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Your data matches 194 different statistics following compositions of up to 3 maps.
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Matching statistic: St000396
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> 1
[1,2] => [.,[.,.]]
=> 1
[2,1] => [[.,.],.]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> 2
[2,3,1] => [[.,[.,.]],.]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> 2
[3,2,1] => [[[.,.],.],.]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> 2
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 2
Description
The register function (or Horton-Strahler number) of a binary tree.
This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.
Matching statistic: St000920
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,2] => [1,0,1,0]
=> 1
[2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2
Description
The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:
$$
\lfloor\log_2(1+height(D))\rfloor
$$
Matching statistic: St000058
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => 1
[2,1] => [1,2] => [1,2] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 1
[1,3,2] => [1,2,3] => [1,2,3] => 1
[2,1,3] => [1,2,3] => [1,2,3] => 1
[2,3,1] => [1,2,3] => [1,2,3] => 1
[3,1,2] => [1,3,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 1
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 2
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 2
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 2
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 2
[3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 2
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 2
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 2
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => 2
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => 2
[4,2,3,1] => [1,4,2,3] => [1,4,3,2] => 2
[4,3,1,2] => [1,4,2,3] => [1,4,3,2] => 2
[4,3,2,1] => [1,4,2,3] => [1,4,3,2] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 2
Description
The order of a permutation.
$\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.
Matching statistic: St000159
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 1
[1,2] => [1,2] => [1,1]
=> 1
[2,1] => [1,2] => [1,1]
=> 1
[1,2,3] => [1,2,3] => [1,1,1]
=> 1
[1,3,2] => [1,3,2] => [2,1]
=> 2
[2,1,3] => [1,3,2] => [2,1]
=> 2
[2,3,1] => [1,2,3] => [1,1,1]
=> 1
[3,1,2] => [1,2,3] => [1,1,1]
=> 1
[3,2,1] => [1,2,3] => [1,1,1]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 2
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 2
[1,3,4,2] => [1,3,4,2] => [2,1,1]
=> 2
[1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 2
[1,4,3,2] => [1,4,2,3] => [2,1,1]
=> 2
[2,1,3,4] => [1,3,4,2] => [2,1,1]
=> 2
[2,1,4,3] => [1,4,2,3] => [2,1,1]
=> 2
[2,3,1,4] => [1,4,2,3] => [2,1,1]
=> 2
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 1
[2,4,1,3] => [1,3,2,4] => [2,1,1]
=> 2
[2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 2
[3,1,2,4] => [1,2,4,3] => [2,1,1]
=> 2
[3,1,4,2] => [1,4,2,3] => [2,1,1]
=> 2
[3,2,1,4] => [1,4,2,3] => [2,1,1]
=> 2
[3,2,4,1] => [1,2,4,3] => [2,1,1]
=> 2
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> 1
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> 1
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> 1
[4,1,3,2] => [1,3,2,4] => [2,1,1]
=> 2
[4,2,1,3] => [1,3,2,4] => [2,1,1]
=> 2
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> 1
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> 1
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 2
[1,2,4,5,3] => [1,2,4,5,3] => [2,1,1,1]
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
=> 2
[1,2,5,4,3] => [1,2,5,3,4] => [2,1,1,1]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [2,1,1,1]
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => [2,1,1,1]
=> 2
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,1,1]
=> 2
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,1,1]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,1,1]
=> 2
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1]
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1]
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,1,1]
=> 2
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,1,1]
=> 2
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000298
(load all 41 compositions to match this statistic)
(load all 41 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1
[1,2] => [1,2] => ([(0,1)],2)
=> 1
[2,1] => [1,2] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[3,2,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,2,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[1,4,3,2] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,1,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[2,4,3,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[3,4,1,2] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 2
[4,1,3,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,2,1,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 2
[4,2,3,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,3,1,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,3,2,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,2,5,4,3] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,3,5,4,2] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,4,2,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,4,3,2,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,4,5,2,3] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000308
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => 1
[2,1] => [1,2] => [2,1] => 1
[1,2,3] => [1,2,3] => [3,2,1] => 1
[1,3,2] => [1,3,2] => [2,3,1] => 2
[2,1,3] => [1,3,2] => [2,3,1] => 2
[2,3,1] => [1,2,3] => [3,2,1] => 1
[3,1,2] => [1,2,3] => [3,2,1] => 1
[3,2,1] => [1,2,3] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 2
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2
[1,3,4,2] => [1,3,4,2] => [2,4,3,1] => 2
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 2
[1,4,3,2] => [1,4,2,3] => [3,2,4,1] => 2
[2,1,3,4] => [1,3,4,2] => [2,4,3,1] => 2
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => 2
[2,3,1,4] => [1,4,2,3] => [3,2,4,1] => 2
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 1
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => 2
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => 2
[3,1,2,4] => [1,2,4,3] => [3,4,2,1] => 2
[3,1,4,2] => [1,4,2,3] => [3,2,4,1] => 2
[3,2,1,4] => [1,4,2,3] => [3,2,4,1] => 2
[3,2,4,1] => [1,2,4,3] => [3,4,2,1] => 2
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 1
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 1
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 1
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => 2
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 2
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 1
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => 2
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => 2
[1,2,5,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => 2
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => 2
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => 2
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => 2
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => 2
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => 2
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => 2
[1,4,3,5,2] => [1,4,2,3,5] => [5,3,2,4,1] => 2
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => 2
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
Matching statistic: St000783
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 1
[1,2] => [1,2] => [1,1]
=> 1
[2,1] => [1,2] => [1,1]
=> 1
[1,2,3] => [1,2,3] => [1,1,1]
=> 1
[1,3,2] => [1,3,2] => [2,1]
=> 2
[2,1,3] => [1,3,2] => [2,1]
=> 2
[2,3,1] => [1,2,3] => [1,1,1]
=> 1
[3,1,2] => [1,2,3] => [1,1,1]
=> 1
[3,2,1] => [1,2,3] => [1,1,1]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 2
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 2
[1,3,4,2] => [1,3,4,2] => [2,1,1]
=> 2
[1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 2
[1,4,3,2] => [1,4,2,3] => [2,1,1]
=> 2
[2,1,3,4] => [1,3,4,2] => [2,1,1]
=> 2
[2,1,4,3] => [1,4,2,3] => [2,1,1]
=> 2
[2,3,1,4] => [1,4,2,3] => [2,1,1]
=> 2
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 1
[2,4,1,3] => [1,3,2,4] => [2,1,1]
=> 2
[2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 2
[3,1,2,4] => [1,2,4,3] => [2,1,1]
=> 2
[3,1,4,2] => [1,4,2,3] => [2,1,1]
=> 2
[3,2,1,4] => [1,4,2,3] => [2,1,1]
=> 2
[3,2,4,1] => [1,2,4,3] => [2,1,1]
=> 2
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> 1
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> 1
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> 1
[4,1,3,2] => [1,3,2,4] => [2,1,1]
=> 2
[4,2,1,3] => [1,3,2,4] => [2,1,1]
=> 2
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> 1
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> 1
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 2
[1,2,4,5,3] => [1,2,4,5,3] => [2,1,1,1]
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
=> 2
[1,2,5,4,3] => [1,2,5,3,4] => [2,1,1,1]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [2,1,1,1]
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => [2,1,1,1]
=> 2
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,1,1]
=> 2
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,1,1]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,1,1]
=> 2
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1]
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1]
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,1,1]
=> 2
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,1,1]
=> 2
Description
The side length of the largest staircase partition fitting into a partition.
For an integer partition $(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer $k$ such that $\lambda_i > k-i$ for $i\in\{1,\dots,k\}$.
In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram.
This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions.
Matching statistic: St001235
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [1] => 1
[1,2] => [[1,2]]
=> [2] => 1
[2,1] => [[1],[2]]
=> [2] => 1
[1,2,3] => [[1,2,3]]
=> [3] => 1
[1,3,2] => [[1,2],[3]]
=> [2,1] => 2
[2,1,3] => [[1,3],[2]]
=> [3] => 1
[2,3,1] => [[1,2],[3]]
=> [2,1] => 2
[3,1,2] => [[1,3],[2]]
=> [3] => 1
[3,2,1] => [[1],[2],[3]]
=> [3] => 1
[1,2,3,4] => [[1,2,3,4]]
=> [4] => 1
[1,2,4,3] => [[1,2,3],[4]]
=> [3,1] => 2
[1,3,2,4] => [[1,2,4],[3]]
=> [2,2] => 2
[1,3,4,2] => [[1,2,3],[4]]
=> [3,1] => 2
[1,4,2,3] => [[1,2,4],[3]]
=> [2,2] => 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [2,2] => 2
[2,1,3,4] => [[1,3,4],[2]]
=> [4] => 1
[2,1,4,3] => [[1,3],[2,4]]
=> [3,1] => 2
[2,3,1,4] => [[1,2,4],[3]]
=> [2,2] => 2
[2,3,4,1] => [[1,2,3],[4]]
=> [3,1] => 2
[2,4,1,3] => [[1,2],[3,4]]
=> [2,2] => 2
[2,4,3,1] => [[1,2],[3],[4]]
=> [2,2] => 2
[3,1,2,4] => [[1,3,4],[2]]
=> [4] => 1
[3,1,4,2] => [[1,3],[2,4]]
=> [3,1] => 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [4] => 1
[3,2,4,1] => [[1,3],[2],[4]]
=> [3,1] => 2
[3,4,1,2] => [[1,2],[3,4]]
=> [2,2] => 2
[3,4,2,1] => [[1,2],[3],[4]]
=> [2,2] => 2
[4,1,2,3] => [[1,3,4],[2]]
=> [4] => 1
[4,1,3,2] => [[1,3],[2],[4]]
=> [3,1] => 2
[4,2,1,3] => [[1,4],[2],[3]]
=> [4] => 1
[4,2,3,1] => [[1,3],[2],[4]]
=> [3,1] => 2
[4,3,1,2] => [[1,4],[2],[3]]
=> [4] => 1
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [4,1] => 2
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [3,2] => 2
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [4,1] => 2
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [3,2] => 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [2,3] => 2
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [2,2,1] => 2
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [3,2] => 2
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [4,1] => 2
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [3,2] => 2
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [3,2] => 2
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [2,3] => 2
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> [2,2,1] => 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [2,3] => 2
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [2,2,1] => 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [3,2] => 2
Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Matching statistic: St001239
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001239: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001239: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1
[1,2] => [2] => [1,1,0,0]
=> 1
[2,1] => [2] => [1,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 2
[2,1,3] => [3] => [1,1,1,0,0,0]
=> 1
[2,3,1] => [3] => [1,1,1,0,0,0]
=> 1
[3,1,2] => [3] => [1,1,1,0,0,0]
=> 1
[3,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
Description
The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.
Matching statistic: St001261
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1
[1,2] => [2] => ([],2)
=> 1
[2,1] => [2] => ([],2)
=> 1
[1,2,3] => [3] => ([],3)
=> 1
[1,3,2] => [1,2] => ([(1,2)],3)
=> 2
[2,1,3] => [3] => ([],3)
=> 1
[2,3,1] => [3] => ([],3)
=> 1
[3,1,2] => [3] => ([],3)
=> 1
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [4] => ([],4)
=> 1
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> 2
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> 2
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> 2
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [4] => ([],4)
=> 1
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [4] => ([],4)
=> 1
[2,3,4,1] => [4] => ([],4)
=> 1
[2,4,1,3] => [4] => ([],4)
=> 1
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [4] => ([],4)
=> 1
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [4] => ([],4)
=> 1
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [4] => ([],4)
=> 1
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> 2
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 2
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 2
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 2
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 2
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> 2
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> 2
Description
The Castelnuovo-Mumford regularity of a graph.
The following 184 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001432The order dimension of the partition. St001734The lettericity of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000397The Strahler number of a rooted tree. St000480The number of lower covers of a partition in dominance order. St000535The rank-width of a graph. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St000013The height of a Dyck path. St000053The number of valleys of the Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000147The largest part of an integer partition. St000166The depth minus 1 of an ordered tree. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000258The burning number of a graph. St000291The number of descents of a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000306The bounce count of a Dyck path. St000328The maximum number of child nodes in a tree. St000381The largest part of an integer composition. St000390The number of runs of ones in a binary word. St000451The length of the longest pattern of the form k 1 2. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000542The number of left-to-right-minima of a permutation. St000679The pruning number of an ordered tree. St000701The protection number of a binary tree. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000862The number of parts of the shifted shape of a permutation. St000903The number of different parts of an integer composition. St000918The 2-limited packing number of a graph. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001093The detour number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001471The magnitude of a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001530The depth of a Dyck path. St001644The dimension of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001716The 1-improper chromatic number of a graph. St001732The number of peaks visible from the left. St001741The largest integer such that all patterns of this size are contained in the permutation. St001962The proper pathwidth of a graph. St000028The number of stack-sorts needed to sort a permutation. St000094The depth of an ordered tree. St000183The side length of the Durfee square of an integer partition. St000225Difference between largest and smallest parts in a partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000292The number of ascents of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000387The matching number of a graph. St000481The number of upper covers of a partition in dominance order. St000552The number of cut vertices of a graph. St000759The smallest missing part in an integer partition. St000845The maximal number of elements covered by an element in a poset. St000897The number of different multiplicities of parts of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001071The beta invariant of the graph. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001092The number of distinct even parts of a partition. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001271The competition number of a graph. St001280The number of parts of an integer partition that are at least two. St001335The cardinality of a minimal cycle-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001512The minimum rank of a graph. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001638The book thickness of a graph. St001665The number of pure excedances of a permutation. St001673The degree of asymmetry of an integer composition. St001728The number of invisible descents of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001743The discrepancy of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000485The length of the longest cycle of a permutation. St000392The length of the longest run of ones in a binary word. St000628The balance of a binary word. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000444The length of the maximal rise of a Dyck path. St000630The length of the shortest palindromic decomposition of a binary word. St000668The least common multiple of the parts of the partition. St001062The maximal size of a block of a set partition. St001128The exponens consonantiae of a partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000442The maximal area to the right of an up step of a Dyck path. St000486The number of cycles of length at least 3 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000659The number of rises of length at least 2 of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001372The length of a longest cyclic run of ones of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000671The maximin edge-connectivity for choosing a subgraph. St000331The number of upper interactions of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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 $eA$. St001330The hat guessing number of a graph. St001568The smallest positive integer that does not appear twice in the 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. St000455The second largest eigenvalue of a graph if it is integral. St000662The staircase size of the code of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000640The rank of the largest boolean interval in a poset. St001896The number of right descents of a signed permutations. St000307The number of rowmotion orbits of a poset. St001624The breadth of a lattice. St001621The number of atoms of a lattice. St000454The largest eigenvalue of a graph if it is integral. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000632The jump number of the poset. St001820The size of the image of the pop stack sorting operator. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001645The pebbling number 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. St000633The size of the automorphism group of a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001569The maximal modular displacement of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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