? = 120 + 1
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> ? = 24 + 1
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 12 + 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 6 + 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,1,1,1]
=> [[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)
=> 2 = 1 + 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? = 720 + 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> ? = 120 + 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? = 48 + 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? = 24 + 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 36 + 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 12 + 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 + 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 8 + 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
The following 33 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001545The second Elser number of a connected graph. St000023The number of inner peaks of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000357The number of occurrences of the pattern 12-3. St000497The lcb statistic of a set partition. St001050The number of terminal closers of a set partition. St001469The holeyness of a permutation. St001520The number of strict 3-descents. St001565The number of arithmetic progressions of length 2 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001935The number of ascents in a parking function. St000075The orbit size of a standard tableau under promotion. St000099The number of valleys of a permutation, including the boundary. St000166The depth minus 1 of an ordered tree. St000461The rix statistic of a permutation. St000522The number of 1-protected nodes of a rooted tree. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001557The number of inversions of the second entry of a permutation. 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. St000090The variation of a composition. St000521The number of distinct subtrees of an ordered tree. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000505The biggest entry in the block containing the 1. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000740The last entry of a permutation. St000971The smallest closer of a set partition. St001497The position of the largest weak excedence of a permutation. St000226The convexity of a permutation. St000231Sum of the maximal elements of the blocks of a set partition.
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