- FindStat

Your data matches 160 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000225
(load all 3 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => 0
[1,2] => [1,2] => [1,1]
 => 0
[2,1] => [1,2] => [1,1]
 => 0
[1,2,3] => [1,2,3] => [1,1,1]
 => 0
[1,3,2] => [1,2,3] => [1,1,1]
 => 0
[2,1,3] => [1,2,3] => [1,1,1]
 => 0
[2,3,1] => [1,2,3] => [1,1,1]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 0
[1,2,4,3] => [1,2,3,4] => [1,1,1,1]
 => 0
[1,3,2,4] => [1,2,3,4] => [1,1,1,1]
 => 0
[1,3,4,2] => [1,2,3,4] => [1,1,1,1]
 => 0
[2,1,3,4] => [1,2,3,4] => [1,1,1,1]
 => 0
[2,1,4,3] => [1,2,3,4] => [1,1,1,1]
 => 0
[2,3,1,4] => [1,2,3,4] => [1,1,1,1]
 => 0
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
 => 0
[4,1,2,3] => [1,4,3,2] => [3,1]
 => 2
[4,2,1,3] => [1,4,3,2] => [3,1]
 => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,5,2,3,4] => [1,2,5,4,3] => [3,1,1]
 => 2
[1,5,3,2,4] => [1,2,5,4,3] => [3,1,1]
 => 2
[2,1,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,5,1,3,4] => [1,2,5,4,3] => [3,1,1]
 => 2
[2,5,3,1,4] => [1,2,5,4,3] => [3,1,1]
 => 2
[3,1,5,2,4] => [1,3,5,4,2] => [3,1,1]
 => 2
[3,1,5,4,2] => [1,3,5,2,4] => [2,1,1,1]
 => 1
[3,2,5,1,4] => [1,3,5,4,2] => [3,1,1]
 => 2
[3,2,5,4,1] => [1,3,5,2,4] => [2,1,1,1]
 => 1
[3,4,5,1,2] => [1,3,5,2,4] => [2,1,1,1]
 => 1
[3,4,5,2,1] => [1,3,5,2,4] => [2,1,1,1]
 => 1
[4,1,2,3,5] => [1,4,3,2,5] => [3,1,1]
 => 2
[4,1,2,5,3] => [1,4,5,3,2] => [3,1,1]
 => 2
[4,1,5,3,2] => [1,4,3,5,2] => [3,1,1]
 => 2
[4,2,1,3,5] => [1,4,3,2,5] => [3,1,1]
 => 2
[4,2,1,5,3] => [1,4,5,3,2] => [3,1,1]
 => 2
[4,2,5,3,1] => [1,4,3,5,2] => [3,1,1]
 => 2
[4,5,1,2,3] => [1,4,2,5,3] => [2,2,1]
 => 1

Description

Difference between largest and smallest parts in a partition.

Matching statistic: St000319
(load all 2 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => 0
[1,2] => [1,2] => [1,1]
 => 0
[2,1] => [1,2] => [1,1]
 => 0
[1,2,3] => [1,2,3] => [1,1,1]
 => 0
[1,3,2] => [1,2,3] => [1,1,1]
 => 0
[2,1,3] => [1,2,3] => [1,1,1]
 => 0
[2,3,1] => [1,2,3] => [1,1,1]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 0
[1,2,4,3] => [1,2,3,4] => [1,1,1,1]
 => 0
[1,3,2,4] => [1,2,3,4] => [1,1,1,1]
 => 0
[1,3,4,2] => [1,2,3,4] => [1,1,1,1]
 => 0
[2,1,3,4] => [1,2,3,4] => [1,1,1,1]
 => 0
[2,1,4,3] => [1,2,3,4] => [1,1,1,1]
 => 0
[2,3,1,4] => [1,2,3,4] => [1,1,1,1]
 => 0
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
 => 0
[4,1,2,3] => [1,4,3,2] => [3,1]
 => 2
[4,2,1,3] => [1,4,3,2] => [3,1]
 => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,5,2,3,4] => [1,2,5,4,3] => [3,1,1]
 => 2
[1,5,3,2,4] => [1,2,5,4,3] => [3,1,1]
 => 2
[2,1,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,5,1,3,4] => [1,2,5,4,3] => [3,1,1]
 => 2
[2,5,3,1,4] => [1,2,5,4,3] => [3,1,1]
 => 2
[3,1,5,2,4] => [1,3,5,4,2] => [3,1,1]
 => 2
[3,1,5,4,2] => [1,3,5,2,4] => [2,1,1,1]
 => 1
[3,2,5,1,4] => [1,3,5,4,2] => [3,1,1]
 => 2
[3,2,5,4,1] => [1,3,5,2,4] => [2,1,1,1]
 => 1
[3,4,5,1,2] => [1,3,5,2,4] => [2,1,1,1]
 => 1
[3,4,5,2,1] => [1,3,5,2,4] => [2,1,1,1]
 => 1
[4,1,2,3,5] => [1,4,3,2,5] => [3,1,1]
 => 2
[4,1,2,5,3] => [1,4,5,3,2] => [3,1,1]
 => 2
[4,1,5,3,2] => [1,4,3,5,2] => [3,1,1]
 => 2
[4,2,1,3,5] => [1,4,3,2,5] => [3,1,1]
 => 2
[4,2,1,5,3] => [1,4,5,3,2] => [3,1,1]
 => 2
[4,2,5,3,1] => [1,4,3,5,2] => [3,1,1]
 => 2
[4,5,1,2,3] => [1,4,2,5,3] => [2,2,1]
 => 1

Description

The spin of an integer partition. The Ferrers shape of an integer partition

$\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of

$\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let

$\lambda = (5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$ The first strip

$(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses

$4$ times, the second strip

$(4,3,3,1) \setminus (2,2)$ crosses

$3$ times, the strip

$(2,2) \setminus (1)$ crosses

$1$ time, and the remaining strip

$(1) \setminus ()$ does not cross. This yields the spin of

$(5,5,4,4,2,1)$ to be

$4+3+1 = 8$ .

Matching statistic: St000320
(load all 2 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => 0
[1,2] => [1,2] => [1,1]
 => 0
[2,1] => [1,2] => [1,1]
 => 0
[1,2,3] => [1,2,3] => [1,1,1]
 => 0
[1,3,2] => [1,2,3] => [1,1,1]
 => 0
[2,1,3] => [1,2,3] => [1,1,1]
 => 0
[2,3,1] => [1,2,3] => [1,1,1]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 0
[1,2,4,3] => [1,2,3,4] => [1,1,1,1]
 => 0
[1,3,2,4] => [1,2,3,4] => [1,1,1,1]
 => 0
[1,3,4,2] => [1,2,3,4] => [1,1,1,1]
 => 0
[2,1,3,4] => [1,2,3,4] => [1,1,1,1]
 => 0
[2,1,4,3] => [1,2,3,4] => [1,1,1,1]
 => 0
[2,3,1,4] => [1,2,3,4] => [1,1,1,1]
 => 0
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
 => 0
[4,1,2,3] => [1,4,3,2] => [3,1]
 => 2
[4,2,1,3] => [1,4,3,2] => [3,1]
 => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,5,2,3,4] => [1,2,5,4,3] => [3,1,1]
 => 2
[1,5,3,2,4] => [1,2,5,4,3] => [3,1,1]
 => 2
[2,1,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[2,5,1,3,4] => [1,2,5,4,3] => [3,1,1]
 => 2
[2,5,3,1,4] => [1,2,5,4,3] => [3,1,1]
 => 2
[3,1,5,2,4] => [1,3,5,4,2] => [3,1,1]
 => 2
[3,1,5,4,2] => [1,3,5,2,4] => [2,1,1,1]
 => 1
[3,2,5,1,4] => [1,3,5,4,2] => [3,1,1]
 => 2
[3,2,5,4,1] => [1,3,5,2,4] => [2,1,1,1]
 => 1
[3,4,5,1,2] => [1,3,5,2,4] => [2,1,1,1]
 => 1
[3,4,5,2,1] => [1,3,5,2,4] => [2,1,1,1]
 => 1
[4,1,2,3,5] => [1,4,3,2,5] => [3,1,1]
 => 2
[4,1,2,5,3] => [1,4,5,3,2] => [3,1,1]
 => 2
[4,1,5,3,2] => [1,4,3,5,2] => [3,1,1]
 => 2
[4,2,1,3,5] => [1,4,3,2,5] => [3,1,1]
 => 2
[4,2,1,5,3] => [1,4,5,3,2] => [3,1,1]
 => 2
[4,2,5,3,1] => [1,4,3,5,2] => [3,1,1]
 => 2
[4,5,1,2,3] => [1,4,2,5,3] => [2,2,1]
 => 1

Description

The dinv adjustment of an integer partition. The Ferrers shape of an integer partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For

$0 \leq j < \lambda_1$ let

$n_j$ be the length of the border strip starting at

$(\lambda_1-j,0)$ . The dinv adjustment is then defined by

$\sum_{j:n_j > 0}(\lambda_1-1-j).$ The following example is taken from Appendix B in [2]: Let

$\lambda=(5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$ and we obtain

$(n_0,\ldots,n_4) = (10,7,0,3,1)$ . The dinv adjustment is thus

$4+3+1+0 = 8$ .

Matching statistic: St000010
(load all 3 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000097
(load all 5 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The order of the largest clique of the graph. A clique in a graph

$G$ is a subset

$U \subseteq V(G)$ such that any pair of vertices in

$U$ are adjacent. I.e. the subgraph induced by

$U$ is a complete graph.

Matching statistic: St000098
(load all 5 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.

Matching statistic: St000147
(load all 3 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest part of an integer partition.

Matching statistic: St001029
(load all 5 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001029: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of the core of a graph. The core of the graph

$G$ is the smallest graph

$C$ such that there is a graph homomorphism from

$G$ to

$C$ and a graph homomorphism from

$C$ to

$G$ .

Matching statistic: St000157
(load all 2 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => [[1]]
 => 0
[1,2] => [1,2] => [2]
 => [[1,2]]
 => 0
[2,1] => [1,2] => [2]
 => [[1,2]]
 => 0
[1,2,3] => [1,2,3] => [3]
 => [[1,2,3]]
 => 0
[1,3,2] => [1,2,3] => [3]
 => [[1,2,3]]
 => 0
[2,1,3] => [1,2,3] => [3]
 => [[1,2,3]]
 => 0
[2,3,1] => [1,2,3] => [3]
 => [[1,2,3]]
 => 0
[1,2,3,4] => [1,2,3,4] => [4]
 => [[1,2,3,4]]
 => 0
[1,2,4,3] => [1,2,3,4] => [4]
 => [[1,2,3,4]]
 => 0
[1,3,2,4] => [1,2,3,4] => [4]
 => [[1,2,3,4]]
 => 0
[1,3,4,2] => [1,2,3,4] => [4]
 => [[1,2,3,4]]
 => 0
[2,1,3,4] => [1,2,3,4] => [4]
 => [[1,2,3,4]]
 => 0
[2,1,4,3] => [1,2,3,4] => [4]
 => [[1,2,3,4]]
 => 0
[2,3,1,4] => [1,2,3,4] => [4]
 => [[1,2,3,4]]
 => 0
[2,3,4,1] => [1,2,3,4] => [4]
 => [[1,2,3,4]]
 => 0
[4,1,2,3] => [1,4,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[4,2,1,3] => [1,4,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => 0
[1,3,2,4,5] => [1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => 0
[1,5,2,3,4] => [1,2,5,4,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 2
[1,5,3,2,4] => [1,2,5,4,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 2
[2,1,3,4,5] => [1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => 0
[2,1,3,5,4] => [1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => 0
[2,1,4,3,5] => [1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => 0
[2,1,4,5,3] => [1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => 0
[2,3,1,4,5] => [1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => 0
[2,3,1,5,4] => [1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => 0
[2,3,4,1,5] => [1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => 0
[2,3,4,5,1] => [1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => 0
[2,5,1,3,4] => [1,2,5,4,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 2
[2,5,3,1,4] => [1,2,5,4,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 2
[3,1,5,2,4] => [1,3,5,4,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 2
[3,1,5,4,2] => [1,3,5,2,4] => [3,2]
 => [[1,2,3],[4,5]]
 => 1
[3,2,5,1,4] => [1,3,5,4,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 2
[3,2,5,4,1] => [1,3,5,2,4] => [3,2]
 => [[1,2,3],[4,5]]
 => 1
[3,4,5,1,2] => [1,3,5,2,4] => [3,2]
 => [[1,2,3],[4,5]]
 => 1
[3,4,5,2,1] => [1,3,5,2,4] => [3,2]
 => [[1,2,3],[4,5]]
 => 1
[4,1,2,3,5] => [1,4,3,2,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 2
[4,1,2,5,3] => [1,4,5,3,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 2
[4,1,5,3,2] => [1,4,3,5,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 2
[4,2,1,3,5] => [1,4,3,2,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 2
[4,2,1,5,3] => [1,4,5,3,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 2
[4,2,5,3,1] => [1,4,3,5,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 2
[4,5,1,2,3] => [1,4,2,5,3] => [3,2]
 => [[1,2,3],[4,5]]
 => 1

Description

The number of descents of a standard tableau. Entry

$i$ of a standard Young tableau is a descent if

$i+1$ appears in a row below the row of

$i$ .

Matching statistic: St000272
(load all 2 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The treewidth of a graph. A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.

The following 150 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000536The pathwidth of a graph. St000741The Colin de Verdière graph invariant. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001777The number of weak descents in an integer composition. St001962The proper pathwidth of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000288The number of ones in a binary word. St000378The diagonal inversion number of an integer partition. St000676The number of odd rises of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000820The number of compositions obtained by rotating the composition. St000822The Hadwiger number of the graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000062The length of the longest increasing subsequence of the permutation. St000744The length of the path to the largest entry in a standard Young tableau. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001323The independence gap of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001644The dimension of a graph. St000528The height of a poset. St000527The width of the poset. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000308The height of the tree associated to a permutation. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000021The number of descents of a permutation. St000080The rank of the poset. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000362The size of a minimal vertex cover of a graph. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001812The biclique partition number of a graph. St001949The rigidity index of a graph. St000087The number of induced subgraphs. St000172The Grundy number of a graph. St000286The number of connected components of the complement of a graph. St000325The width of the tree associated to a permutation. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000926The clique-coclique number of a graph. St000991The number of right-to-left minima of a permutation. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001581The achromatic number of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000632The jump number of the poset. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000307The number of rowmotion orbits of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000100The number of linear extensions of a poset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001960The number of descents of a permutation minus one if its first entry is not one. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001964The interval resolution global dimension of a poset. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001896The number of right descents of a signed permutations. St001720The minimal length of a chain of small intervals in a lattice. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001864The number of excedances of a signed permutation. St001894The depth of a signed permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001095The number of non-isomorphic posets with precisely one further covering relation. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001862The number of crossings of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001893The flag descent of a signed permutation. St001490The number of connected components of a skew partition. St000635The number of strictly order preserving maps of a poset into itself. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001892The flag excedance statistic of a signed permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.