- FindStat

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

Matching statistic: St001491

Mapping

Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2]
 => 0 => 1 => 1 => 1
[2,1]
 => 01 => 10 => 01 => 1
[4]
 => 0 => 1 => 1 => 1
[2,2]
 => 00 => 11 => 11 => 2
[2,1,1]
 => 011 => 100 => 001 => 1
[4,1]
 => 01 => 10 => 01 => 1
[3,2]
 => 10 => 01 => 10 => 1
[2,2,1]
 => 001 => 110 => 101 => 2
[2,1,1,1]
 => 0111 => 1000 => 0001 => 1
[6]
 => 0 => 1 => 1 => 1
[4,2]
 => 00 => 11 => 11 => 2
[4,1,1]
 => 011 => 100 => 001 => 1
[3,2,1]
 => 101 => 010 => 100 => 1
[2,2,2]
 => 000 => 111 => 111 => 3
[2,2,1,1]
 => 0011 => 1100 => 1001 => 2
[6,1]
 => 01 => 10 => 01 => 1
[5,2]
 => 10 => 01 => 10 => 1
[4,3]
 => 01 => 10 => 01 => 1
[4,2,1]
 => 001 => 110 => 101 => 2
[4,1,1,1]
 => 0111 => 1000 => 0001 => 1
[3,2,2]
 => 100 => 011 => 110 => 1
[3,2,1,1]
 => 1011 => 0100 => 1000 => 1
[2,2,2,1]
 => 0001 => 1110 => 1101 => 2
[8]
 => 0 => 1 => 1 => 1
[6,2]
 => 00 => 11 => 11 => 2
[6,1,1]
 => 011 => 100 => 001 => 1
[5,2,1]
 => 101 => 010 => 100 => 1
[4,4]
 => 00 => 11 => 11 => 2
[4,3,1]
 => 011 => 100 => 001 => 1
[4,2,2]
 => 000 => 111 => 111 => 3
[4,2,1,1]
 => 0011 => 1100 => 1001 => 2
[3,3,2]
 => 110 => 001 => 010 => 1
[3,2,2,1]
 => 1001 => 0110 => 1100 => 1
[2,2,2,2]
 => 0000 => 1111 => 1111 => 4
[8,1]
 => 01 => 10 => 01 => 1
[7,2]
 => 10 => 01 => 10 => 1
[6,3]
 => 01 => 10 => 01 => 1
[6,2,1]
 => 001 => 110 => 101 => 2
[6,1,1,1]
 => 0111 => 1000 => 0001 => 1
[5,4]
 => 10 => 01 => 10 => 1
[5,2,2]
 => 100 => 011 => 110 => 1
[5,2,1,1]
 => 1011 => 0100 => 1000 => 1
[4,4,1]
 => 001 => 110 => 101 => 2
[4,3,2]
 => 010 => 101 => 011 => 1
[4,3,1,1]
 => 0111 => 1000 => 0001 => 1
[4,2,2,1]
 => 0001 => 1110 => 1101 => 2
[3,3,2,1]
 => 1101 => 0010 => 0100 => 1
[3,2,2,2]
 => 1000 => 0111 => 1110 => 2
[10]
 => 0 => 1 => 1 => 1
[8,2]
 => 00 => 11 => 11 => 2

Description

The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let

$A_n=K[x]/(x^n)$ . We associate to a nonempty subset S of an (n-1)-set the module

$M_S$ , which is the direct sum of

$A_n$ -modules with indecomposable non-projective direct summands of dimension

$i$ when

$i$ is in

$S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of

$M_S$ . We decode the subset as a binary word so that for example the subset

$S=\{1,3 \}$ of

$\{1,2,3 \}$ is decoded as 101.

Matching statistic: St000237
(load all 2 compositions to match this statistic)

Mapping

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 80%

Values

[2]
 => [[1,2]]
 => [1,2] => [2,1] => 1
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => [2,1,3] => 1
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => [4,3,2,1] => 1
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [2,1,4,3] => 2
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [2,1,3,4] => 1
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => [4,3,2,1,5] => 1
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => [3,2,1,5,4] => 1
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [2,1,4,3,5] => 2
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [2,1,3,4,5] => 1
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 1
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [4,3,2,1,6,5] => 2
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => [4,3,2,1,5,6] => 1
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => [3,2,1,5,4,6] => 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => [2,1,4,3,6,5] => 3
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => [2,1,4,3,5,6] => 2
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => [6,5,4,3,2,1,7] => 1
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => [5,4,3,2,1,7,6] => 1
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => [4,3,2,1,7,6,5] => ? = 1
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => [4,3,2,1,6,5,7] => ? = 2
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => [4,3,2,1,5,6,7] => 1
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => [3,2,1,5,4,7,6] => ? = 1
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => [3,2,1,5,4,6,7] => ? = 1
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => [2,1,4,3,6,5,7] => ? = 2
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => 1
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7] => 2
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => [8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8] => 1
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8] => ? = 1
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => 2
[4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => [8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8] => ? = 1
[4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7] => 3
[4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => [8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8] => ? = 2
[3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => [7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7] => ? = 1
[3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => [8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8] => ? = 1
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => 4
[8,1]
 => [[1,2,3,4,5,6,7,8],[9]]
 => [9,1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1,9] => 1
[7,2]
 => [[1,2,3,4,5,6,7],[8,9]]
 => [8,9,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,9,8] => ? = 1
[6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => [7,8,9,1,2,3,4,5,6] => [6,5,4,3,2,1,9,8,7] => ? = 1
[6,2,1]
 => [[1,2,3,4,5,6],[7,8],[9]]
 => [9,7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7,9] => ? = 2
[6,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9]]
 => [9,8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8,9] => 1
[5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => [6,7,8,9,1,2,3,4,5] => [5,4,3,2,1,9,8,7,6] => ? = 1
[5,2,2]
 => [[1,2,3,4,5],[6,7],[8,9]]
 => [8,9,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,9,8] => ? = 1
[5,2,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9]]
 => [9,8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8,9] => ? = 1
[4,4,1]
 => [[1,2,3,4],[5,6,7,8],[9]]
 => [9,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,9] => ? = 2
[4,3,2]
 => [[1,2,3,4],[5,6,7],[8,9]]
 => [8,9,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,9,8] => ? = 1
[4,3,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9]]
 => [9,8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8,9] => ? = 1
[4,2,2,1]
 => [[1,2,3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7,9] => ? = 2
[3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9]]
 => [9,7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7,9] => ? = 1
[3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => [8,9,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,9,8] => ? = 2
[10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => [1,2,3,4,5,6,7,8,9,10] => [10,9,8,7,6,5,4,3,2,1] => 1
[8,2]
 => [[1,2,3,4,5,6,7,8],[9,10]]
 => [9,10,1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1,10,9] => 2
[8,1,1]
 => [[1,2,3,4,5,6,7,8],[9],[10]]
 => [10,9,1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1,9,10] => 1
[7,2,1]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => [10,8,9,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,9,8,10] => ? = 1
[6,4]
 => [[1,2,3,4,5,6],[7,8,9,10]]
 => [7,8,9,10,1,2,3,4,5,6] => [6,5,4,3,2,1,10,9,8,7] => 2
[6,3,1]
 => [[1,2,3,4,5,6],[7,8,9],[10]]
 => [10,7,8,9,1,2,3,4,5,6] => [6,5,4,3,2,1,9,8,7,10] => ? = 1
[6,2,2]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => [9,10,7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7,10,9] => 3
[6,2,1,1]
 => [[1,2,3,4,5,6],[7,8],[9],[10]]
 => [10,9,7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7,9,10] => ? = 2
[5,4,1]
 => [[1,2,3,4,5],[6,7,8,9],[10]]
 => [10,6,7,8,9,1,2,3,4,5] => [5,4,3,2,1,9,8,7,6,10] => ? = 1
[5,3,2]
 => [[1,2,3,4,5],[6,7,8],[9,10]]
 => [9,10,6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6,10,9] => ? = 1
[5,2,2,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10]]
 => [10,8,9,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,9,8,10] => ? = 1
[4,4,2]
 => [[1,2,3,4],[5,6,7,8],[9,10]]
 => [9,10,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,10,9] => 3
[4,4,1,1]
 => [[1,2,3,4],[5,6,7,8],[9],[10]]
 => [10,9,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,9,10] => ? = 2
[4,3,3]
 => [[1,2,3,4],[5,6,7],[8,9,10]]
 => [8,9,10,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,10,9,8] => ? = 1
[4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,9,8,10] => ? = 0
[4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => [9,10,7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7,10,9] => 4
[3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [9,10,7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7,10,9] => ? = 2
[10,1]
 => [[1,2,3,4,5,6,7,8,9,10],[11]]
 => ? => ? => ? = 1
[9,2]
 => [[1,2,3,4,5,6,7,8,9],[10,11]]
 => ? => ? => ? = 1
[8,3]
 => [[1,2,3,4,5,6,7,8],[9,10,11]]
 => ? => ? => ? = 1
[8,2,1]
 => [[1,2,3,4,5,6,7,8],[9,10],[11]]
 => ? => ? => ? = 2
[8,1,1,1]
 => [[1,2,3,4,5,6,7,8],[9],[10],[11]]
 => ? => ? => ? = 1
[7,4]
 => [[1,2,3,4,5,6,7],[8,9,10,11]]
 => ? => ? => ? = 1
[7,2,2]
 => [[1,2,3,4,5,6,7],[8,9],[10,11]]
 => ? => ? => ? = 1
[7,2,1,1]
 => [[1,2,3,4,5,6,7],[8,9],[10],[11]]
 => ? => ? => ? = 1
[6,5]
 => [[1,2,3,4,5,6],[7,8,9,10,11]]
 => ? => ? => ? = 1
[6,4,1]
 => [[1,2,3,4,5,6],[7,8,9,10],[11]]
 => ? => ? => ? = 2
[6,3,2]
 => [[1,2,3,4,5,6],[7,8,9],[10,11]]
 => ? => ? => ? = 1
[6,3,1,1]
 => [[1,2,3,4,5,6],[7,8,9],[10],[11]]
 => ? => ? => ? = 1
[6,2,2,1]
 => [[1,2,3,4,5,6],[7,8],[9,10],[11]]
 => ? => ? => ? = 2
[5,4,2]
 => [[1,2,3,4,5],[6,7,8,9],[10,11]]
 => [10,11,6,7,8,9,1,2,3,4,5] => [5,4,3,2,1,9,8,7,6,11,10] => ? = 1
[5,4,1,1]
 => [[1,2,3,4,5],[6,7,8,9],[10],[11]]
 => [11,10,6,7,8,9,1,2,3,4,5] => [5,4,3,2,1,9,8,7,6,10,11] => ? = 1
[5,3,2,1]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11]]
 => [11,9,10,6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6,10,9,11] => ? = 1
[5,2,2,2]
 => [[1,2,3,4,5],[6,7],[8,9],[10,11]]
 => [10,11,8,9,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,9,8,11,10] => ? = 2
[4,4,3]
 => [[1,2,3,4],[5,6,7,8],[9,10,11]]
 => [9,10,11,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,11,10,9] => ? = 2
[12]
 => [[1,2,3,4,5,6,7,8,9,10,11,12]]
 => [1,2,3,4,5,6,7,8,9,10,11,12] => [12,11,10,9,8,7,6,5,4,3,2,1] => 1
[6,6]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12]]
 => [7,8,9,10,11,12,1,2,3,4,5,6] => [6,5,4,3,2,1,12,11,10,9,8,7] => 2
[6,4,2]
 => [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
 => [11,12,7,8,9,10,1,2,3,4,5,6] => [6,5,4,3,2,1,10,9,8,7,12,11] => 3
[4,4,2,2]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
 => [11,12,9,10,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,10,9,12,11] => 4

Description

The number of small exceedances. This is the number of indices

$i$ such that

$\pi_i=i+1$ .

Matching statistic: St001465

Mapping

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 80%

Values

[2]
 => [[1,2]]
 => [1,2] => [2,1] => 1
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => [2,1,3] => 1
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => [4,3,2,1] => 1
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [2,1,4,3] => 2
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [2,1,3,4] => 1
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => [4,3,2,1,5] => 1
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => [3,2,1,5,4] => 1
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [2,1,4,3,5] => 2
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [2,1,3,4,5] => 1
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 1
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [4,3,2,1,6,5] => 2
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => [4,3,2,1,5,6] => 1
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => [3,2,1,5,4,6] => 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => [2,1,4,3,6,5] => 3
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => [2,1,4,3,5,6] => 2
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => [6,5,4,3,2,1,7] => 1
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => [5,4,3,2,1,7,6] => 1
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => [4,3,2,1,7,6,5] => ? = 1
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => [4,3,2,1,6,5,7] => ? = 2
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => [4,3,2,1,5,6,7] => 1
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => [3,2,1,5,4,7,6] => ? = 1
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => [3,2,1,5,4,6,7] => ? = 1
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => [2,1,4,3,6,5,7] => ? = 2
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => 1
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7] => 2
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => [8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8] => 1
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8] => ? = 1
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => 2
[4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => [8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8] => ? = 1
[4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7] => 3
[4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => [8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8] => ? = 2
[3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => [7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7] => ? = 1
[3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => [8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8] => ? = 1
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => 4
[8,1]
 => [[1,2,3,4,5,6,7,8],[9]]
 => [9,1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1,9] => 1
[7,2]
 => [[1,2,3,4,5,6,7],[8,9]]
 => [8,9,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,9,8] => ? = 1
[6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => [7,8,9,1,2,3,4,5,6] => [6,5,4,3,2,1,9,8,7] => ? = 1
[6,2,1]
 => [[1,2,3,4,5,6],[7,8],[9]]
 => [9,7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7,9] => ? = 2
[6,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9]]
 => [9,8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8,9] => 1
[5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => [6,7,8,9,1,2,3,4,5] => [5,4,3,2,1,9,8,7,6] => ? = 1
[5,2,2]
 => [[1,2,3,4,5],[6,7],[8,9]]
 => [8,9,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,9,8] => ? = 1
[5,2,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9]]
 => [9,8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8,9] => ? = 1
[4,4,1]
 => [[1,2,3,4],[5,6,7,8],[9]]
 => [9,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,9] => ? = 2
[4,3,2]
 => [[1,2,3,4],[5,6,7],[8,9]]
 => [8,9,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,9,8] => ? = 1
[4,3,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9]]
 => [9,8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8,9] => ? = 1
[4,2,2,1]
 => [[1,2,3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7,9] => ? = 2
[3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9]]
 => [9,7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7,9] => ? = 1
[3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => [8,9,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,9,8] => ? = 2
[10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => [1,2,3,4,5,6,7,8,9,10] => [10,9,8,7,6,5,4,3,2,1] => 1
[8,2]
 => [[1,2,3,4,5,6,7,8],[9,10]]
 => [9,10,1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1,10,9] => 2
[8,1,1]
 => [[1,2,3,4,5,6,7,8],[9],[10]]
 => [10,9,1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1,9,10] => 1
[7,2,1]
 => [[1,2,3,4,5,6,7],[8,9],[10]]
 => [10,8,9,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,9,8,10] => ? = 1
[6,4]
 => [[1,2,3,4,5,6],[7,8,9,10]]
 => [7,8,9,10,1,2,3,4,5,6] => [6,5,4,3,2,1,10,9,8,7] => 2
[6,3,1]
 => [[1,2,3,4,5,6],[7,8,9],[10]]
 => [10,7,8,9,1,2,3,4,5,6] => [6,5,4,3,2,1,9,8,7,10] => ? = 1
[6,2,2]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => [9,10,7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7,10,9] => 3
[6,2,1,1]
 => [[1,2,3,4,5,6],[7,8],[9],[10]]
 => [10,9,7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7,9,10] => ? = 2
[5,4,1]
 => [[1,2,3,4,5],[6,7,8,9],[10]]
 => [10,6,7,8,9,1,2,3,4,5] => [5,4,3,2,1,9,8,7,6,10] => ? = 1
[5,3,2]
 => [[1,2,3,4,5],[6,7,8],[9,10]]
 => [9,10,6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6,10,9] => ? = 1
[5,2,2,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10]]
 => [10,8,9,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,9,8,10] => ? = 1
[4,4,2]
 => [[1,2,3,4],[5,6,7,8],[9,10]]
 => [9,10,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,10,9] => 3
[4,4,1,1]
 => [[1,2,3,4],[5,6,7,8],[9],[10]]
 => [10,9,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,9,10] => ? = 2
[4,3,3]
 => [[1,2,3,4],[5,6,7],[8,9,10]]
 => [8,9,10,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,10,9,8] => ? = 1
[4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,9,8,10] => ? = 0
[4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => [9,10,7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7,10,9] => 4
[3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [9,10,7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7,10,9] => ? = 2
[10,1]
 => [[1,2,3,4,5,6,7,8,9,10],[11]]
 => ? => ? => ? = 1
[9,2]
 => [[1,2,3,4,5,6,7,8,9],[10,11]]
 => ? => ? => ? = 1
[8,3]
 => [[1,2,3,4,5,6,7,8],[9,10,11]]
 => ? => ? => ? = 1
[8,2,1]
 => [[1,2,3,4,5,6,7,8],[9,10],[11]]
 => ? => ? => ? = 2
[8,1,1,1]
 => [[1,2,3,4,5,6,7,8],[9],[10],[11]]
 => ? => ? => ? = 1
[7,4]
 => [[1,2,3,4,5,6,7],[8,9,10,11]]
 => ? => ? => ? = 1
[7,2,2]
 => [[1,2,3,4,5,6,7],[8,9],[10,11]]
 => ? => ? => ? = 1
[7,2,1,1]
 => [[1,2,3,4,5,6,7],[8,9],[10],[11]]
 => ? => ? => ? = 1
[6,5]
 => [[1,2,3,4,5,6],[7,8,9,10,11]]
 => ? => ? => ? = 1
[6,4,1]
 => [[1,2,3,4,5,6],[7,8,9,10],[11]]
 => ? => ? => ? = 2
[6,3,2]
 => [[1,2,3,4,5,6],[7,8,9],[10,11]]
 => ? => ? => ? = 1
[6,3,1,1]
 => [[1,2,3,4,5,6],[7,8,9],[10],[11]]
 => ? => ? => ? = 1
[6,2,2,1]
 => [[1,2,3,4,5,6],[7,8],[9,10],[11]]
 => ? => ? => ? = 2
[5,4,2]
 => [[1,2,3,4,5],[6,7,8,9],[10,11]]
 => [10,11,6,7,8,9,1,2,3,4,5] => [5,4,3,2,1,9,8,7,6,11,10] => ? = 1
[5,4,1,1]
 => [[1,2,3,4,5],[6,7,8,9],[10],[11]]
 => [11,10,6,7,8,9,1,2,3,4,5] => [5,4,3,2,1,9,8,7,6,10,11] => ? = 1
[5,3,2,1]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11]]
 => [11,9,10,6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6,10,9,11] => ? = 1
[5,2,2,2]
 => [[1,2,3,4,5],[6,7],[8,9],[10,11]]
 => [10,11,8,9,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,9,8,11,10] => ? = 2
[4,4,3]
 => [[1,2,3,4],[5,6,7,8],[9,10,11]]
 => [9,10,11,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,11,10,9] => ? = 2
[12]
 => [[1,2,3,4,5,6,7,8,9,10,11,12]]
 => [1,2,3,4,5,6,7,8,9,10,11,12] => [12,11,10,9,8,7,6,5,4,3,2,1] => 1
[6,6]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12]]
 => [7,8,9,10,11,12,1,2,3,4,5,6] => [6,5,4,3,2,1,12,11,10,9,8,7] => 2
[6,4,2]
 => [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
 => [11,12,7,8,9,10,1,2,3,4,5,6] => [6,5,4,3,2,1,10,9,8,7,12,11] => 3
[4,4,2,2]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
 => [11,12,9,10,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,10,9,12,11] => 4

Description

The number of adjacent transpositions in the cycle decomposition of a permutation.