searching the database
Your data matches 44 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001279
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 0 = 1 - 1
[2]
=> [1,1]
=> 0 = 1 - 1
[1,1]
=> [2]
=> 2 = 3 - 1
[3]
=> [1,1,1]
=> 0 = 1 - 1
[2,1]
=> [2,1]
=> 2 = 3 - 1
[1,1,1]
=> [3]
=> 3 = 4 - 1
[4]
=> [1,1,1,1]
=> 0 = 1 - 1
[3,1]
=> [2,1,1]
=> 2 = 3 - 1
[2,2]
=> [2,2]
=> 4 = 5 - 1
[2,1,1]
=> [3,1]
=> 3 = 4 - 1
[1,1,1,1]
=> [4]
=> 4 = 5 - 1
[5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[4,1]
=> [2,1,1,1]
=> 2 = 3 - 1
[3,2]
=> [2,2,1]
=> 4 = 5 - 1
[3,1,1]
=> [3,1,1]
=> 3 = 4 - 1
[2,2,1]
=> [3,2]
=> 5 = 6 - 1
[2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,1,1,1,1]
=> [5]
=> 5 = 6 - 1
[6]
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
[5,1]
=> [2,1,1,1,1]
=> 2 = 3 - 1
[4,2]
=> [2,2,1,1]
=> 4 = 5 - 1
[4,1,1]
=> [3,1,1,1]
=> 3 = 4 - 1
[3,3]
=> [2,2,2]
=> 6 = 7 - 1
[3,2,1]
=> [3,2,1]
=> 5 = 6 - 1
[3,1,1,1]
=> [4,1,1]
=> 4 = 5 - 1
[2,2,2]
=> [3,3]
=> 6 = 7 - 1
[2,2,1,1]
=> [4,2]
=> 6 = 7 - 1
[2,1,1,1,1]
=> [5,1]
=> 5 = 6 - 1
[1,1,1,1,1,1]
=> [6]
=> 6 = 7 - 1
[7]
=> [1,1,1,1,1,1,1]
=> 0 = 1 - 1
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St001468
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001468: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001468: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1] => 1
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 1
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 3
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 3
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [2,3,1] => 4
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 1
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 3
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [4,3,2,1] => 5
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => 4
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [3,4,1,2] => 5
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 3
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [4,3,2,1,5] => 5
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => 4
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,5,4,3,1] => 6
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [3,4,1,2,5] => 5
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [3,4,5,1,2] => 6
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 3
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [4,3,2,1,5,6] => 5
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,3,1,4,5,6] => 4
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [6,5,4,3,2,1] => 7
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,5,4,3,1,6] => 6
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [3,4,1,2,5,6] => 5
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [4,3,5,6,2,1] => 7
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,6,1,5,4,2] => 7
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [3,4,5,1,2,6] => 6
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [4,5,6,1,2,3] => 7
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 1
Description
The smallest fixpoint of a permutation.
A fixpoint of a permutation of length $n$ if an index $i$ such that $\pi(i) = i$, and we set $\pi(n+1) = n+1$.
Matching statistic: St000394
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 5 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4 = 5 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 5 = 6 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5 = 6 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 4 = 5 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 6 = 7 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 5 = 6 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 6 = 7 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 6 = 7 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 5 = 6 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 6 = 7 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000915
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000915: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000915: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => ([],1)
=> 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [1,2] => ([],2)
=> 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 2 = 3 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 2 = 3 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 3 = 4 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 2 = 3 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 4 = 5 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 4 = 5 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 2 = 3 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 5 = 6 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 6 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> 0 = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> 2 = 3 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 3 = 4 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6 = 7 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6 = 7 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 6 = 7 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 6 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 6 = 7 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ([],7)
=> 0 = 1 - 1
Description
The Ore degree of a graph.
This is the maximal Ore degree of an edge, which is the sum of the degrees of its two endpoints.
Matching statistic: St000673
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000673: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000673: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1] => ? = 1 - 1
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 2 = 3 - 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 2 = 3 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [2,3,1] => 3 = 4 - 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [4,3,2,1] => 4 = 5 - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => 3 = 4 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [3,4,1,2] => 4 = 5 - 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 2 = 3 - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [4,3,2,1,5] => 4 = 5 - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => 3 = 4 - 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,5,4,3,1] => 5 = 6 - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [3,4,1,2,5] => 4 = 5 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [3,4,5,1,2] => 5 = 6 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 2 = 3 - 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [4,3,2,1,5,6] => 4 = 5 - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,3,1,4,5,6] => 3 = 4 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [6,5,4,3,2,1] => 6 = 7 - 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,5,4,3,1,6] => 5 = 6 - 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [3,4,1,2,5,6] => 4 = 5 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [4,3,5,6,2,1] => 6 = 7 - 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,6,1,5,4,2] => 6 = 7 - 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [3,4,5,1,2,6] => 5 = 6 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [4,5,6,1,2,3] => 6 = 7 - 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0 = 1 - 1
Description
The number of non-fixed points of a permutation.
In other words, this statistic is $n$ minus the number of fixed points ([[St000022]]) of $\pi$.
Matching statistic: St000718
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> 0 = 1 - 1
[2]
=> [[1,2]]
=> [1,2] => ([],2)
=> 0 = 1 - 1
[1,1]
=> [[1],[2]]
=> [2,1] => ([(0,1)],2)
=> 2 = 3 - 1
[3]
=> [[1,2,3]]
=> [1,2,3] => ([],3)
=> 0 = 1 - 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => ([(1,2)],3)
=> 2 = 3 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> 2 = 3 - 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 5 - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => ([],5)
=> 0 = 1 - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 2 = 3 - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 5 - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [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)
=> 5 = 6 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([],6)
=> 0 = 1 - 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => ([(4,5)],6)
=> 2 = 3 - 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 5 - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6 = 7 - 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [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)
=> 6 = 7 - 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 7 - 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [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)
=> 5 = 6 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [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)
=> 6 = 7 - 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => ([],7)
=> 0 = 1 - 1
Description
The largest Laplacian eigenvalue of a graph if it is integral.
This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral.
Various results are collected in Section 3.9 of [1]
Matching statistic: St001278
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001278: Dyck paths ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001278: Dyck paths ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 5 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 5 = 6 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [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]
=> ? = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [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]
=> ? = 3 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 4 = 5 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 4 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 6 = 7 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 5 = 6 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 5 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 6 = 7 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 6 = 7 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 6 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 7 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
Description
The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra.
The statistic is also equal to the number of non-projective torsionless indecomposable modules in the corresponding Nakayama algebra.
See theorem 5.8. in the reference for a motivation.
Matching statistic: St001232
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 5 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 5 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? = 6 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [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 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [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]
=> 2 = 3 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 5 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 7 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 6 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 7 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 7 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6 = 7 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001090
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St001090: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St001090: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [(1,2)]
=> [2,1] => 1
[2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 1
[1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 3
[3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 3
[1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 4
[4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? = 3
[2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 5
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? = 4
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 5
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => ? = 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? = 5
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => ? = 4
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 6
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => ? = 5
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => ? = 6
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => ? = 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,11,12,10,9] => ? = 3
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => ? = 5
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [2,1,4,3,6,5,9,11,8,12,10,7] => ? = 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 7
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => ? = 6
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [2,1,4,3,7,9,6,11,8,12,10,5] => ? = 5
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? = 7
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => ? = 7
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [2,1,5,7,4,9,6,11,8,12,10,3] => ? = 6
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> [3,5,2,7,4,9,6,11,8,12,10,1] => ? = 7
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? = 1
Description
The number of pop-stack-sorts needed to sort a permutation.
The pop-stack sorting operator is defined as follows. Process the permutation $\pi$ from left to right. If the stack is empty or its top element is smaller than the current element, empty the stack completely and append its elements to the output in reverse order. Next, push the current element onto the stack. After having processed the last entry, append the stack to the output in reverse order.
A permutation is $t$-pop-stack sortable if it is sortable using $t$ pop-stacks in series.
Matching statistic: St000864
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 67%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [(1,2)]
=> [2,1] => 1
[2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 1
[1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> [4,3,2,1] => 3
[3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 3
[1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 4
[4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? = 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => ? = 3
[2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 5
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => ? = 4
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => ? = 5
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => ? = 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,10,9,8,7] => ? = 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => ? = 5
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,10,7,6,9,8,5] => ? = 4
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => ? = 6
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,10,5,4,7,6,9,8,3] => ? = 5
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [10,3,2,5,4,7,6,9,8,1] => ? = 6
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => ? = 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,12,11,10,9] => ? = 3
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,10,9,8,7,6,5] => ? = 5
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [2,1,4,3,6,5,12,9,8,11,10,7] => ? = 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => ? = 7
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,10,7,6,5,4,9,8,3] => ? = 6
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [2,1,4,3,12,7,6,9,8,11,10,5] => ? = 5
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => ? = 7
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [10,5,4,3,2,7,6,9,8,1] => ? = 7
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [2,1,12,5,4,7,6,9,8,11,10,3] => ? = 6
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> [12,3,2,5,4,7,6,9,8,11,10,1] => ? = 7
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? = 1
Description
The number of circled entries of the shifted recording tableau of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of circled entries in $Q$.
The following 34 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000993The multiplicity of the largest part of an integer partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001536The number of cyclic misalignments of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001268The size of the largest ordinal summand in the poset. St001557The number of inversions of the second entry of a permutation. St000939The number of characters of the symmetric group whose value on the partition is positive. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000284The Plancherel distribution on integer partitions. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. St000941The number of characters of the symmetric group whose value on the partition is even. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!