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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000342
(load all 4 compositions to match this statistic)

Mapping

St000342: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => 5
[2,1] => 4
[1,2,3] => 14
[1,3,2] => 13
[2,1,3] => 13
[2,3,1] => 11
[3,1,2] => 11
[3,2,1] => 10
[1,2,4,3] => 29
[1,3,2,4] => 29
[1,3,4,2] => 27
[1,4,2,3] => 27
[1,4,3,2] => 26
[2,1,3,4] => 29
[2,1,4,3] => 28
[2,3,1,4] => 27
[2,3,4,1] => 24
[2,4,1,3] => 25
[2,4,3,1] => 23
[3,1,2,4] => 27
[3,1,4,2] => 25
[3,2,1,4] => 26
[3,2,4,1] => 23
[3,4,1,2] => 22
[3,4,2,1] => 21
[4,1,2,3] => 24
[4,1,3,2] => 23
[4,2,1,3] => 23
[4,2,3,1] => 21
[4,3,1,2] => 21
[4,3,2,1] => 20

Description

The cosine of a permutation. For a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ , this is given by

$\sum_{i=1}^n (i\pi_i)$ . The name comes from the observation that this equals

$\frac{n(n+1)(2n+1)}{6}\cos(\theta)$ where

$\theta$ is the angle between the vector

$(\pi_1,\ldots,\pi_n)$ and the vector

$(1,\ldots,n)$ , see [1].

Matching statistic: St001168
(load all 4 compositions to match this statistic)

Mapping

St001168: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => 5
[2,1] => 4
[1,2,3] => 14
[1,3,2] => 13
[2,1,3] => 13
[2,3,1] => 11
[3,1,2] => 11
[3,2,1] => 10
[1,2,4,3] => 29
[1,3,2,4] => 29
[1,3,4,2] => 27
[1,4,2,3] => 27
[1,4,3,2] => 26
[2,1,3,4] => 29
[2,1,4,3] => 28
[2,3,1,4] => 27
[2,3,4,1] => 24
[2,4,1,3] => 25
[2,4,3,1] => 23
[3,1,2,4] => 27
[3,1,4,2] => 25
[3,2,1,4] => 26
[3,2,4,1] => 23
[3,4,1,2] => 22
[3,4,2,1] => 21
[4,1,2,3] => 24
[4,1,3,2] => 23
[4,2,1,3] => 23
[4,2,3,1] => 21
[4,3,1,2] => 21
[4,3,2,1] => 20

Description

The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ .

Matching statistic: St000114

Mapping

Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00076: Semistandard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000114: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [[1,0],[0,1]]
 => [[1,1],[2]]
 => [[2,1],[2]]
 => 5
[2,1] => [[0,1],[1,0]]
 => [[1,2],[2]]
 => [[2,1],[1]]
 => 4
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => [[3,2,1],[3,2],[3]]
 => 14
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => [[3,2,1],[3,1],[3]]
 => 13
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => [[3,2,1],[3,2],[2]]
 => 13
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => [[3,2,1],[2,1],[2]]
 => 11
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => [[3,2,1],[3,1],[1]]
 => 11
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => [[3,2,1],[2,1],[1]]
 => 10
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => [[4,3,2,1],[4,3,1],[4,3],[4]]
 => 29
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => [[4,3,2,1],[4,3,2],[4,2],[4]]
 => 29
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,1],[2,2,4],[3,4],[4]]
 => [[4,3,2,1],[4,2,1],[4,2],[4]]
 => 27
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,1],[2,3,3],[3,4],[4]]
 => [[4,3,2,1],[4,3,1],[4,1],[4]]
 => 27
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,1],[2,3,4],[3,4],[4]]
 => [[4,3,2,1],[4,2,1],[4,1],[4]]
 => 26
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => [[4,3,2,1],[4,3,2],[4,3],[3]]
 => 29
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => [[4,3,2,1],[4,3,1],[4,3],[3]]
 => 28
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,3],[2,2,3],[3,3],[4]]
 => [[4,3,2,1],[4,3,2],[3,2],[3]]
 => 27
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,4],[2,2,4],[3,4],[4]]
 => [[4,3,2,1],[3,2,1],[3,2],[3]]
 => 24
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,3],[2,3,3],[3,4],[4]]
 => [[4,3,2,1],[4,3,1],[3,1],[3]]
 => 25
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,4],[2,3,4],[3,4],[4]]
 => [[4,3,2,1],[3,2,1],[3,1],[3]]
 => 23
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,2],[2,2,3],[3,3],[4]]
 => [[4,3,2,1],[4,3,2],[4,2],[2]]
 => 27
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,2],[2,2,4],[3,4],[4]]
 => [[4,3,2,1],[4,2,1],[4,2],[2]]
 => 25
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,3],[2,2,3],[3,3],[4]]
 => [[4,3,2,1],[4,3,2],[3,2],[2]]
 => 26
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,4],[2,2,4],[3,4],[4]]
 => [[4,3,2,1],[3,2,1],[3,2],[2]]
 => 23
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,3],[2,3,4],[3,4],[4]]
 => [[4,3,2,1],[4,2,1],[2,1],[2]]
 => 22
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,4],[2,3,4],[3,4],[4]]
 => [[4,3,2,1],[3,2,1],[2,1],[2]]
 => 21
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,2],[2,3,3],[3,4],[4]]
 => [[4,3,2,1],[4,3,1],[4,1],[1]]
 => 24
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,2],[2,3,4],[3,4],[4]]
 => [[4,3,2,1],[4,2,1],[4,1],[1]]
 => 23
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,3],[2,3,3],[3,4],[4]]
 => [[4,3,2,1],[4,3,1],[3,1],[1]]
 => 23
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,4],[2,3,4],[3,4],[4]]
 => [[4,3,2,1],[3,2,1],[3,1],[1]]
 => 21
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
 => [[1,2,3,3],[2,3,4],[3,4],[4]]
 => [[4,3,2,1],[4,2,1],[2,1],[1]]
 => 21
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => [[1,2,3,4],[2,3,4],[3,4],[4]]
 => [[4,3,2,1],[3,2,1],[2,1],[1]]
 => 20

Description

The sum of the entries of the Gelfand-Tsetlin pattern.

Matching statistic: St000008

Mapping

Mp00305: Permutations —parking function⟶ Parking functions
Mp00319: Parking functions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 38%

Values

[1,2] => [1,2] => [1,2] => [2,1] => 2 = 5 - 3
[2,1] => [2,1] => [2,1] => [1,2] => 1 = 4 - 3
[1,2,3] => [1,2,3] => [1,2,3] => [2,2,1,1] => 11 = 14 - 3
[1,3,2] => [1,3,2] => [1,3,2] => [2,1,2,1] => 10 = 13 - 3
[2,1,3] => [2,1,3] => [2,1,3] => [1,3,1,1] => 10 = 13 - 3
[2,3,1] => [2,3,1] => [2,3,1] => [1,2,1,2] => 8 = 11 - 3
[3,1,2] => [3,1,2] => [3,1,2] => [1,1,3,1] => 8 = 11 - 3
[3,2,1] => [3,2,1] => [3,2,1] => [1,1,2,2] => 7 = 10 - 3
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [2,2,1,1,2,1,1] => ? = 29 - 3
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [2,1,2,2,1,1,1] => ? = 29 - 3
[1,3,4,2] => [1,3,4,2] => [1,3,4,2] => [2,1,2,1,1,2,1] => ? = 27 - 3
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [2,1,1,2,2,1,1] => ? = 27 - 3
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [2,1,1,2,1,2,1] => ? = 26 - 3
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [1,3,1,2,1,1,1] => ? = 29 - 3
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [1,3,1,1,2,1,1] => ? = 28 - 3
[2,3,1,4] => [2,3,1,4] => [2,3,1,4] => [1,2,1,3,1,1,1] => ? = 27 - 3
[2,3,4,1] => [2,3,4,1] => [2,3,4,1] => [1,2,1,2,1,1,2] => ? = 24 - 3
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => [1,2,1,1,3,1,1] => ? = 25 - 3
[2,4,3,1] => [2,4,3,1] => [2,4,3,1] => [1,2,1,1,2,1,2] => ? = 23 - 3
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => [1,1,3,2,1,1,1] => ? = 27 - 3
[3,1,4,2] => [3,1,4,2] => [3,1,4,2] => [1,1,3,1,1,2,1] => ? = 25 - 3
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [1,1,2,3,1,1,1] => ? = 26 - 3
[3,2,4,1] => [3,2,4,1] => [3,2,4,1] => [1,1,2,2,1,1,2] => ? = 23 - 3
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => [1,1,2,1,1,3,1] => ? = 22 - 3
[3,4,2,1] => [3,4,2,1] => [3,4,2,1] => [1,1,2,1,1,2,2] => ? = 21 - 3
[4,1,2,3] => [4,1,2,3] => [4,1,2,3] => [1,1,1,3,2,1,1] => ? = 24 - 3
[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => [1,1,1,3,1,2,1] => ? = 23 - 3
[4,2,1,3] => [4,2,1,3] => [4,2,1,3] => [1,1,1,2,3,1,1] => ? = 23 - 3
[4,2,3,1] => [4,2,3,1] => [4,2,3,1] => [1,1,1,2,2,1,2] => ? = 21 - 3
[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => [1,1,1,2,1,3,1] => ? = 21 - 3
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,2,1,2,2] => ? = 20 - 3

Description

The major index of the composition. The descents of a composition

$[c_1,c_2,\dots,c_k]$ are the partial sums

$c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$ , excluding the sum of all parts. The major index of a composition is the sum of its descents. For details about the major index see [[Permutations/Descents-Major]].

Matching statistic: St000304

Mapping

Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000304: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 38%

Values

[1,2] => [[1,0],[0,1]]
 => [[1,1],[2]]
 => [3,1,2] => 2 = 5 - 3
[2,1] => [[0,1],[1,0]]
 => [[1,2],[2]]
 => [2,1,3] => 1 = 4 - 3
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => [6,4,5,1,2,3] => 11 = 14 - 3
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => [5,4,6,1,2,3] => 10 = 13 - 3
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => [6,3,4,1,2,5] => 10 = 13 - 3
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => [4,3,5,1,2,6] => 8 = 11 - 3
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => [[1,2,2],[2,3],[3]]
 => [5,2,6,1,3,4] => 8 = 11 - 3
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => [4,2,5,1,3,6] => 7 = 10 - 3
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => [9,8,10,5,6,7,1,2,3,4] => ? = 29 - 3
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => [10,7,8,5,6,9,1,2,3,4] => ? = 29 - 3
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,1],[2,2,4],[3,4],[4]]
 => [8,7,9,5,6,10,1,2,3,4] => ? = 27 - 3
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,1],[2,3,3],[3,4],[4]]
 => [9,6,10,5,7,8,1,2,3,4] => ? = 27 - 3
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,1],[2,3,4],[3,4],[4]]
 => [8,6,9,5,7,10,1,2,3,4] => ? = 26 - 3
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => [10,8,9,4,5,6,1,2,3,7] => ? = 29 - 3
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => [9,8,10,4,5,6,1,2,3,7] => ? = 28 - 3
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,3],[2,2,3],[3,3],[4]]
 => [10,6,7,4,5,8,1,2,3,9] => ? = 27 - 3
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,4],[2,2,4],[3,4],[4]]
 => [7,6,8,4,5,9,1,2,3,10] => ? = 24 - 3
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => [[1,1,1,3],[2,3,3],[3,4],[4]]
 => [9,5,10,4,6,7,1,2,3,8] => ? = 25 - 3
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,4],[2,3,4],[3,4],[4]]
 => [7,5,8,4,6,9,1,2,3,10] => ? = 23 - 3
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,2],[2,2,3],[3,3],[4]]
 => [10,7,8,3,4,9,1,2,5,6] => ? = 27 - 3
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,2],[2,2,4],[3,4],[4]]
 => [8,7,9,3,4,10,1,2,5,6] => ? = 25 - 3
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,3],[2,2,3],[3,3],[4]]
 => [10,6,7,3,4,8,1,2,5,9] => ? = 26 - 3
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,4],[2,2,4],[3,4],[4]]
 => [7,6,8,3,4,9,1,2,5,10] => ? = 23 - 3
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,3],[2,3,4],[3,4],[4]]
 => [8,4,9,3,5,10,1,2,6,7] => ? = 22 - 3
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => [[1,1,3,4],[2,3,4],[3,4],[4]]
 => [7,4,8,3,5,9,1,2,6,10] => ? = 21 - 3
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,2],[2,3,3],[3,4],[4]]
 => [9,6,10,2,7,8,1,3,4,5] => ? = 24 - 3
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,2],[2,3,4],[3,4],[4]]
 => [8,6,9,2,7,10,1,3,4,5] => ? = 23 - 3
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => [[1,2,2,3],[2,3,3],[3,4],[4]]
 => [9,5,10,2,6,7,1,3,4,8] => ? = 23 - 3
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => [[1,2,2,4],[2,3,4],[3,4],[4]]
 => [7,5,8,2,6,9,1,3,4,10] => ? = 21 - 3
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
 => [[1,2,3,3],[2,3,4],[3,4],[4]]
 => [8,4,9,2,5,10,1,3,6,7] => ? = 21 - 3
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => [[1,2,3,4],[2,3,4],[3,4],[4]]
 => [7,4,8,2,5,9,1,3,6,10] => ? = 20 - 3

Description

The load of a permutation. The definition of the load of a finite word in a totally ordered alphabet can be found in [1], for permutations, it is given by the major index [[St000004]] of the reverse [[Mp00064]] of the inverse [[Mp00066]] permutation.