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Matching statistic: St001176
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,3] => [1,3,2] => [2,1]
=> [1]
=> 0
[1,2,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[1,3,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[1,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[1,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[1,4,3,2] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[2,1,3,4] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[2,1,4,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[2,3,1,4] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[2,4,1,3] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[3,1,2,4] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[3,1,4,2] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[3,2,1,4] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[3,2,4,1] => [1,2,4,3] => [3,1]
=> [1]
=> 0
[4,1,3,2] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[4,2,1,3] => [1,3,2,4] => [3,1]
=> [1]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 0
[1,2,4,5,3] => [1,2,4,5,3] => [4,1]
=> [1]
=> 0
[1,2,5,3,4] => [1,2,5,3,4] => [4,1]
=> [1]
=> 0
[1,2,5,4,3] => [1,2,5,3,4] => [4,1]
=> [1]
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> [1]
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 0
[1,3,4,2,5] => [1,3,4,2,5] => [4,1]
=> [1]
=> 0
[1,3,4,5,2] => [1,3,4,5,2] => [4,1]
=> [1]
=> 0
[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 0
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 0
[1,4,2,3,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> 0
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 0
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
=> [2]
=> 0
[1,4,3,5,2] => [1,4,2,3,5] => [4,1]
=> [1]
=> 0
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> 0
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 0
[1,5,2,3,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> 0
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [1,5,2,3,4] => [4,1]
=> [1]
=> 0
[1,5,4,2,3] => [1,5,2,3,4] => [4,1]
=> [1]
=> 0
[1,5,4,3,2] => [1,5,2,3,4] => [4,1]
=> [1]
=> 0
[2,1,3,4,5] => [1,3,4,5,2] => [4,1]
=> [1]
=> 0
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 0
[2,1,4,3,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> 0
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> 0
[2,1,5,3,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> 0
[2,1,5,4,3] => [1,5,2,3,4] => [4,1]
=> [1]
=> 0
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> 0
[2,3,1,5,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> 0
[2,3,4,1,5] => [1,5,2,3,4] => [4,1]
=> [1]
=> 0
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000319
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,3,2] => [1,3,2] => [2,1]
=> [2,1]
=> 1 = 0 + 1
[2,1,3] => [1,3,2] => [2,1]
=> [2,1]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[1,3,4,2] => [1,3,4,2] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[1,4,2,3] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[1,4,3,2] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[2,1,3,4] => [1,3,4,2] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[2,1,4,3] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[2,3,1,4] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[3,1,2,4] => [1,2,4,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[3,1,4,2] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[3,2,1,4] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[3,2,4,1] => [1,2,4,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[4,2,1,3] => [1,3,2,4] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[1,5,2,3,4] => [1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> [3,1,1]
=> 2 = 1 + 1
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
=> [3,1,1]
=> 2 = 1 + 1
[1,5,3,4,2] => [1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,5,4,2,3] => [1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,5,4,3,2] => [1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[2,1,3,4,5] => [1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[2,1,4,3,5] => [1,4,2,3,5] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[2,1,5,3,4] => [1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[2,1,5,4,3] => [1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[2,3,1,5,4] => [1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[2,3,4,1,5] => [1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,11,10] => [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[2,1,3,4,5,6,7,8,9,10,11] => [1,3,4,5,6,7,8,9,10,11,2] => [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[10,1,2,3,4,5,6,7,8,11,9] => [1,2,3,4,5,6,7,8,11,9,10] => [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[10,9,8,7,6,5,4,3,2,1,11] => [1,11,2,3,4,5,6,7,8,9,10] => [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,11,10,9,8,7,6,5,4,3,2] => [1,11,2,3,4,5,6,7,8,9,10] => [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[10,1,2,3,4,5,6,7,8,9,11] => [1,2,3,4,5,6,7,8,9,11,10] => [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,11,2,3,4,5,6,7,8,9,10] => [1,11,2,3,4,5,6,7,8,9,10] => [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[2,1,8,9,10,11,12,7,6,5,4,3] => [1,8,9,10,11,12,2,3,4,5,6,7] => [7,5]
=> [2,2,2,2,2,1,1]
=> ? = 0 + 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
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,3,2] => [1,3,2] => [2,1]
=> [2,1]
=> 1 = 0 + 1
[2,1,3] => [1,3,2] => [2,1]
=> [2,1]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[1,3,4,2] => [1,3,4,2] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[1,4,2,3] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[1,4,3,2] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[2,1,3,4] => [1,3,4,2] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[2,1,4,3] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[2,3,1,4] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[3,1,2,4] => [1,2,4,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[3,1,4,2] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[3,2,1,4] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[3,2,4,1] => [1,2,4,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[4,2,1,3] => [1,3,2,4] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[1,5,2,3,4] => [1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> [3,1,1]
=> 2 = 1 + 1
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
=> [3,1,1]
=> 2 = 1 + 1
[1,5,3,4,2] => [1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,5,4,2,3] => [1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,5,4,3,2] => [1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[2,1,3,4,5] => [1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[2,1,4,3,5] => [1,4,2,3,5] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[2,1,5,3,4] => [1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[2,1,5,4,3] => [1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2,2,1]
=> 1 = 0 + 1
[2,3,1,5,4] => [1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[2,3,4,1,5] => [1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,11,10] => [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[2,1,3,4,5,6,7,8,9,10,11] => [1,3,4,5,6,7,8,9,10,11,2] => [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[10,1,2,3,4,5,6,7,8,11,9] => [1,2,3,4,5,6,7,8,11,9,10] => [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[10,9,8,7,6,5,4,3,2,1,11] => [1,11,2,3,4,5,6,7,8,9,10] => [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,11,10,9,8,7,6,5,4,3,2] => [1,11,2,3,4,5,6,7,8,9,10] => [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[10,1,2,3,4,5,6,7,8,9,11] => [1,2,3,4,5,6,7,8,9,11,10] => [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,11,2,3,4,5,6,7,8,9,10] => [1,11,2,3,4,5,6,7,8,9,10] => [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[2,1,8,9,10,11,12,7,6,5,4,3] => [1,8,9,10,11,12,2,3,4,5,6,7] => [7,5]
=> [2,2,2,2,2,1,1]
=> ? = 0 + 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: St000097
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[1,3,2] => [1,3,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[2,1,3] => [1,3,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,3,4,2] => [1,3,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,4,2,3] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,4,3,2] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[2,1,3,4] => [1,3,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[2,1,4,3] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[2,3,1,4] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[2,4,1,3] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[3,1,2,4] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[3,1,4,2] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[3,2,1,4] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[3,2,4,1] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[4,1,3,2] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[4,2,1,3] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,2,5,4,3] => [1,2,5,3,4] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,4,3,5,2] => [1,4,2,3,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,4,5,3,2] => [1,4,5,2,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,5,2,4,3] => [1,5,2,4,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[1,5,3,2,4] => [1,5,2,4,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[1,5,3,4,2] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,5,4,2,3] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,5,4,3,2] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,1,3,4,5] => [1,3,4,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,1,3,5,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,1,4,3,5] => [1,4,2,3,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,1,4,5,3] => [1,4,5,2,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,1,5,3,4] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,1,5,4,3] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,3,1,4,5] => [1,4,5,2,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,3,1,5,4] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,3,4,1,5] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[5,4,3,2,6,7,8,1] => [1,2,6,7,8,3,4,5] => [2,6,7,8,3,4,5,1] => ?
=> ? = 0 + 3
[3,4,5,2,6,7,8,1] => [1,2,6,7,8,3,4,5] => [2,6,7,8,3,4,5,1] => ?
=> ? = 0 + 3
[5,4,6,3,7,8,1,2] => [1,2,3,7,8,4,6,5] => [2,3,6,8,7,4,5,1] => ?
=> ? = 1 + 3
[6,4,3,5,7,8,1,2] => [1,2,3,5,7,8,4,6] => [2,3,7,4,8,5,6,1] => ?
=> ? = 0 + 3
[5,4,3,6,7,8,1,2] => [1,2,3,6,7,8,4,5] => [2,3,7,8,4,5,6,1] => ?
=> ? = 0 + 3
[4,5,3,6,7,8,1,2] => [1,2,3,6,7,8,4,5] => [2,3,7,8,4,5,6,1] => ?
=> ? = 0 + 3
[6,7,5,8,3,1,2,4] => [1,2,4,3,5,8,6,7] => [2,4,3,5,7,8,6,1] => ?
=> ? = 0 + 3
[6,5,7,8,2,1,3,4] => [1,3,4,2,5,7,8,6] => [4,2,3,5,8,6,7,1] => ?
=> ? = 0 + 3
[7,8,5,4,2,1,3,6] => [1,3,6,2,4,5,7,8] => [4,2,5,6,3,7,8,1] => ?
=> ? = 0 + 3
[8,7,5,2,1,3,4,6] => [1,3,4,6,2,5,7,8] => [5,2,3,6,4,7,8,1] => ?
=> ? = 0 + 3
[7,8,5,2,1,3,4,6] => [1,3,4,6,2,5,7,8] => [5,2,3,6,4,7,8,1] => ?
=> ? = 0 + 3
[7,8,4,2,1,3,5,6] => [1,3,5,6,2,4,7,8] => [5,2,6,3,4,7,8,1] => ?
=> ? = 0 + 3
[8,7,3,2,1,4,5,6] => [1,4,5,6,2,3,7,8] => [5,6,2,3,4,7,8,1] => ?
=> ? = 0 + 3
[7,8,3,2,1,4,5,6] => [1,4,5,6,2,3,7,8] => [5,6,2,3,4,7,8,1] => ?
=> ? = 0 + 3
[8,3,1,2,4,5,6,7] => [1,2,4,5,6,7,3,8] => [2,7,3,4,5,6,8,1] => ?
=> ? = 0 + 3
[6,5,7,4,2,1,3,8] => [1,3,8,2,4,5,7,6] => [4,2,5,6,8,7,3,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 3
[7,5,4,6,2,1,3,8] => [1,3,8,2,4,6,5,7] => [4,2,5,7,6,8,3,1] => ?
=> ? = 1 + 3
[7,6,4,3,2,1,5,8] => [1,5,8,2,3,4,6,7] => [4,5,6,2,7,8,3,1] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[7,5,3,1,2,4,6,8] => [1,2,4,6,8,3,5,7] => [2,6,3,7,4,8,5,1] => ?
=> ? = 0 + 3
[7,4,3,2,1,5,6,8] => [1,5,6,8,2,3,4,7] => [5,6,7,2,3,8,4,1] => ([(0,6),(0,7),(1,3),(1,4),(1,5),(1,7),(2,3),(2,4),(2,5),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[7,3,1,2,4,5,6,8] => [1,2,4,5,6,8,3,7] => [2,7,3,4,5,8,6,1] => ?
=> ? = 0 + 3
[6,5,4,3,2,1,7,8] => [1,7,8,2,3,4,5,6] => [4,5,6,7,8,2,3,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[4,2,3,5,6,1,7,8] => [1,7,8,2,3,5,6,4] => [4,5,8,6,7,2,3,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 3
[2,3,4,5,6,1,7,8] => [1,7,8,2,3,4,5,6] => [4,5,6,7,8,2,3,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[5,4,6,3,1,2,7,8] => [1,2,7,8,3,4,6,5] => [2,5,6,8,7,3,4,1] => ?
=> ? = 1 + 3
[5,6,4,2,1,3,7,8] => [1,3,7,8,2,4,5,6] => [5,2,6,7,8,3,4,1] => ?
=> ? = 0 + 3
[4,5,6,1,2,3,7,8] => [1,2,3,7,8,4,5,6] => [2,3,6,7,8,4,5,1] => ?
=> ? = 0 + 3
[5,6,3,2,1,4,7,8] => [1,4,7,8,2,3,5,6] => [5,6,2,7,8,3,4,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[6,4,3,2,1,5,7,8] => [1,5,7,8,2,3,4,6] => [5,6,7,2,8,3,4,1] => ?
=> ? = 0 + 3
[6,4,3,1,2,5,7,8] => [1,2,5,7,8,3,4,6] => [2,6,7,3,8,4,5,1] => ?
=> ? = 0 + 3
[6,4,1,2,3,5,7,8] => [1,2,3,5,7,8,4,6] => [2,3,7,4,8,5,6,1] => ?
=> ? = 0 + 3
[6,2,3,1,4,5,7,8] => [1,4,5,7,8,2,3,6] => [6,7,2,3,8,4,5,1] => ?
=> ? = 0 + 3
[3,2,1,4,5,6,7,8] => [1,4,5,6,7,8,2,3] => [7,8,2,3,4,5,6,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[3,5,6,8,7,4,2,1] => [1,2,3,5,6,8,4,7] => [2,3,7,4,5,8,6,1] => ?
=> ? = 0 + 3
[2,6,8,7,5,4,3,1] => [1,2,6,8,3,4,5,7] => [2,5,6,7,3,8,4,1] => ?
=> ? = 0 + 3
[2,4,6,8,7,5,3,1] => [1,2,4,6,8,3,5,7] => [2,6,3,7,4,8,5,1] => ?
=> ? = 0 + 3
[2,7,6,5,4,8,3,1] => [1,2,7,3,4,8,5,6] => [2,4,5,7,8,3,6,1] => ?
=> ? = 0 + 3
[2,3,5,6,8,7,4,1] => [1,2,3,5,6,8,4,7] => [2,3,7,4,5,8,6,1] => ?
=> ? = 0 + 3
[3,4,2,5,7,8,6,1] => [1,2,5,7,8,3,4,6] => [2,6,7,3,8,4,5,1] => ?
=> ? = 0 + 3
[5,4,3,2,6,8,7,1] => [1,2,6,8,3,4,5,7] => [2,5,6,7,3,8,4,1] => ?
=> ? = 0 + 3
[3,4,2,6,7,5,8,1] => [1,2,6,7,3,4,5,8] => [2,5,6,7,3,4,8,1] => ?
=> ? = 0 + 3
[3,2,5,4,7,6,8,1] => [1,2,5,3,4,7,6,8] => [2,4,5,3,7,6,8,1] => ?
=> ? = 0 + 3
[1,7,8,6,5,4,3,2] => [1,7,8,2,3,4,5,6] => [4,5,6,7,8,2,3,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[1,6,8,7,5,4,3,2] => [1,6,8,2,3,4,5,7] => [4,5,6,7,2,8,3,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[1,5,8,7,6,4,3,2] => [1,5,8,2,3,4,6,7] => [4,5,6,2,7,8,3,1] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[1,4,8,7,6,5,3,2] => [1,4,8,2,3,5,6,7] => [4,5,2,6,7,8,3,1] => ?
=> ? = 0 + 3
[1,4,7,8,6,5,3,2] => [1,4,7,8,2,3,5,6] => [5,6,2,7,8,3,4,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[1,3,5,7,8,6,4,2] => [1,3,5,7,8,2,4,6] => [6,2,7,3,8,4,5,1] => ([(0,6),(0,7),(1,4),(1,5),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[1,3,6,5,7,8,4,2] => [1,3,6,2,4,5,7,8] => [4,2,5,6,3,7,8,1] => ?
=> ? = 0 + 3
[1,3,5,6,7,8,4,2] => [1,3,5,6,7,8,2,4] => [7,2,8,3,4,5,6,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
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: St000093
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Values
[1,3,2] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[2,1,3] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,3,4,2] => [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,4,3,2] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,1,3,4] => [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,3,1,4] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,1,2,4] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,1,4,2] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,2,1,4] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,2,4,1] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,5,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,3,5,2] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,5,3,2] => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,5,2,3,4] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,5,2,4,3] => [1,5,2,4,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,5,3,2,4] => [1,5,2,4,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,5,3,4,2] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,5,4,2,3] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,5,4,3,2] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,3,4,5] => [1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,3,5,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,4,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,4,5,3] => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,5,3,4] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,5,4,3] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,1,4,5] => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,1,5,4] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,4,1,5] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[5,4,6,3,7,2,8,1] => [1,2,8,3,7,4,6,5] => [5,6,4,7,3,8,2,1] => ?
=> ? = 2 + 2
[7,6,5,3,2,4,8,1] => [1,2,4,8,3,5,6,7] => [7,6,5,3,8,4,2,1] => ?
=> ? = 0 + 2
[5,4,6,3,7,8,1,2] => [1,2,3,7,8,4,6,5] => [5,6,4,8,7,3,2,1] => ?
=> ? = 1 + 2
[5,4,3,6,7,8,1,2] => [1,2,3,6,7,8,4,5] => [5,4,8,7,6,3,2,1] => ?
=> ? = 0 + 2
[4,5,3,6,7,8,1,2] => [1,2,3,6,7,8,4,5] => [5,4,8,7,6,3,2,1] => ?
=> ? = 0 + 2
[6,5,7,8,2,1,3,4] => [1,3,4,2,5,7,8,6] => [6,8,7,5,2,4,3,1] => ?
=> ? = 0 + 2
[8,7,6,4,2,1,3,5] => [1,3,5,2,4,6,7,8] => [8,7,6,4,2,5,3,1] => ?
=> ? = 0 + 2
[6,7,8,4,2,1,3,5] => [1,3,5,2,4,6,7,8] => [8,7,6,4,2,5,3,1] => ?
=> ? = 0 + 2
[8,7,6,2,1,3,4,5] => [1,3,4,5,2,6,7,8] => [8,7,6,2,5,4,3,1] => ?
=> ? = 0 + 2
[6,7,8,2,1,3,4,5] => [1,3,4,5,2,6,7,8] => [8,7,6,2,5,4,3,1] => ?
=> ? = 0 + 2
[7,8,5,4,2,1,3,6] => [1,3,6,2,4,5,7,8] => [8,7,5,4,2,6,3,1] => ?
=> ? = 0 + 2
[8,7,4,3,2,1,5,6] => [1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ?
=> ? = 0 + 2
[7,8,4,3,2,1,5,6] => [1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ?
=> ? = 0 + 2
[7,8,2,3,4,1,5,6] => [1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ?
=> ? = 0 + 2
[8,7,3,2,1,4,5,6] => [1,4,5,6,2,3,7,8] => [8,7,3,2,6,5,4,1] => ?
=> ? = 0 + 2
[7,8,3,2,1,4,5,6] => [1,4,5,6,2,3,7,8] => [8,7,3,2,6,5,4,1] => ?
=> ? = 0 + 2
[7,8,2,1,3,4,5,6] => [1,3,4,5,6,2,7,8] => [8,7,2,6,5,4,3,1] => ?
=> ? = 0 + 2
[8,6,5,2,3,1,4,7] => [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ?
=> ? = 0 + 2
[8,5,6,2,3,1,4,7] => [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ?
=> ? = 0 + 2
[7,5,4,3,6,2,1,8] => [1,8,2,3,6,4,5,7] => [7,5,4,6,3,2,8,1] => ?
=> ? = 1 + 2
[7,4,5,2,3,6,1,8] => [1,8,2,3,6,4,5,7] => [7,5,4,6,3,2,8,1] => ?
=> ? = 1 + 2
[5,4,3,2,6,7,1,8] => [1,8,2,6,7,3,4,5] => [5,4,3,7,6,2,8,1] => ([(0,7),(1,2),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 2
[3,2,4,5,6,7,1,8] => [1,8,2,4,5,6,7,3] => [3,7,6,5,4,2,8,1] => ?
=> ? = 1 + 2
[7,6,5,4,2,1,3,8] => [1,3,8,2,4,5,6,7] => [7,6,5,4,2,8,3,1] => ?
=> ? = 0 + 2
[6,5,7,4,2,1,3,8] => [1,3,8,2,4,5,7,6] => [6,7,5,4,2,8,3,1] => ?
=> ? = 1 + 2
[7,5,4,6,2,1,3,8] => [1,3,8,2,4,6,5,7] => [7,5,6,4,2,8,3,1] => ?
=> ? = 1 + 2
[7,4,5,6,2,1,3,8] => [1,3,8,2,4,5,6,7] => [7,6,5,4,2,8,3,1] => ?
=> ? = 0 + 2
[4,5,6,7,2,1,3,8] => [1,3,8,2,4,5,6,7] => [7,6,5,4,2,8,3,1] => ?
=> ? = 0 + 2
[5,6,7,3,1,2,4,8] => [1,2,4,8,3,5,6,7] => [7,6,5,3,8,4,2,1] => ?
=> ? = 0 + 2
[5,6,7,2,1,3,4,8] => [1,3,4,8,2,5,6,7] => [7,6,5,2,8,4,3,1] => ?
=> ? = 0 + 2
[7,6,4,3,2,1,5,8] => [1,5,8,2,3,4,6,7] => [7,6,4,3,2,8,5,1] => ([(0,1),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[7,4,3,2,1,5,6,8] => [1,5,6,8,2,3,4,7] => [7,4,3,2,8,6,5,1] => ?
=> ? = 0 + 2
[7,2,1,3,4,5,6,8] => [1,3,4,5,6,8,2,7] => [7,2,8,6,5,4,3,1] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,5,4,3,2,1,7,8] => [1,7,8,2,3,4,5,6] => [6,5,4,3,2,8,7,1] => ([(0,1),(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[4,2,3,5,6,1,7,8] => [1,7,8,2,3,5,6,4] => [4,6,5,3,2,8,7,1] => ([(0,1),(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 2
[2,3,4,5,6,1,7,8] => [1,7,8,2,3,4,5,6] => [6,5,4,3,2,8,7,1] => ([(0,1),(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[5,4,3,6,1,2,7,8] => [1,2,7,8,3,6,4,5] => [5,4,6,3,8,7,2,1] => ?
=> ? = 1 + 2
[5,6,4,2,1,3,7,8] => [1,3,7,8,2,4,5,6] => [6,5,4,2,8,7,3,1] => ?
=> ? = 0 + 2
[4,5,6,1,2,3,7,8] => [1,2,3,7,8,4,5,6] => [6,5,4,8,7,3,2,1] => ?
=> ? = 0 + 2
[5,6,3,2,1,4,7,8] => [1,4,7,8,2,3,5,6] => [6,5,3,2,8,7,4,1] => ([(0,1),(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,4,3,2,1,5,7,8] => [1,5,7,8,2,3,4,6] => [6,4,3,2,8,7,5,1] => ([(0,1),(0,5),(0,7),(1,5),(1,7),(2,3),(2,4),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,4,3,1,2,5,7,8] => [1,2,5,7,8,3,4,6] => [6,4,3,8,7,5,2,1] => ?
=> ? = 0 + 2
[6,2,3,1,4,5,7,8] => [1,4,5,7,8,2,3,6] => [6,3,2,8,7,5,4,1] => ([(0,1),(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[5,4,3,2,1,6,7,8] => [1,6,7,8,2,3,4,5] => [5,4,3,2,8,7,6,1] => ([(0,1),(0,2),(0,7),(1,2),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[4,5,2,3,1,6,7,8] => [1,6,7,8,2,3,4,5] => [5,4,3,2,8,7,6,1] => ([(0,1),(0,2),(0,7),(1,2),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[5,2,3,4,1,6,7,8] => [1,6,7,8,2,3,4,5] => [5,4,3,2,8,7,6,1] => ([(0,1),(0,2),(0,7),(1,2),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[2,3,4,5,1,6,7,8] => [1,6,7,8,2,3,4,5] => [5,4,3,2,8,7,6,1] => ([(0,1),(0,2),(0,7),(1,2),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[4,3,2,1,5,6,7,8] => [1,5,6,7,8,2,3,4] => [4,3,2,8,7,6,5,1] => ([(0,1),(0,2),(0,7),(1,2),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[3,2,4,1,5,6,7,8] => [1,5,6,7,8,2,4,3] => [3,4,2,8,7,6,5,1] => ([(0,2),(0,7),(1,2),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 2
[2,3,4,1,5,6,7,8] => [1,5,6,7,8,2,3,4] => [4,3,2,8,7,6,5,1] => ([(0,1),(0,2),(0,7),(1,2),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000098
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Values
[1,3,2] => [1,3,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[2,1,3] => [1,3,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,3,4,2] => [1,3,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,4,2,3] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,4,3,2] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[2,1,3,4] => [1,3,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[2,1,4,3] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[2,3,1,4] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[2,4,1,3] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[3,1,2,4] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[3,1,4,2] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[3,2,1,4] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[3,2,4,1] => [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[4,1,3,2] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[4,2,1,3] => [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,2,5,4,3] => [1,2,5,3,4] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,4,3,5,2] => [1,4,2,3,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,4,5,3,2] => [1,4,5,2,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,5,2,4,3] => [1,5,2,4,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[1,5,3,2,4] => [1,5,2,4,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[1,5,3,4,2] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,5,4,2,3] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,5,4,3,2] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,1,3,4,5] => [1,3,4,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,1,3,5,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,1,4,3,5] => [1,4,2,3,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,1,4,5,3] => [1,4,5,2,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,1,5,3,4] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,1,5,4,3] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,3,1,4,5] => [1,4,5,2,3] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,3,1,5,4] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,3,4,1,5] => [1,5,2,3,4] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[7,5,4,6,8,3,2,1] => [1,2,3,4,6,8,5,7] => [2,3,4,7,5,8,6,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[7,6,4,3,5,8,2,1] => [1,2,3,5,8,4,6,7] => [2,3,6,4,7,8,5,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[8,6,5,7,3,2,4,1] => [1,2,4,3,5,7,6,8] => [2,4,3,5,7,6,8,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[7,6,5,3,2,4,8,1] => [1,2,4,8,3,5,6,7] => [2,5,3,6,7,8,4,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[5,4,3,2,6,7,8,1] => [1,2,6,7,8,3,4,5] => [2,6,7,8,3,4,5,1] => ?
=> ? = 0 + 3
[3,4,5,2,6,7,8,1] => [1,2,6,7,8,3,4,5] => [2,6,7,8,3,4,5,1] => ?
=> ? = 0 + 3
[4,3,2,5,6,7,8,1] => [1,2,5,6,7,8,3,4] => [2,7,8,3,4,5,6,1] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[3,4,2,5,6,7,8,1] => [1,2,5,6,7,8,3,4] => [2,7,8,3,4,5,6,1] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[5,4,6,3,7,8,1,2] => [1,2,3,7,8,4,6,5] => [2,3,6,8,7,4,5,1] => ?
=> ? = 1 + 3
[5,6,3,4,7,8,1,2] => [1,2,3,4,7,8,5,6] => [2,3,4,7,8,5,6,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[6,4,3,5,7,8,1,2] => [1,2,3,5,7,8,4,6] => [2,3,7,4,8,5,6,1] => ?
=> ? = 0 + 3
[5,4,3,6,7,8,1,2] => [1,2,3,6,7,8,4,5] => [2,3,7,8,4,5,6,1] => ?
=> ? = 0 + 3
[4,5,3,6,7,8,1,2] => [1,2,3,6,7,8,4,5] => [2,3,7,8,4,5,6,1] => ?
=> ? = 0 + 3
[7,6,8,5,4,2,1,3] => [1,3,2,4,5,6,8,7] => [3,2,4,5,6,8,7,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[6,5,4,7,8,1,2,3] => [1,2,3,4,7,8,5,6] => [2,3,4,7,8,5,6,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[7,6,5,8,3,2,1,4] => [1,4,2,3,5,8,6,7] => [3,4,2,5,7,8,6,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[6,7,5,8,2,3,1,4] => [1,4,2,3,5,8,6,7] => [3,4,2,5,7,8,6,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[7,5,6,8,2,3,1,4] => [1,4,2,3,5,6,8,7] => [3,4,2,5,6,8,7,1] => ([(0,7),(1,7),(2,3),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[6,7,5,8,3,1,2,4] => [1,2,4,3,5,8,6,7] => [2,4,3,5,7,8,6,1] => ?
=> ? = 0 + 3
[7,5,6,8,3,1,2,4] => [1,2,4,3,5,6,8,7] => [2,4,3,5,6,8,7,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[6,5,7,8,2,1,3,4] => [1,3,4,2,5,7,8,6] => [4,2,3,5,8,6,7,1] => ?
=> ? = 0 + 3
[8,7,6,4,2,1,3,5] => [1,3,5,2,4,6,7,8] => [4,2,5,3,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[6,7,8,4,2,1,3,5] => [1,3,5,2,4,6,7,8] => [4,2,5,3,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[8,7,6,3,2,1,4,5] => [1,4,5,2,3,6,7,8] => [4,5,2,3,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[6,7,8,3,2,1,4,5] => [1,4,5,2,3,6,7,8] => [4,5,2,3,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[8,7,6,2,3,1,4,5] => [1,4,5,2,3,6,7,8] => [4,5,2,3,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[6,7,8,2,3,1,4,5] => [1,4,5,2,3,6,7,8] => [4,5,2,3,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[7,8,5,4,2,1,3,6] => [1,3,6,2,4,5,7,8] => [4,2,5,6,3,7,8,1] => ?
=> ? = 0 + 3
[8,7,5,2,1,3,4,6] => [1,3,4,6,2,5,7,8] => [5,2,3,6,4,7,8,1] => ?
=> ? = 0 + 3
[7,8,5,2,1,3,4,6] => [1,3,4,6,2,5,7,8] => [5,2,3,6,4,7,8,1] => ?
=> ? = 0 + 3
[8,7,4,3,2,1,5,6] => [1,5,6,2,3,4,7,8] => [4,5,6,2,3,7,8,1] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[7,8,4,3,2,1,5,6] => [1,5,6,2,3,4,7,8] => [4,5,6,2,3,7,8,1] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[7,8,2,3,4,1,5,6] => [1,5,6,2,3,4,7,8] => [4,5,6,2,3,7,8,1] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[7,8,3,4,1,2,5,6] => [1,2,5,6,3,4,7,8] => [2,5,6,3,4,7,8,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[7,8,4,2,1,3,5,6] => [1,3,5,6,2,4,7,8] => [5,2,6,3,4,7,8,1] => ?
=> ? = 0 + 3
[8,7,3,2,1,4,5,6] => [1,4,5,6,2,3,7,8] => [5,6,2,3,4,7,8,1] => ?
=> ? = 0 + 3
[7,8,3,2,1,4,5,6] => [1,4,5,6,2,3,7,8] => [5,6,2,3,4,7,8,1] => ?
=> ? = 0 + 3
[8,6,5,4,2,1,3,7] => [1,3,7,2,4,5,6,8] => [4,2,5,6,7,3,8,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[8,6,5,2,3,1,4,7] => [1,4,7,2,3,5,6,8] => [4,5,2,6,7,3,8,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[8,5,6,2,3,1,4,7] => [1,4,7,2,3,5,6,8] => [4,5,2,6,7,3,8,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[8,5,6,2,1,3,4,7] => [1,3,4,7,2,5,6,8] => [5,2,3,6,7,4,8,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[8,6,4,2,1,3,5,7] => [1,3,5,7,2,4,6,8] => [5,2,6,3,7,4,8,1] => ([(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
[8,4,3,2,1,5,6,7] => [1,5,6,7,2,3,4,8] => [5,6,7,2,3,4,8,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[8,2,3,4,1,5,6,7] => [1,5,6,7,2,3,4,8] => [5,6,7,2,3,4,8,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[8,3,2,1,4,5,6,7] => [1,4,5,6,7,2,3,8] => [6,7,2,3,4,5,8,1] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[8,2,3,1,4,5,6,7] => [1,4,5,6,7,2,3,8] => [6,7,2,3,4,5,8,1] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 3
[8,3,1,2,4,5,6,7] => [1,2,4,5,6,7,3,8] => [2,7,3,4,5,6,8,1] => ?
=> ? = 0 + 3
[6,5,7,4,2,1,3,8] => [1,3,8,2,4,5,7,6] => [4,2,5,6,8,7,3,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 3
[7,5,4,6,2,1,3,8] => [1,3,8,2,4,6,5,7] => [4,2,5,7,6,8,3,1] => ?
=> ? = 1 + 3
[5,6,7,3,1,2,4,8] => [1,2,4,8,3,5,6,7] => [2,5,3,6,7,8,4,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 3
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: St000786
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Values
[1,3,2] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[2,1,3] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,3,4,2] => [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,4,3,2] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,1,3,4] => [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,3,1,4] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,1,2,4] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,1,4,2] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,2,1,4] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,2,4,1] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,5,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,3,5,2] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,5,3,2] => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,5,2,3,4] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,5,2,4,3] => [1,5,2,4,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,5,3,2,4] => [1,5,2,4,3] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,5,3,4,2] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,5,4,2,3] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,5,4,3,2] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,3,4,5] => [1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,3,5,4] => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,4,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,4,5,3] => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,5,3,4] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,5,4,3] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,1,4,5] => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,1,5,4] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,4,1,5] => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[7,5,4,6,8,3,2,1] => [1,2,3,4,6,8,5,7] => [7,5,8,6,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[7,6,4,3,5,8,2,1] => [1,2,3,5,8,4,6,7] => [7,6,4,8,5,3,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[8,6,5,7,3,2,4,1] => [1,2,4,3,5,7,6,8] => [8,6,7,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[5,4,6,3,7,2,8,1] => [1,2,8,3,7,4,6,5] => [5,6,4,7,3,8,2,1] => ?
=> ? = 2 + 2
[7,6,5,3,2,4,8,1] => [1,2,4,8,3,5,6,7] => [7,6,5,3,8,4,2,1] => ?
=> ? = 0 + 2
[5,4,3,2,6,7,8,1] => [1,2,6,7,8,3,4,5] => [5,4,3,8,7,6,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[3,4,5,2,6,7,8,1] => [1,2,6,7,8,3,4,5] => [5,4,3,8,7,6,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[4,3,2,5,6,7,8,1] => [1,2,5,6,7,8,3,4] => [4,3,8,7,6,5,2,1] => ([(0,1),(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[3,4,2,5,6,7,8,1] => [1,2,5,6,7,8,3,4] => [4,3,8,7,6,5,2,1] => ([(0,1),(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[5,4,6,3,7,8,1,2] => [1,2,3,7,8,4,6,5] => [5,6,4,8,7,3,2,1] => ?
=> ? = 1 + 2
[5,6,3,4,7,8,1,2] => [1,2,3,4,7,8,5,6] => [6,5,8,7,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,4,3,5,7,8,1,2] => [1,2,3,5,7,8,4,6] => [6,4,8,7,5,3,2,1] => ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[5,4,3,6,7,8,1,2] => [1,2,3,6,7,8,4,5] => [5,4,8,7,6,3,2,1] => ?
=> ? = 0 + 2
[4,5,3,6,7,8,1,2] => [1,2,3,6,7,8,4,5] => [5,4,8,7,6,3,2,1] => ?
=> ? = 0 + 2
[7,6,8,5,4,2,1,3] => [1,3,2,4,5,6,8,7] => [7,8,6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,5,4,7,8,1,2,3] => [1,2,3,4,7,8,5,6] => [6,5,8,7,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[7,6,5,8,3,2,1,4] => [1,4,2,3,5,8,6,7] => [7,6,8,5,3,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,7,5,8,2,3,1,4] => [1,4,2,3,5,8,6,7] => [7,6,8,5,3,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[7,5,6,8,2,3,1,4] => [1,4,2,3,5,6,8,7] => [7,8,6,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,7,5,8,3,1,2,4] => [1,2,4,3,5,8,6,7] => [7,6,8,5,3,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[7,5,6,8,3,1,2,4] => [1,2,4,3,5,6,8,7] => [7,8,6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,5,7,8,2,1,3,4] => [1,3,4,2,5,7,8,6] => [6,8,7,5,2,4,3,1] => ?
=> ? = 0 + 2
[8,7,6,4,2,1,3,5] => [1,3,5,2,4,6,7,8] => [8,7,6,4,2,5,3,1] => ?
=> ? = 0 + 2
[6,7,8,4,2,1,3,5] => [1,3,5,2,4,6,7,8] => [8,7,6,4,2,5,3,1] => ?
=> ? = 0 + 2
[8,7,6,3,2,1,4,5] => [1,4,5,2,3,6,7,8] => [8,7,6,3,2,5,4,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,7,8,3,2,1,4,5] => [1,4,5,2,3,6,7,8] => [8,7,6,3,2,5,4,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[8,7,6,2,3,1,4,5] => [1,4,5,2,3,6,7,8] => [8,7,6,3,2,5,4,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,7,8,2,3,1,4,5] => [1,4,5,2,3,6,7,8] => [8,7,6,3,2,5,4,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[8,7,6,2,1,3,4,5] => [1,3,4,5,2,6,7,8] => [8,7,6,2,5,4,3,1] => ?
=> ? = 0 + 2
[6,7,8,2,1,3,4,5] => [1,3,4,5,2,6,7,8] => [8,7,6,2,5,4,3,1] => ?
=> ? = 0 + 2
[7,8,5,4,2,1,3,6] => [1,3,6,2,4,5,7,8] => [8,7,5,4,2,6,3,1] => ?
=> ? = 0 + 2
[8,7,5,2,1,3,4,6] => [1,3,4,6,2,5,7,8] => [8,7,5,2,6,4,3,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[7,8,5,2,1,3,4,6] => [1,3,4,6,2,5,7,8] => [8,7,5,2,6,4,3,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[8,7,4,3,2,1,5,6] => [1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ?
=> ? = 0 + 2
[7,8,4,3,2,1,5,6] => [1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ?
=> ? = 0 + 2
[7,8,2,3,4,1,5,6] => [1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ?
=> ? = 0 + 2
[7,8,3,4,1,2,5,6] => [1,2,5,6,3,4,7,8] => [8,7,4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[7,8,4,2,1,3,5,6] => [1,3,5,6,2,4,7,8] => [8,7,4,2,6,5,3,1] => ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[8,7,3,2,1,4,5,6] => [1,4,5,6,2,3,7,8] => [8,7,3,2,6,5,4,1] => ?
=> ? = 0 + 2
[7,8,3,2,1,4,5,6] => [1,4,5,6,2,3,7,8] => [8,7,3,2,6,5,4,1] => ?
=> ? = 0 + 2
[7,8,2,1,3,4,5,6] => [1,3,4,5,6,2,7,8] => [8,7,2,6,5,4,3,1] => ?
=> ? = 0 + 2
[8,6,5,4,2,1,3,7] => [1,3,7,2,4,5,6,8] => [8,6,5,4,2,7,3,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[8,6,5,2,3,1,4,7] => [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ?
=> ? = 0 + 2
[8,5,6,2,3,1,4,7] => [1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ?
=> ? = 0 + 2
[8,5,6,2,1,3,4,7] => [1,3,4,7,2,5,6,8] => [8,6,5,2,7,4,3,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[8,6,4,2,1,3,5,7] => [1,3,5,7,2,4,6,8] => [8,6,4,2,7,5,3,1] => ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[8,4,3,2,1,5,6,7] => [1,5,6,7,2,3,4,8] => [8,4,3,2,7,6,5,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[8,2,3,4,1,5,6,7] => [1,5,6,7,2,3,4,8] => [8,4,3,2,7,6,5,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[8,3,2,1,4,5,6,7] => [1,4,5,6,7,2,3,8] => [8,3,2,7,6,5,4,1] => ([(0,1),(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[8,2,3,1,4,5,6,7] => [1,4,5,6,7,2,3,8] => [8,3,2,7,6,5,4,1] => ([(0,1),(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0 + 2
Description
The maximal number of occurrences of a colour in a proper colouring of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
Matching statistic: St001336
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001336: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001336: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Values
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> 0
[2,1,3] => [1,3,2] => ([(1,2)],3)
=> 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
[1,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
[1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
[2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[2,3,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
=> 0
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> 0
[3,1,2,4] => [1,2,4,3] => ([(2,3)],4)
=> 0
[3,1,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[3,2,4,1] => [1,2,4,3] => ([(2,3)],4)
=> 0
[4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
=> 0
[4,2,1,3] => [1,3,2,4] => ([(2,3)],4)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 0
[1,2,4,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
[1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
[1,2,5,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0
[1,3,4,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 0
[1,3,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,3,5,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 0
[1,3,5,4,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 0
[1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
[1,4,2,5,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 0
[1,4,3,2,5] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 0
[1,4,3,5,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
[1,4,5,2,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[1,4,5,3,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[1,5,2,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,5,2,4,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,3,2,4] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,3,4,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,5,4,2,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,1,3,5,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 0
[2,1,4,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
[2,1,4,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[2,1,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,1,5,4,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,3,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[2,3,1,5,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,3,4,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[7,6,8,5,4,3,2,1] => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 0
[7,6,5,8,4,3,2,1] => [1,2,3,4,5,8,6,7] => ([(5,7),(6,7)],8)
=> ? = 0
[7,5,6,8,4,3,2,1] => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 0
[6,5,7,8,4,3,2,1] => [1,2,3,4,5,7,8,6] => ([(5,7),(6,7)],8)
=> ? = 0
[7,6,5,4,8,3,2,1] => [1,2,3,4,8,5,6,7] => ([(4,7),(5,7),(6,7)],8)
=> ? = 0
[6,5,7,4,8,3,2,1] => [1,2,3,4,8,5,7,6] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[7,6,4,5,8,3,2,1] => [1,2,3,4,5,8,6,7] => ([(5,7),(6,7)],8)
=> ? = 0
[7,5,4,6,8,3,2,1] => [1,2,3,4,6,8,5,7] => ([(4,7),(5,6),(6,7)],8)
=> ? = 0
[7,4,5,6,8,3,2,1] => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 0
[8,7,5,4,3,6,2,1] => [1,2,3,6,4,5,7,8] => ([(5,7),(6,7)],8)
=> ? = 0
[7,6,5,4,3,8,2,1] => [1,2,3,8,4,5,6,7] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0
[6,4,5,7,3,8,2,1] => [1,2,3,8,4,5,7,6] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[5,4,6,7,3,8,2,1] => [1,2,3,8,4,6,7,5] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[4,5,6,7,3,8,2,1] => [1,2,3,8,4,5,6,7] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0
[7,6,5,3,4,8,2,1] => [1,2,3,4,8,5,6,7] => ([(4,7),(5,7),(6,7)],8)
=> ? = 0
[7,6,4,3,5,8,2,1] => [1,2,3,5,8,4,6,7] => ([(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 0
[7,6,3,4,5,8,2,1] => [1,2,3,4,5,8,6,7] => ([(5,7),(6,7)],8)
=> ? = 0
[6,3,4,5,7,8,2,1] => [1,2,3,4,5,7,8,6] => ([(5,7),(6,7)],8)
=> ? = 0
[4,3,5,6,7,8,2,1] => [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0
[8,7,6,5,3,2,4,1] => [1,2,4,3,5,6,7,8] => ([(6,7)],8)
=> ? = 0
[8,6,5,7,3,2,4,1] => [1,2,4,3,5,7,6,8] => ([(4,7),(5,6)],8)
=> ? = 0
[5,6,7,8,3,2,4,1] => [1,2,4,3,5,6,7,8] => ([(6,7)],8)
=> ? = 0
[7,8,3,2,4,5,6,1] => [1,2,4,5,6,3,7,8] => ([(4,7),(5,7),(6,7)],8)
=> ? = 0
[7,6,5,4,3,2,8,1] => [1,2,8,3,4,5,6,7] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0
[6,7,5,4,3,2,8,1] => [1,2,8,3,4,5,6,7] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0
[6,5,7,4,3,2,8,1] => [1,2,8,3,4,5,7,6] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[6,4,5,7,3,2,8,1] => [1,2,8,3,4,5,7,6] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[4,5,6,7,3,2,8,1] => [1,2,8,3,4,5,6,7] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0
[6,5,4,3,7,2,8,1] => [1,2,8,3,7,4,5,6] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[5,4,6,3,7,2,8,1] => [1,2,8,3,7,4,6,5] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[7,6,5,4,2,3,8,1] => [1,2,3,8,4,5,6,7] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0
[6,7,4,5,2,3,8,1] => [1,2,3,8,4,5,6,7] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0
[6,4,5,7,2,3,8,1] => [1,2,3,8,4,5,7,6] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[4,5,6,7,2,3,8,1] => [1,2,3,8,4,5,6,7] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0
[7,6,5,3,2,4,8,1] => [1,2,4,8,3,5,6,7] => ([(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 0
[7,6,5,2,3,4,8,1] => [1,2,3,4,8,5,6,7] => ([(4,7),(5,7),(6,7)],8)
=> ? = 0
[6,7,2,3,4,5,8,1] => [1,2,3,4,5,8,6,7] => ([(5,7),(6,7)],8)
=> ? = 0
[5,4,3,2,6,7,8,1] => [1,2,6,7,8,3,4,5] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0
[3,4,5,2,6,7,8,1] => [1,2,6,7,8,3,4,5] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0
[5,2,3,4,6,7,8,1] => [1,2,3,4,6,7,8,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 0
[4,3,2,5,6,7,8,1] => [1,2,5,6,7,8,3,4] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 0
[3,4,2,5,6,7,8,1] => [1,2,5,6,7,8,3,4] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 0
[4,2,3,5,6,7,8,1] => [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0
[3,2,4,5,6,7,8,1] => [1,2,4,5,6,7,8,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0
[6,5,7,8,4,3,1,2] => [1,2,3,4,5,7,8,6] => ([(5,7),(6,7)],8)
=> ? = 0
[7,6,5,8,3,4,1,2] => [1,2,3,4,5,8,6,7] => ([(5,7),(6,7)],8)
=> ? = 0
[6,7,5,8,3,4,1,2] => [1,2,3,4,5,8,6,7] => ([(5,7),(6,7)],8)
=> ? = 0
[7,5,6,8,3,4,1,2] => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 0
[6,5,7,8,3,4,1,2] => [1,2,3,4,5,7,8,6] => ([(5,7),(6,7)],8)
=> ? = 0
[7,8,5,4,3,6,1,2] => [1,2,3,6,4,5,7,8] => ([(5,7),(6,7)],8)
=> ? = 0
Description
The minimal number of vertices in a graph whose complement is triangle-free.
Matching statistic: St001029
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001029: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001029: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Values
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> 2 = 0 + 2
[2,1,3] => [1,3,2] => ([(1,2)],3)
=> 2 = 0 + 2
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2 = 0 + 2
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 2 = 0 + 2
[1,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,3,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
=> 2 = 0 + 2
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> 2 = 0 + 2
[3,1,2,4] => [1,2,4,3] => ([(2,3)],4)
=> 2 = 0 + 2
[3,1,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,2,4,1] => [1,2,4,3] => ([(2,3)],4)
=> 2 = 0 + 2
[4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
=> 2 = 0 + 2
[4,2,1,3] => [1,3,2,4] => ([(2,3)],4)
=> 2 = 0 + 2
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 2 = 0 + 2
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 2 = 0 + 2
[1,2,4,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,5,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 2 = 0 + 2
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2 = 0 + 2
[1,3,4,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,5,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,3,5,4,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,2,5,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,4,3,2,5] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,4,3,5,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,5,2,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,4,5,3,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,5,2,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,5,2,4,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,5,3,2,4] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,5,3,4,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,5,4,2,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,3,5,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[2,1,4,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,4,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 0 + 2
[2,1,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,5,4,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 0 + 2
[2,3,1,5,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,4,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[7,6,8,5,4,3,2,1] => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 0 + 2
[7,6,5,8,4,3,2,1] => [1,2,3,4,5,8,6,7] => ([(5,7),(6,7)],8)
=> ? = 0 + 2
[7,5,6,8,4,3,2,1] => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 0 + 2
[6,5,7,8,4,3,2,1] => [1,2,3,4,5,7,8,6] => ([(5,7),(6,7)],8)
=> ? = 0 + 2
[7,6,5,4,8,3,2,1] => [1,2,3,4,8,5,6,7] => ([(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,5,7,4,8,3,2,1] => [1,2,3,4,8,5,7,6] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 2
[7,6,4,5,8,3,2,1] => [1,2,3,4,5,8,6,7] => ([(5,7),(6,7)],8)
=> ? = 0 + 2
[7,5,4,6,8,3,2,1] => [1,2,3,4,6,8,5,7] => ([(4,7),(5,6),(6,7)],8)
=> ? = 0 + 2
[7,4,5,6,8,3,2,1] => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 0 + 2
[8,7,5,4,3,6,2,1] => [1,2,3,6,4,5,7,8] => ([(5,7),(6,7)],8)
=> ? = 0 + 2
[7,6,5,4,3,8,2,1] => [1,2,3,8,4,5,6,7] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,4,5,7,3,8,2,1] => [1,2,3,8,4,5,7,6] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 2
[5,4,6,7,3,8,2,1] => [1,2,3,8,4,6,7,5] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 2
[4,5,6,7,3,8,2,1] => [1,2,3,8,4,5,6,7] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[7,6,5,3,4,8,2,1] => [1,2,3,4,8,5,6,7] => ([(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[7,6,4,3,5,8,2,1] => [1,2,3,5,8,4,6,7] => ([(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 0 + 2
[7,6,3,4,5,8,2,1] => [1,2,3,4,5,8,6,7] => ([(5,7),(6,7)],8)
=> ? = 0 + 2
[6,3,4,5,7,8,2,1] => [1,2,3,4,5,7,8,6] => ([(5,7),(6,7)],8)
=> ? = 0 + 2
[4,3,5,6,7,8,2,1] => [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[8,7,6,5,3,2,4,1] => [1,2,4,3,5,6,7,8] => ([(6,7)],8)
=> ? = 0 + 2
[8,6,5,7,3,2,4,1] => [1,2,4,3,5,7,6,8] => ([(4,7),(5,6)],8)
=> ? = 0 + 2
[5,6,7,8,3,2,4,1] => [1,2,4,3,5,6,7,8] => ([(6,7)],8)
=> ? = 0 + 2
[7,8,3,2,4,5,6,1] => [1,2,4,5,6,3,7,8] => ([(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[7,6,5,4,3,2,8,1] => [1,2,8,3,4,5,6,7] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,7,5,4,3,2,8,1] => [1,2,8,3,4,5,6,7] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,5,7,4,3,2,8,1] => [1,2,8,3,4,5,7,6] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 2
[6,4,5,7,3,2,8,1] => [1,2,8,3,4,5,7,6] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 2
[4,5,6,7,3,2,8,1] => [1,2,8,3,4,5,6,7] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,5,4,3,7,2,8,1] => [1,2,8,3,7,4,5,6] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 2
[5,4,6,3,7,2,8,1] => [1,2,8,3,7,4,6,5] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 2
[7,6,5,4,2,3,8,1] => [1,2,3,8,4,5,6,7] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,7,4,5,2,3,8,1] => [1,2,3,8,4,5,6,7] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,4,5,7,2,3,8,1] => [1,2,3,8,4,5,7,6] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 2
[4,5,6,7,2,3,8,1] => [1,2,3,8,4,5,6,7] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[7,6,5,3,2,4,8,1] => [1,2,4,8,3,5,6,7] => ([(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 0 + 2
[7,6,5,2,3,4,8,1] => [1,2,3,4,8,5,6,7] => ([(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,7,2,3,4,5,8,1] => [1,2,3,4,5,8,6,7] => ([(5,7),(6,7)],8)
=> ? = 0 + 2
[5,4,3,2,6,7,8,1] => [1,2,6,7,8,3,4,5] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0 + 2
[3,4,5,2,6,7,8,1] => [1,2,6,7,8,3,4,5] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0 + 2
[5,2,3,4,6,7,8,1] => [1,2,3,4,6,7,8,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[4,3,2,5,6,7,8,1] => [1,2,5,6,7,8,3,4] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 0 + 2
[3,4,2,5,6,7,8,1] => [1,2,5,6,7,8,3,4] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 0 + 2
[4,2,3,5,6,7,8,1] => [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[3,2,4,5,6,7,8,1] => [1,2,4,5,6,7,8,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,5,7,8,4,3,1,2] => [1,2,3,4,5,7,8,6] => ([(5,7),(6,7)],8)
=> ? = 0 + 2
[7,6,5,8,3,4,1,2] => [1,2,3,4,5,8,6,7] => ([(5,7),(6,7)],8)
=> ? = 0 + 2
[6,7,5,8,3,4,1,2] => [1,2,3,4,5,8,6,7] => ([(5,7),(6,7)],8)
=> ? = 0 + 2
[7,5,6,8,3,4,1,2] => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 0 + 2
[6,5,7,8,3,4,1,2] => [1,2,3,4,5,7,8,6] => ([(5,7),(6,7)],8)
=> ? = 0 + 2
[7,8,5,4,3,6,1,2] => [1,2,3,6,4,5,7,8] => ([(5,7),(6,7)],8)
=> ? = 0 + 2
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: St000062
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Values
[1,3,2] => [1,3,2] => [3,1,2] => [1,2] => 2 = 0 + 2
[2,1,3] => [1,3,2] => [3,1,2] => [1,2] => 2 = 0 + 2
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => [3,1,2] => 2 = 0 + 2
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [2,3,1] => 2 = 0 + 2
[1,3,4,2] => [1,3,4,2] => [4,2,1,3] => [2,1,3] => 2 = 0 + 2
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => [1,3,2] => 2 = 0 + 2
[1,4,3,2] => [1,4,2,3] => [4,1,3,2] => [1,3,2] => 2 = 0 + 2
[2,1,3,4] => [1,3,4,2] => [4,2,1,3] => [2,1,3] => 2 = 0 + 2
[2,1,4,3] => [1,4,2,3] => [4,1,3,2] => [1,3,2] => 2 = 0 + 2
[2,3,1,4] => [1,4,2,3] => [4,1,3,2] => [1,3,2] => 2 = 0 + 2
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [2,3,1] => 2 = 0 + 2
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [3,1,2] => 2 = 0 + 2
[3,1,2,4] => [1,2,4,3] => [4,3,1,2] => [3,1,2] => 2 = 0 + 2
[3,1,4,2] => [1,4,2,3] => [4,1,3,2] => [1,3,2] => 2 = 0 + 2
[3,2,1,4] => [1,4,2,3] => [4,1,3,2] => [1,3,2] => 2 = 0 + 2
[3,2,4,1] => [1,2,4,3] => [4,3,1,2] => [3,1,2] => 2 = 0 + 2
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [2,3,1] => 2 = 0 + 2
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [2,3,1] => 2 = 0 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => [4,3,1,2] => 2 = 0 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [4,2,3,1] => 2 = 0 + 2
[1,2,4,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [4,2,1,3] => 2 = 0 + 2
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => [4,1,3,2] => 2 = 0 + 2
[1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => [4,1,3,2] => 2 = 0 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => [3,4,2,1] => 2 = 0 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => [3,4,1,2] => 2 = 0 + 2
[1,3,4,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => [3,2,4,1] => 2 = 0 + 2
[1,3,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => [3,2,1,4] => 2 = 0 + 2
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => [3,1,4,2] => 2 = 0 + 2
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,4,2] => [3,1,4,2] => 2 = 0 + 2
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => [2,4,3,1] => 2 = 0 + 2
[1,4,2,5,3] => [1,4,2,5,3] => [5,2,4,1,3] => [2,4,1,3] => 2 = 0 + 2
[1,4,3,2,5] => [1,4,2,5,3] => [5,2,4,1,3] => [2,4,1,3] => 2 = 0 + 2
[1,4,3,5,2] => [1,4,2,3,5] => [5,2,4,3,1] => [2,4,3,1] => 2 = 0 + 2
[1,4,5,2,3] => [1,4,5,2,3] => [5,2,1,4,3] => [2,1,4,3] => 2 = 0 + 2
[1,4,5,3,2] => [1,4,5,2,3] => [5,2,1,4,3] => [2,1,4,3] => 2 = 0 + 2
[1,5,2,3,4] => [1,5,2,3,4] => [5,1,4,3,2] => [1,4,3,2] => 2 = 0 + 2
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => [1,4,2,3] => 3 = 1 + 2
[1,5,3,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => [1,4,2,3] => 3 = 1 + 2
[1,5,3,4,2] => [1,5,2,3,4] => [5,1,4,3,2] => [1,4,3,2] => 2 = 0 + 2
[1,5,4,2,3] => [1,5,2,3,4] => [5,1,4,3,2] => [1,4,3,2] => 2 = 0 + 2
[1,5,4,3,2] => [1,5,2,3,4] => [5,1,4,3,2] => [1,4,3,2] => 2 = 0 + 2
[2,1,3,4,5] => [1,3,4,5,2] => [5,3,2,1,4] => [3,2,1,4] => 2 = 0 + 2
[2,1,3,5,4] => [1,3,5,2,4] => [5,3,1,4,2] => [3,1,4,2] => 2 = 0 + 2
[2,1,4,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => [2,4,3,1] => 2 = 0 + 2
[2,1,4,5,3] => [1,4,5,2,3] => [5,2,1,4,3] => [2,1,4,3] => 2 = 0 + 2
[2,1,5,3,4] => [1,5,2,3,4] => [5,1,4,3,2] => [1,4,3,2] => 2 = 0 + 2
[2,1,5,4,3] => [1,5,2,3,4] => [5,1,4,3,2] => [1,4,3,2] => 2 = 0 + 2
[2,3,1,4,5] => [1,4,5,2,3] => [5,2,1,4,3] => [2,1,4,3] => 2 = 0 + 2
[2,3,1,5,4] => [1,5,2,3,4] => [5,1,4,3,2] => [1,4,3,2] => 2 = 0 + 2
[2,3,4,1,5] => [1,5,2,3,4] => [5,1,4,3,2] => [1,4,3,2] => 2 = 0 + 2
[7,6,8,5,4,3,2,1] => [1,2,3,4,5,6,8,7] => [8,7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => ? = 0 + 2
[7,6,5,8,4,3,2,1] => [1,2,3,4,5,8,6,7] => [8,7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => ? = 0 + 2
[7,5,6,8,4,3,2,1] => [1,2,3,4,5,6,8,7] => [8,7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => ? = 0 + 2
[6,5,7,8,4,3,2,1] => [1,2,3,4,5,7,8,6] => [8,7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => ? = 0 + 2
[7,6,5,4,8,3,2,1] => [1,2,3,4,8,5,6,7] => [8,7,6,5,1,4,3,2] => [7,6,5,1,4,3,2] => ? = 0 + 2
[6,5,7,4,8,3,2,1] => [1,2,3,4,8,5,7,6] => [8,7,6,5,1,4,2,3] => [7,6,5,1,4,2,3] => ? = 1 + 2
[7,6,4,5,8,3,2,1] => [1,2,3,4,5,8,6,7] => [8,7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => ? = 0 + 2
[7,5,4,6,8,3,2,1] => [1,2,3,4,6,8,5,7] => [8,7,6,5,3,1,4,2] => [7,6,5,3,1,4,2] => ? = 0 + 2
[7,4,5,6,8,3,2,1] => [1,2,3,4,5,6,8,7] => [8,7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => ? = 0 + 2
[8,7,5,4,3,6,2,1] => [1,2,3,6,4,5,7,8] => [8,7,6,3,5,4,2,1] => ? => ? = 0 + 2
[7,6,5,4,3,8,2,1] => [1,2,3,8,4,5,6,7] => [8,7,6,1,5,4,3,2] => [7,6,1,5,4,3,2] => ? = 0 + 2
[6,4,5,7,3,8,2,1] => [1,2,3,8,4,5,7,6] => [8,7,6,1,5,4,2,3] => ? => ? = 1 + 2
[5,4,6,7,3,8,2,1] => [1,2,3,8,4,6,7,5] => [8,7,6,1,5,3,2,4] => ? => ? = 1 + 2
[4,5,6,7,3,8,2,1] => [1,2,3,8,4,5,6,7] => [8,7,6,1,5,4,3,2] => [7,6,1,5,4,3,2] => ? = 0 + 2
[7,6,5,3,4,8,2,1] => [1,2,3,4,8,5,6,7] => [8,7,6,5,1,4,3,2] => [7,6,5,1,4,3,2] => ? = 0 + 2
[7,6,4,3,5,8,2,1] => [1,2,3,5,8,4,6,7] => [8,7,6,4,1,5,3,2] => ? => ? = 0 + 2
[7,6,3,4,5,8,2,1] => [1,2,3,4,5,8,6,7] => [8,7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => ? = 0 + 2
[6,3,4,5,7,8,2,1] => [1,2,3,4,5,7,8,6] => [8,7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => ? = 0 + 2
[4,3,5,6,7,8,2,1] => [1,2,3,5,6,7,8,4] => [8,7,6,4,3,2,1,5] => [7,6,4,3,2,1,5] => ? = 0 + 2
[8,7,6,5,3,2,4,1] => [1,2,4,3,5,6,7,8] => [8,7,5,6,4,3,2,1] => ? => ? = 0 + 2
[8,6,5,7,3,2,4,1] => [1,2,4,3,5,7,6,8] => [8,7,5,6,4,2,3,1] => [7,5,6,4,2,3,1] => ? = 0 + 2
[5,6,7,8,3,2,4,1] => [1,2,4,3,5,6,7,8] => [8,7,5,6,4,3,2,1] => ? => ? = 0 + 2
[7,8,3,2,4,5,6,1] => [1,2,4,5,6,3,7,8] => [8,7,5,4,3,6,2,1] => [7,5,4,3,6,2,1] => ? = 0 + 2
[7,6,5,4,3,2,8,1] => [1,2,8,3,4,5,6,7] => [8,7,1,6,5,4,3,2] => ? => ? = 0 + 2
[6,7,5,4,3,2,8,1] => [1,2,8,3,4,5,6,7] => [8,7,1,6,5,4,3,2] => ? => ? = 0 + 2
[6,5,7,4,3,2,8,1] => [1,2,8,3,4,5,7,6] => [8,7,1,6,5,4,2,3] => ? => ? = 1 + 2
[6,4,5,7,3,2,8,1] => [1,2,8,3,4,5,7,6] => [8,7,1,6,5,4,2,3] => ? => ? = 1 + 2
[4,5,6,7,3,2,8,1] => [1,2,8,3,4,5,6,7] => [8,7,1,6,5,4,3,2] => ? => ? = 0 + 2
[6,5,4,3,7,2,8,1] => [1,2,8,3,7,4,5,6] => [8,7,1,6,2,5,4,3] => [7,1,6,2,5,4,3] => ? = 1 + 2
[5,4,6,3,7,2,8,1] => [1,2,8,3,7,4,6,5] => [8,7,1,6,2,5,3,4] => ? => ? = 2 + 2
[7,6,5,4,2,3,8,1] => [1,2,3,8,4,5,6,7] => [8,7,6,1,5,4,3,2] => [7,6,1,5,4,3,2] => ? = 0 + 2
[6,7,4,5,2,3,8,1] => [1,2,3,8,4,5,6,7] => [8,7,6,1,5,4,3,2] => [7,6,1,5,4,3,2] => ? = 0 + 2
[6,4,5,7,2,3,8,1] => [1,2,3,8,4,5,7,6] => [8,7,6,1,5,4,2,3] => ? => ? = 1 + 2
[4,5,6,7,2,3,8,1] => [1,2,3,8,4,5,6,7] => [8,7,6,1,5,4,3,2] => [7,6,1,5,4,3,2] => ? = 0 + 2
[7,6,5,3,2,4,8,1] => [1,2,4,8,3,5,6,7] => [8,7,5,1,6,4,3,2] => ? => ? = 0 + 2
[7,6,5,2,3,4,8,1] => [1,2,3,4,8,5,6,7] => [8,7,6,5,1,4,3,2] => [7,6,5,1,4,3,2] => ? = 0 + 2
[6,7,2,3,4,5,8,1] => [1,2,3,4,5,8,6,7] => [8,7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => ? = 0 + 2
[5,4,3,2,6,7,8,1] => [1,2,6,7,8,3,4,5] => [8,7,3,2,1,6,5,4] => ? => ? = 0 + 2
[3,4,5,2,6,7,8,1] => [1,2,6,7,8,3,4,5] => [8,7,3,2,1,6,5,4] => ? => ? = 0 + 2
[5,2,3,4,6,7,8,1] => [1,2,3,4,6,7,8,5] => [8,7,6,5,3,2,1,4] => [7,6,5,3,2,1,4] => ? = 0 + 2
[4,3,2,5,6,7,8,1] => [1,2,5,6,7,8,3,4] => [8,7,4,3,2,1,6,5] => ? => ? = 0 + 2
[3,4,2,5,6,7,8,1] => [1,2,5,6,7,8,3,4] => [8,7,4,3,2,1,6,5] => ? => ? = 0 + 2
[4,2,3,5,6,7,8,1] => [1,2,3,5,6,7,8,4] => [8,7,6,4,3,2,1,5] => [7,6,4,3,2,1,5] => ? = 0 + 2
[3,2,4,5,6,7,8,1] => [1,2,4,5,6,7,8,3] => [8,7,5,4,3,2,1,6] => ? => ? = 0 + 2
[6,5,7,8,4,3,1,2] => [1,2,3,4,5,7,8,6] => [8,7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => ? = 0 + 2
[7,6,5,8,3,4,1,2] => [1,2,3,4,5,8,6,7] => [8,7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => ? = 0 + 2
[6,7,5,8,3,4,1,2] => [1,2,3,4,5,8,6,7] => [8,7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => ? = 0 + 2
[7,5,6,8,3,4,1,2] => [1,2,3,4,5,6,8,7] => [8,7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => ? = 0 + 2
[6,5,7,8,3,4,1,2] => [1,2,3,4,5,7,8,6] => [8,7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => ? = 0 + 2
[7,8,5,4,3,6,1,2] => [1,2,3,6,4,5,7,8] => [8,7,6,3,5,4,2,1] => ? => ? = 0 + 2
Description
The length of the longest increasing subsequence of the permutation.
The following 235 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001108The 2-dynamic chromatic number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St000528The height of a poset. St001343The dimension of the reduced incidence algebra of a poset. St000527The width of the poset. St000647The number of big descents of a permutation. St001323The independence gap of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000779The tier of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000451The length of the longest pattern of the form k 1 2. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000095The number of triangles of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000344The number of strongly connected outdegree sequences of a graph. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000379The number of Hamiltonian cycles in a graph. St000462The major index minus the number of excedences of a permutation. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001335The cardinality of a minimal cycle-isolating set of a graph. St001638The book thickness of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001728The number of invisible descents of a permutation. St001736The total number of cycles in a graph. St001797The number of overfull subgraphs of a graph. 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. St000272The treewidth of a graph. St000310The minimal degree of a vertex of a graph. St000362The size of a minimal vertex cover of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000454The largest eigenvalue of a graph if it is integral. St000482The (zero)-forcing number of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000778The metric dimension of a graph. St000948The chromatic discriminant 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. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001281The normalized isoperimetric number of a graph. St001345The Hamming dimension of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001391The disjunction number of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001592The maximal number of simple paths between any two different vertices of a graph. St001644The dimension of a graph. St001665The number of pure excedances of a permutation. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001716The 1-improper chromatic number of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001812The biclique partition number of a graph. St001869The maximum cut size of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth 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. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000636The hull number of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000722The number of different neighbourhoods in a graph. St000822The Hadwiger number of the graph. St000926The clique-coclique 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. St001330The hat guessing 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. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. 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. St001706The number of closed sets in a graph. St000422The energy of a graph, if it is integral. St000759The smallest missing part in an integer partition. St000475The number of parts equal to 1 in a partition. St000929The constant term of the character polynomial of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St001651The Frankl number of a lattice. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. 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. St001720The minimal length of a chain of small intervals in a lattice. St000068The number of minimal elements in a poset. St001866The nesting alignments of a signed permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001557The number of inversions of the second entry of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. 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. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001618The cardinality of the Frattini sublattice of a lattice. 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 $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of 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. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001896The number of right descents of a signed permutations. St001893The flag descent of a signed permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001249Sum of the odd parts of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001383The BG-rank of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000781The number of proper colouring schemes of a Ferrers diagram. St001128The exponens consonantiae of a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. 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. St001851The number of Hecke atoms of a signed permutation. St000312The number of leaves in a graph. St001889The size of the connectivity set of a signed permutation. St001625The Möbius invariant of a lattice. St001429The number of negative entries in a signed permutation. St000850The number of 1/2-balanced pairs in a poset. St000633The size of the automorphism group of a poset. St001399The distinguishing number of a poset. St001472The permanent of the Coxeter matrix of the poset. St001964The interval resolution global dimension of a poset. St000627The exponent of a binary word. St000878The number of ones minus the number of zeros of a binary word. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001095The number of non-isomorphic posets with precisely one further covering relation. St000181The number of connected components of the Hasse diagram for the poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001890The maximum magnitude of the Möbius function of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001490The number of connected components of a skew partition. St001877Number of indecomposable injective modules with projective dimension 2. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001754The number of tolerances of a finite lattice. St000635The number of strictly order preserving maps of a poset into itself. St001333The cardinality of a minimal edge-isolating set of a graph.
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