- FindStat

Your data matches 140 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000259
(load all 4 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [1] => [1] => ([],1)
 => 0
[[1],[2]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 3
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
[[1],[2],[3],[4],[5]]
 => [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)
 => 1
[[1,3,5],[2],[4],[6]]
 => [6,4,2,1,3,5] => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 3
[[1,3,4],[2],[5],[6]]
 => [6,5,2,1,3,4] => [4,1,2,6,5,3] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 3
[[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [5,4,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,4],[2],[3],[5],[6]]
 => [6,5,3,2,1,4] => [5,4,1,6,3,2] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,3],[2],[4],[5],[6]]
 => [6,5,4,2,1,3] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1],[2],[3],[4],[5],[6]]
 => [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)
 => 1
[[1,3,4,6],[2],[5],[7]]
 => [7,5,2,1,3,4,6] => [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 5
[[1,4,6],[2],[3],[5],[7]]
 => [7,5,3,2,1,4,6] => [5,4,1,6,2,7,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 3
[[1,3,6],[2],[4],[5],[7]]
 => [7,5,4,2,1,3,6] => [5,1,6,4,2,7,3] => ([(0,6),(1,5),(2,3),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
[[1,4,5],[2],[3],[6],[7]]
 => [7,6,3,2,1,4,5] => [5,4,1,2,7,6,3] => ([(0,1),(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
[[1,3,5],[2],[4],[6],[7]]
 => [7,6,4,2,1,3,5] => [5,1,6,2,7,4,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 3
[[1,3,4],[2],[5],[6],[7]]
 => [7,6,5,2,1,3,4] => [5,1,2,7,6,4,3] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 3
[[1,6],[2],[3],[4],[5],[7]]
 => [7,5,4,3,2,1,6] => [6,5,4,3,1,7,2] => ([(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)
 => 3
[[1,5],[2],[3],[4],[6],[7]]
 => [7,6,4,3,2,1,5] => [6,5,4,1,7,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
[[1,4],[2],[3],[5],[6],[7]]
 => [7,6,5,3,2,1,4] => [6,5,1,7,4,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
[[1,3],[2],[4],[5],[6],[7]]
 => [7,6,5,4,2,1,3] => [6,1,7,5,4,3,2] => ([(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)
 => 3
[[1],[2],[3],[4],[5],[6],[7]]
 => [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)
 => 1

Description

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

Matching statistic: St001500
(load all 2 compositions to match this statistic)

Mapping

Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001500: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%

Values

[[1]]
 => [1] => [1,0]
 => [1,0]
 => ? = 0
[[1],[2]]
 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1
[[1],[2],[3]]
 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[[1,3],[2],[4]]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[[1],[2],[3],[4]]
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[[1,4],[2],[3],[5]]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 3
[[1,3],[2],[4],[5]]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[[1,3,5],[2],[4],[6]]
 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 3
[[1,3,4],[2],[5],[6]]
 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 3
[[1,5],[2],[3],[4],[6]]
 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 3
[[1,4],[2],[3],[5],[6]]
 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 3
[[1,3],[2],[4],[5],[6]]
 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 3
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => [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
[[1,3,4,6],[2],[5],[7]]
 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => 5
[[1,4,6],[2],[3],[5],[7]]
 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => 3
[[1,3,6],[2],[4],[5],[7]]
 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => 3
[[1,4,5],[2],[3],[6],[7]]
 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => 3
[[1,3,5],[2],[4],[6],[7]]
 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => 3
[[1,3,4],[2],[5],[6],[7]]
 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => 3
[[1,6],[2],[3],[4],[5],[7]]
 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => 3
[[1,5],[2],[3],[4],[6],[7]]
 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => 3
[[1,4],[2],[3],[5],[6],[7]]
 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => 3
[[1,3],[2],[4],[5],[6],[7]]
 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => 3
[[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1] => [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

Description

The global dimension of magnitude 1 Nakayama algebras. We use the code below to translate them to Dyck paths.

Matching statistic: St000673

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000673: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%

Values

[[1]]
 => [1] => [.,.]
 => [1] => ? = 0 - 1
[[1],[2]]
 => [2,1] => [[.,.],.]
 => [1,2] => 0 = 1 - 1
[[1],[2],[3]]
 => [3,2,1] => [[[.,.],.],.]
 => [1,2,3] => 0 = 1 - 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 2 = 3 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,2,3,4] => 0 = 1 - 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => 2 = 3 - 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => 2 = 3 - 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
 => [6,4,2,1,3,5] => [[[[.,.],.],.],[.,[.,.]]]
 => [1,2,3,6,5,4] => 2 = 3 - 1
[[1,3,4],[2],[5],[6]]
 => [6,5,2,1,3,4] => [[[[.,.],.],.],[.,[.,.]]]
 => [1,2,3,6,5,4] => 2 = 3 - 1
[[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [[[[[.,.],.],.],.],[.,.]]
 => [1,2,3,4,6,5] => 2 = 3 - 1
[[1,4],[2],[3],[5],[6]]
 => [6,5,3,2,1,4] => [[[[[.,.],.],.],.],[.,.]]
 => [1,2,3,4,6,5] => 2 = 3 - 1
[[1,3],[2],[4],[5],[6]]
 => [6,5,4,2,1,3] => [[[[[.,.],.],.],.],[.,.]]
 => [1,2,3,4,6,5] => 2 = 3 - 1
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
 => [1,2,3,4,5,6] => 0 = 1 - 1
[[1,3,4,6],[2],[5],[7]]
 => [7,5,2,1,3,4,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
 => [1,2,3,7,6,5,4] => 4 = 5 - 1
[[1,4,6],[2],[3],[5],[7]]
 => [7,5,3,2,1,4,6] => [[[[[.,.],.],.],.],[.,[.,.]]]
 => [1,2,3,4,7,6,5] => 2 = 3 - 1
[[1,3,6],[2],[4],[5],[7]]
 => [7,5,4,2,1,3,6] => [[[[[.,.],.],.],.],[.,[.,.]]]
 => [1,2,3,4,7,6,5] => 2 = 3 - 1
[[1,4,5],[2],[3],[6],[7]]
 => [7,6,3,2,1,4,5] => [[[[[.,.],.],.],.],[.,[.,.]]]
 => [1,2,3,4,7,6,5] => 2 = 3 - 1
[[1,3,5],[2],[4],[6],[7]]
 => [7,6,4,2,1,3,5] => [[[[[.,.],.],.],.],[.,[.,.]]]
 => [1,2,3,4,7,6,5] => 2 = 3 - 1
[[1,3,4],[2],[5],[6],[7]]
 => [7,6,5,2,1,3,4] => [[[[[.,.],.],.],.],[.,[.,.]]]
 => [1,2,3,4,7,6,5] => 2 = 3 - 1
[[1,6],[2],[3],[4],[5],[7]]
 => [7,5,4,3,2,1,6] => [[[[[[.,.],.],.],.],.],[.,.]]
 => [1,2,3,4,5,7,6] => 2 = 3 - 1
[[1,5],[2],[3],[4],[6],[7]]
 => [7,6,4,3,2,1,5] => [[[[[[.,.],.],.],.],.],[.,.]]
 => [1,2,3,4,5,7,6] => 2 = 3 - 1
[[1,4],[2],[3],[5],[6],[7]]
 => [7,6,5,3,2,1,4] => [[[[[[.,.],.],.],.],.],[.,.]]
 => [1,2,3,4,5,7,6] => 2 = 3 - 1
[[1,3],[2],[4],[5],[6],[7]]
 => [7,6,5,4,2,1,3] => [[[[[[.,.],.],.],.],.],[.,.]]
 => [1,2,3,4,5,7,6] => 2 = 3 - 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],.]
 => [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: St000691

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%

Values

[[1]]
 =>  =>  => ? => ? = 0 - 1
[[1],[2]]
 => 1 => 1 => 1 => 0 = 1 - 1
[[1],[2],[3]]
 => 11 => 11 => 11 => 0 = 1 - 1
[[1,3],[2],[4]]
 => 101 => 001 => 010 => 2 = 3 - 1
[[1],[2],[3],[4]]
 => 111 => 111 => 111 => 0 = 1 - 1
[[1,4],[2],[3],[5]]
 => 1101 => 0011 => 0110 => 2 = 3 - 1
[[1,3],[2],[4],[5]]
 => 1011 => 0001 => 0010 => 2 = 3 - 1
[[1],[2],[3],[4],[5]]
 => 1111 => 1111 => 1111 => 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
 => 10101 => 11001 => 10011 => 2 = 3 - 1
[[1,3,4],[2],[5],[6]]
 => 10011 => 11101 => 11011 => 2 = 3 - 1
[[1,5],[2],[3],[4],[6]]
 => 11101 => 00111 => 01110 => 2 = 3 - 1
[[1,4],[2],[3],[5],[6]]
 => 11011 => 00011 => 00110 => 2 = 3 - 1
[[1,3],[2],[4],[5],[6]]
 => 10111 => 00001 => 00010 => 2 = 3 - 1
[[1],[2],[3],[4],[5],[6]]
 => 11111 => 11111 => 11111 => 0 = 1 - 1
[[1,3,4,6],[2],[5],[7]]
 => 100101 => 001101 => 011010 => 4 = 5 - 1
[[1,4,6],[2],[3],[5],[7]]
 => 110101 => 110011 => 100111 => 2 = 3 - 1
[[1,3,6],[2],[4],[5],[7]]
 => 101101 => 110001 => 100011 => 2 = 3 - 1
[[1,4,5],[2],[3],[6],[7]]
 => 110011 => 111011 => 110111 => 2 = 3 - 1
[[1,3,5],[2],[4],[6],[7]]
 => 101011 => 111001 => 110011 => 2 = 3 - 1
[[1,3,4],[2],[5],[6],[7]]
 => 100111 => 111101 => 111011 => 2 = 3 - 1
[[1,6],[2],[3],[4],[5],[7]]
 => 111101 => 001111 => 011110 => 2 = 3 - 1
[[1,5],[2],[3],[4],[6],[7]]
 => 111011 => 000111 => 001110 => 2 = 3 - 1
[[1,4],[2],[3],[5],[6],[7]]
 => 110111 => 000011 => 000110 => 2 = 3 - 1
[[1,3],[2],[4],[5],[6],[7]]
 => 101111 => 000001 => 000010 => 2 = 3 - 1
[[1],[2],[3],[4],[5],[6],[7]]
 => 111111 => 111111 => 111111 => 0 = 1 - 1

Description

The number of changes of a binary word. This is the number of indices

$i$ such that

$w_i \neq w_{i+1}$ .

Matching statistic: St001093

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001093: Graphs ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [1] => [1] => ([],1)
 => 1 = 0 + 1
[[1],[2]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 4 = 3 + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[[1],[2],[3],[4],[5]]
 => [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)
 => 2 = 1 + 1
[[1,3,5],[2],[4],[6]]
 => [6,4,2,1,3,5] => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[[1,3,4],[2],[5],[6]]
 => [6,5,2,1,3,4] => [4,1,2,6,5,3] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 4 = 3 + 1
[[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [5,4,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[[1,4],[2],[3],[5],[6]]
 => [6,5,3,2,1,4] => [5,4,1,6,3,2] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[[1,3],[2],[4],[5],[6]]
 => [6,5,4,2,1,3] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[[1],[2],[3],[4],[5],[6]]
 => [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)
 => 2 = 1 + 1
[[1,3,4,6],[2],[5],[7]]
 => [7,5,2,1,3,4,6] => [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 6 = 5 + 1
[[1,4,6],[2],[3],[5],[7]]
 => [7,5,3,2,1,4,6] => [5,4,1,6,2,7,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 4 = 3 + 1
[[1,3,6],[2],[4],[5],[7]]
 => [7,5,4,2,1,3,6] => [5,1,6,4,2,7,3] => ([(0,6),(1,5),(2,3),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 4 = 3 + 1
[[1,4,5],[2],[3],[6],[7]]
 => [7,6,3,2,1,4,5] => [5,4,1,2,7,6,3] => ([(0,1),(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[[1,3,5],[2],[4],[6],[7]]
 => [7,6,4,2,1,3,5] => [5,1,6,2,7,4,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 4 = 3 + 1
[[1,3,4],[2],[5],[6],[7]]
 => [7,6,5,2,1,3,4] => [5,1,2,7,6,4,3] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 4 = 3 + 1
[[1,6],[2],[3],[4],[5],[7]]
 => [7,5,4,3,2,1,6] => [6,5,4,3,1,7,2] => ([(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 = 3 + 1
[[1,5],[2],[3],[4],[6],[7]]
 => [7,6,4,3,2,1,5] => [6,5,4,1,7,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4 = 3 + 1
[[1,4],[2],[3],[5],[6],[7]]
 => [7,6,5,3,2,1,4] => [6,5,1,7,4,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4 = 3 + 1
[[1,3],[2],[4],[5],[6],[7]]
 => [7,6,5,4,2,1,3] => [6,1,7,5,4,3,2] => ([(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 = 3 + 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [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)
 => 2 = 1 + 1

Description

The detour number of a graph. This is the number of vertices in a longest induced path in a graph. Note that [1] defines the detour number as the number of edges in a longest induced path, which is unsuitable for the empty graph.

Matching statistic: St001502
(load all 2 compositions to match this statistic)

Mapping

Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001502: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%

Values

[[1]]
 => [1] => [1,0]
 => [1,0]
 => ? = 0 - 1
[[1],[2]]
 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[[1],[2],[3]]
 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[[1,3],[2],[4]]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[1],[2],[3],[4]]
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[[1,4],[2],[3],[5]]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[[1,3],[2],[4],[5]]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 3 - 1
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 2 = 3 - 1
[[1,3,4],[2],[5],[6]]
 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2 = 3 - 1
[[1,5],[2],[3],[4],[6]]
 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 2 = 3 - 1
[[1,4],[2],[3],[5],[6]]
 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 2 = 3 - 1
[[1,3],[2],[4],[5],[6]]
 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 2 = 3 - 1
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[[1,3,4,6],[2],[5],[7]]
 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => 4 = 5 - 1
[[1,4,6],[2],[3],[5],[7]]
 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => 2 = 3 - 1
[[1,3,6],[2],[4],[5],[7]]
 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => 2 = 3 - 1
[[1,4,5],[2],[3],[6],[7]]
 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => 2 = 3 - 1
[[1,3,5],[2],[4],[6],[7]]
 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => 2 = 3 - 1
[[1,3,4],[2],[5],[6],[7]]
 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => 2 = 3 - 1
[[1,6],[2],[3],[4],[5],[7]]
 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => 2 = 3 - 1
[[1,5],[2],[3],[4],[6],[7]]
 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => 2 = 3 - 1
[[1,4],[2],[3],[5],[6],[7]]
 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => 2 = 3 - 1
[[1,3],[2],[4],[5],[6],[7]]
 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => 2 = 3 - 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1] => [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]
 => 0 = 1 - 1

Description

The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. We use the code below to translate them to Dyck paths. The algebras where the statistic returns 0 are exactly the higher Auslander algebras and are of special interest. It seems like they are counted by the number of divisors function.

Matching statistic: St000401
(load all 9 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000401: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 84%●distinct values known / distinct values provided: 75%

Values

[[1]]
 => [1] => [1] => [1] => 1 = 0 + 1
[[1],[2]]
 => [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 2 = 1 + 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,3,4,2] => [1,4,3,2] => 4 = 3 + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 2 = 1 + 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [1,3,4,5,2] => [1,5,3,4,2] => 4 = 3 + 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [1,2,4,5,3] => [1,2,5,4,3] => 4 = 3 + 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 2 = 1 + 1
[[1,3,5],[2],[4],[6]]
 => [6,4,2,1,3,5] => [1,3,5,6,4,2] => [1,6,5,4,3,2] => 4 = 3 + 1
[[1,3,4],[2],[5],[6]]
 => [6,5,2,1,3,4] => [1,2,5,6,4,3] => [1,2,6,5,4,3] => 4 = 3 + 1
[[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => 4 = 3 + 1
[[1,4],[2],[3],[5],[6]]
 => [6,5,3,2,1,4] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => 4 = 3 + 1
[[1,3],[2],[4],[5],[6]]
 => [6,5,4,2,1,3] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => 4 = 3 + 1
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 2 = 1 + 1
[[1,3,4,6],[2],[5],[7]]
 => [7,5,2,1,3,4,6] => [1,3,6,7,5,4,2] => [1,7,6,5,4,3,2] => ? = 5 + 1
[[1,4,6],[2],[3],[5],[7]]
 => [7,5,3,2,1,4,6] => [1,3,5,6,7,4,2] => [1,7,6,4,5,3,2] => ? = 3 + 1
[[1,3,6],[2],[4],[5],[7]]
 => [7,5,4,2,1,3,6] => [1,3,4,6,7,5,2] => [1,7,3,6,5,4,2] => ? = 3 + 1
[[1,4,5],[2],[3],[6],[7]]
 => [7,6,3,2,1,4,5] => [1,2,5,6,7,4,3] => [1,2,7,6,5,4,3] => 4 = 3 + 1
[[1,3,5],[2],[4],[6],[7]]
 => [7,6,4,2,1,3,5] => [1,2,4,6,7,5,3] => [1,2,7,6,5,4,3] => 4 = 3 + 1
[[1,3,4],[2],[5],[6],[7]]
 => [7,6,5,2,1,3,4] => [1,2,3,6,7,5,4] => [1,2,3,7,6,5,4] => 4 = 3 + 1
[[1,6],[2],[3],[4],[5],[7]]
 => [7,5,4,3,2,1,6] => [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => ? = 3 + 1
[[1,5],[2],[3],[4],[6],[7]]
 => [7,6,4,3,2,1,5] => [1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => 4 = 3 + 1
[[1,4],[2],[3],[5],[6],[7]]
 => [7,6,5,3,2,1,4] => [1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => 4 = 3 + 1
[[1,3],[2],[4],[5],[6],[7]]
 => [7,6,5,4,2,1,3] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => 4 = 3 + 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 2 = 1 + 1

Description

The size of the symmetry class of a permutation. The symmetry class of a permutation

$\pi$ is the set of all permutations that can be obtained from

$\pi$ by the three elementary operations '''inverse''' ([[Mp00066]]), '''reverse''' ([[Mp00064]]), and '''complement''' ([[Mp00069]]). Two elements in the same symmetry class are also in the same Wilf-equivalence class, see for example [2].

Matching statistic: St001005
(load all 10 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St001005: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 80%●distinct values known / distinct values provided: 50%

Values

[[1]]
 => [1] => [1] => [1] => ? = 0 - 1
[[1],[2]]
 => [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,3,4,2] => [1,4,3,2] => 2 = 3 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [1,3,4,5,2] => [1,5,3,4,2] => 2 = 3 - 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [1,2,4,5,3] => [1,2,5,4,3] => 2 = 3 - 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
 => [6,4,2,1,3,5] => [1,3,5,6,4,2] => [1,6,5,4,3,2] => 2 = 3 - 1
[[1,3,4],[2],[5],[6]]
 => [6,5,2,1,3,4] => [1,2,5,6,4,3] => [1,2,6,5,4,3] => 2 = 3 - 1
[[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => 2 = 3 - 1
[[1,4],[2],[3],[5],[6]]
 => [6,5,3,2,1,4] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => 2 = 3 - 1
[[1,3],[2],[4],[5],[6]]
 => [6,5,4,2,1,3] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => 2 = 3 - 1
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[[1,3,4,6],[2],[5],[7]]
 => [7,5,2,1,3,4,6] => [1,3,6,7,5,4,2] => [1,7,6,5,4,3,2] => ? = 5 - 1
[[1,4,6],[2],[3],[5],[7]]
 => [7,5,3,2,1,4,6] => [1,3,5,6,7,4,2] => [1,7,6,4,5,3,2] => ? = 3 - 1
[[1,3,6],[2],[4],[5],[7]]
 => [7,5,4,2,1,3,6] => [1,3,4,6,7,5,2] => [1,7,3,6,5,4,2] => ? = 3 - 1
[[1,4,5],[2],[3],[6],[7]]
 => [7,6,3,2,1,4,5] => [1,2,5,6,7,4,3] => [1,2,7,6,5,4,3] => 2 = 3 - 1
[[1,3,5],[2],[4],[6],[7]]
 => [7,6,4,2,1,3,5] => [1,2,4,6,7,5,3] => [1,2,7,6,5,4,3] => 2 = 3 - 1
[[1,3,4],[2],[5],[6],[7]]
 => [7,6,5,2,1,3,4] => [1,2,3,6,7,5,4] => [1,2,3,7,6,5,4] => 2 = 3 - 1
[[1,6],[2],[3],[4],[5],[7]]
 => [7,5,4,3,2,1,6] => [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => ? = 3 - 1
[[1,5],[2],[3],[4],[6],[7]]
 => [7,6,4,3,2,1,5] => [1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => 2 = 3 - 1
[[1,4],[2],[3],[5],[6],[7]]
 => [7,6,5,3,2,1,4] => [1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => 2 = 3 - 1
[[1,3],[2],[4],[5],[6],[7]]
 => [7,6,5,4,2,1,3] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => 2 = 3 - 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0 = 1 - 1

Description

The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both.

Matching statistic: St000956

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 76%●distinct values known / distinct values provided: 50%

Values

[[1]]
 => [1] => [1] => [1] => ? = 0 - 1
[[1],[2]]
 => [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,3,2,4] => [3,1,2,4] => 2 = 3 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => 2 = 3 - 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [1,3,2,4,5] => [3,1,2,4,5] => 2 = 3 - 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
 => [6,4,2,1,3,5] => [1,3,5,2,4,6] => [3,1,2,5,4,6] => 2 = 3 - 1
[[1,3,4],[2],[5],[6]]
 => [6,5,2,1,3,4] => [1,3,4,2,5,6] => [3,1,4,2,5,6] => 2 = 3 - 1
[[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [1,5,2,3,4,6] => [1,2,5,3,4,6] => 2 = 3 - 1
[[1,4],[2],[3],[5],[6]]
 => [6,5,3,2,1,4] => [1,4,2,3,5,6] => [1,4,2,3,5,6] => 2 = 3 - 1
[[1,3],[2],[4],[5],[6]]
 => [6,5,4,2,1,3] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => 2 = 3 - 1
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[[1,3,4,6],[2],[5],[7]]
 => [7,5,2,1,3,4,6] => [1,3,4,6,2,5,7] => [3,1,4,2,6,5,7] => ? = 5 - 1
[[1,4,6],[2],[3],[5],[7]]
 => [7,5,3,2,1,4,6] => [1,4,6,2,3,5,7] => [1,4,2,3,6,5,7] => 2 = 3 - 1
[[1,3,6],[2],[4],[5],[7]]
 => [7,5,4,2,1,3,6] => [1,3,6,2,4,5,7] => [3,1,2,4,6,5,7] => ? = 3 - 1
[[1,4,5],[2],[3],[6],[7]]
 => [7,6,3,2,1,4,5] => [1,4,5,2,3,6,7] => [1,4,2,5,3,6,7] => 2 = 3 - 1
[[1,3,5],[2],[4],[6],[7]]
 => [7,6,4,2,1,3,5] => [1,3,5,2,4,6,7] => [3,1,2,5,4,6,7] => ? = 3 - 1
[[1,3,4],[2],[5],[6],[7]]
 => [7,6,5,2,1,3,4] => [1,3,4,2,5,6,7] => [3,1,4,2,5,6,7] => ? = 3 - 1
[[1,6],[2],[3],[4],[5],[7]]
 => [7,5,4,3,2,1,6] => [1,6,2,3,4,5,7] => [1,2,3,6,4,5,7] => 2 = 3 - 1
[[1,5],[2],[3],[4],[6],[7]]
 => [7,6,4,3,2,1,5] => [1,5,2,3,4,6,7] => [1,2,5,3,4,6,7] => 2 = 3 - 1
[[1,4],[2],[3],[5],[6],[7]]
 => [7,6,5,3,2,1,4] => [1,4,2,3,5,6,7] => [1,4,2,3,5,6,7] => 2 = 3 - 1
[[1,3],[2],[4],[5],[6],[7]]
 => [7,6,5,4,2,1,3] => [1,3,2,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 3 - 1
[[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0 = 1 - 1

Description

The maximal displacement of a permutation. This is

$\max\{ |\pi(i)-i| \mid 1 \leq i \leq n\}$ for a permutation

$\pi$ of

$\{1,\ldots,n\}$ . This statistic without the absolute value is the maximal drop size [[St000141]].

Matching statistic: St001364

Mapping

Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001364: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 72%●distinct values known / distinct values provided: 50%

Values

[[1]]
 => [1]
 => [1]
 => []
 => ? = 0 - 2
[[1],[2]]
 => [1,1]
 => [2]
 => []
 => ? = 1 - 2
[[1],[2],[3]]
 => [1,1,1]
 => [3]
 => []
 => ? = 1 - 2
[[1,3],[2],[4]]
 => [2,1,1]
 => [3,1]
 => [1]
 => 1 = 3 - 2
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [4]
 => []
 => ? = 1 - 2
[[1,4],[2],[3],[5]]
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 1 = 3 - 2
[[1,3],[2],[4],[5]]
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 1 = 3 - 2
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [5]
 => []
 => ? = 1 - 2
[[1,3,5],[2],[4],[6]]
 => [3,1,1,1]
 => [4,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,3,4],[2],[5],[6]]
 => [3,1,1,1]
 => [4,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,5],[2],[3],[4],[6]]
 => [2,1,1,1,1]
 => [5,1]
 => [1]
 => 1 = 3 - 2
[[1,4],[2],[3],[5],[6]]
 => [2,1,1,1,1]
 => [5,1]
 => [1]
 => 1 = 3 - 2
[[1,3],[2],[4],[5],[6]]
 => [2,1,1,1,1]
 => [5,1]
 => [1]
 => 1 = 3 - 2
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => [6]
 => []
 => ? = 1 - 2
[[1,3,4,6],[2],[5],[7]]
 => [4,1,1,1]
 => [4,1,1,1]
 => [1,1,1]
 => 3 = 5 - 2
[[1,4,6],[2],[3],[5],[7]]
 => [3,1,1,1,1]
 => [5,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,3,6],[2],[4],[5],[7]]
 => [3,1,1,1,1]
 => [5,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,4,5],[2],[3],[6],[7]]
 => [3,1,1,1,1]
 => [5,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,3,5],[2],[4],[6],[7]]
 => [3,1,1,1,1]
 => [5,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,3,4],[2],[5],[6],[7]]
 => [3,1,1,1,1]
 => [5,1,1]
 => [1,1]
 => 1 = 3 - 2
[[1,6],[2],[3],[4],[5],[7]]
 => [2,1,1,1,1,1]
 => [6,1]
 => [1]
 => 1 = 3 - 2
[[1,5],[2],[3],[4],[6],[7]]
 => [2,1,1,1,1,1]
 => [6,1]
 => [1]
 => 1 = 3 - 2
[[1,4],[2],[3],[5],[6],[7]]
 => [2,1,1,1,1,1]
 => [6,1]
 => [1]
 => 1 = 3 - 2
[[1,3],[2],[4],[5],[6],[7]]
 => [2,1,1,1,1,1]
 => [6,1]
 => [1]
 => 1 = 3 - 2
[[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1]
 => [7]
 => []
 => ? = 1 - 2

Description

The number of permutations whose cube equals a fixed permutation of given cycle type. For example, the permutation

$\pi=412365$ has cycle type

$(4,2)$ and

$234165$ is the unique permutation whose cube is

$\pi$ .

The following 130 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001527The cyclic permutation representation number of an integer partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000662The staircase size of the code of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000646The number of big ascents of a permutation. St000653The last descent of a permutation. St000794The mak of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000619The number of cyclic descents of a permutation. St000670The reversal length of a permutation. St000354The number of recoils of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000638The number of up-down runs of a permutation. St000021The number of descents of a permutation. St000154The sum of the descent bottoms of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000864The number of circled entries of the shifted recording tableau of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001375The pancake length of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000711The number of big exceedences of a permutation. St000798The makl of a permutation. St000836The number of descents of distance 2 of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000060The greater neighbor of the maximum. St000061The number of nodes on the left branch of a binary tree. St000472The sum of the ascent bottoms of a permutation. St000485The length of the longest cycle of a permutation. St000837The number of ascents of distance 2 of a permutation. St000989The number of final rises of a permutation. St001388The number of non-attacking neighbors of a permutation. St000652The maximal difference between successive positions of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000402Half the size of the symmetry class of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001488The number of corners of a skew partition. 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

$eA$ . St000545The number of parabolic double cosets with minimal element being the given permutation. St000831The number of indices that are either descents or recoils. St001060The distinguishing index of a graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. 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. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000477The weight of a partition according to Alladi. St000928The sum of the coefficients of the character polynomial of an integer partition. St000307The number of rowmotion orbits of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000632The jump number of the poset. St001569The maximal modular displacement of a permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000260The radius of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001557The number of inversions of the second entry of a permutation. St001668The number of points of the poset minus the width of the poset. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000100The number of linear extensions of a poset. St000526The number of posets with combinatorially isomorphic order polytopes. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001964The interval resolution global dimension of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001413Half the length of the longest even length palindromic prefix of a binary word. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001555The order of a signed permutation. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001890The maximum magnitude of the Möbius function of a poset. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000907The number of maximal antichains of minimal length in a poset. St001424The number of distinct squares in a binary word. 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. St001927Sparre Andersen's number of positives of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St001624The breadth of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St000298The order dimension or Dushnik-Miller dimension of a poset. St000679The pruning number of an ordered tree. St001330The hat guessing number of a graph. St001399The distinguishing number of a poset. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001621The number of atoms of a lattice. St000717The number of ordinal summands of a poset. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset.