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Your data matches 219 different statistics following compositions of up to 3 maps.
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Matching statistic: St001174
(load all 264 compositions to match this statistic)
(load all 264 compositions to match this statistic)
St001174: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 0
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
[1,4,5,3,2] => 1
Description
The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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
[1,2] => [.,[.,.]]
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
=> 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 2 = 1 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> 2 = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> 2 = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 2 = 1 + 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> 1 = 0 + 1
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
[1,2] => [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [1,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,3] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[2,3,1] => [1,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,2] => [1,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,1] => [1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
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: St000392
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0 => 0
[2,1] => [1,2] => 0 => 0
[1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,3,2] => 01 => 1
[2,1,3] => [1,3,2] => 01 => 1
[2,3,1] => [1,2,3] => 00 => 0
[3,1,2] => [1,2,3] => 00 => 0
[3,2,1] => [1,2,3] => 00 => 0
[1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,2,4,3] => 001 => 1
[1,3,2,4] => [1,3,2,4] => 010 => 1
[1,3,4,2] => [1,3,4,2] => 001 => 1
[1,4,2,3] => [1,4,2,3] => 010 => 1
[1,4,3,2] => [1,4,2,3] => 010 => 1
[2,1,3,4] => [1,3,4,2] => 001 => 1
[2,1,4,3] => [1,4,2,3] => 010 => 1
[2,3,1,4] => [1,4,2,3] => 010 => 1
[2,3,4,1] => [1,2,3,4] => 000 => 0
[2,4,1,3] => [1,3,2,4] => 010 => 1
[2,4,3,1] => [1,2,4,3] => 001 => 1
[3,1,2,4] => [1,2,4,3] => 001 => 1
[3,1,4,2] => [1,4,2,3] => 010 => 1
[3,2,1,4] => [1,4,2,3] => 010 => 1
[3,2,4,1] => [1,2,4,3] => 001 => 1
[3,4,1,2] => [1,2,3,4] => 000 => 0
[3,4,2,1] => [1,2,3,4] => 000 => 0
[4,1,2,3] => [1,2,3,4] => 000 => 0
[4,1,3,2] => [1,3,2,4] => 010 => 1
[4,2,1,3] => [1,3,2,4] => 010 => 1
[4,2,3,1] => [1,2,3,4] => 000 => 0
[4,3,1,2] => [1,2,3,4] => 000 => 0
[4,3,2,1] => [1,2,3,4] => 000 => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,2,4,5,3] => [1,2,4,5,3] => 0001 => 1
[1,2,5,3,4] => [1,2,5,3,4] => 0010 => 1
[1,2,5,4,3] => [1,2,5,3,4] => 0010 => 1
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 1
[1,3,4,2,5] => [1,3,4,2,5] => 0010 => 1
[1,3,4,5,2] => [1,3,4,5,2] => 0001 => 1
[1,3,5,2,4] => [1,3,5,2,4] => 0010 => 1
[1,3,5,4,2] => [1,3,5,2,4] => 0010 => 1
[1,4,2,3,5] => [1,4,2,3,5] => 0100 => 1
[1,4,2,5,3] => [1,4,2,5,3] => 0101 => 1
[1,4,3,2,5] => [1,4,2,5,3] => 0101 => 1
[1,4,3,5,2] => [1,4,2,3,5] => 0100 => 1
[1,4,5,2,3] => [1,4,5,2,3] => 0010 => 1
[1,4,5,3,2] => [1,4,5,2,3] => 0010 => 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000480
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,1]
=> 0
[2,1] => [1,2] => [1,1]
=> 0
[1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> 1
[2,1,3] => [1,3,2] => [2,1]
=> 1
[2,3,1] => [1,2,3] => [1,1,1]
=> 0
[3,1,2] => [1,2,3] => [1,1,1]
=> 0
[3,2,1] => [1,2,3] => [1,1,1]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[1,3,4,2] => [1,3,4,2] => [2,1,1]
=> 1
[1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 1
[1,4,3,2] => [1,4,2,3] => [2,1,1]
=> 1
[2,1,3,4] => [1,3,4,2] => [2,1,1]
=> 1
[2,1,4,3] => [1,4,2,3] => [2,1,1]
=> 1
[2,3,1,4] => [1,4,2,3] => [2,1,1]
=> 1
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,4,1,3] => [1,3,2,4] => [2,1,1]
=> 1
[2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 1
[3,1,2,4] => [1,2,4,3] => [2,1,1]
=> 1
[3,1,4,2] => [1,4,2,3] => [2,1,1]
=> 1
[3,2,1,4] => [1,4,2,3] => [2,1,1]
=> 1
[3,2,4,1] => [1,2,4,3] => [2,1,1]
=> 1
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> 0
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> 0
[4,1,3,2] => [1,3,2,4] => [2,1,1]
=> 1
[4,2,1,3] => [1,3,2,4] => [2,1,1]
=> 1
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> 0
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,1,1,1]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,1,1,1]
=> 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,1,1,1]
=> 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1]
=> 1
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1]
=> 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,1,1]
=> 1
[1,4,5,3,2] => [1,4,5,2,3] => [2,1,1,1]
=> 1
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is
$$
\frac{1}{2}(\sqrt{1+8n}-3)
$$
and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St000535
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000535: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000535: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => ([],2)
=> 0
[2,1] => [2] => ([],2)
=> 0
[1,2,3] => [3] => ([],3)
=> 0
[1,3,2] => [1,2] => ([(1,2)],3)
=> 1
[2,1,3] => [3] => ([],3)
=> 0
[2,3,1] => [3] => ([],3)
=> 0
[3,1,2] => [3] => ([],3)
=> 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [4] => ([],4)
=> 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [4] => ([],4)
=> 0
[2,3,4,1] => [4] => ([],4)
=> 0
[2,4,1,3] => [4] => ([],4)
=> 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [4] => ([],4)
=> 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [4] => ([],4)
=> 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [4] => ([],4)
=> 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> 1
[1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The rank-width of a graph.
Matching statistic: St000628
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000628: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
St000628: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0 => 0
[2,1] => [1,2] => 0 => 0
[1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,3,2] => 01 => 1
[2,1,3] => [1,3,2] => 01 => 1
[2,3,1] => [1,2,3] => 00 => 0
[3,1,2] => [1,2,3] => 00 => 0
[3,2,1] => [1,2,3] => 00 => 0
[1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,2,4,3] => 001 => 1
[1,3,2,4] => [1,3,2,4] => 010 => 1
[1,3,4,2] => [1,3,4,2] => 001 => 1
[1,4,2,3] => [1,4,2,3] => 010 => 1
[1,4,3,2] => [1,4,2,3] => 010 => 1
[2,1,3,4] => [1,3,4,2] => 001 => 1
[2,1,4,3] => [1,4,2,3] => 010 => 1
[2,3,1,4] => [1,4,2,3] => 010 => 1
[2,3,4,1] => [1,2,3,4] => 000 => 0
[2,4,1,3] => [1,3,2,4] => 010 => 1
[2,4,3,1] => [1,2,4,3] => 001 => 1
[3,1,2,4] => [1,2,4,3] => 001 => 1
[3,1,4,2] => [1,4,2,3] => 010 => 1
[3,2,1,4] => [1,4,2,3] => 010 => 1
[3,2,4,1] => [1,2,4,3] => 001 => 1
[3,4,1,2] => [1,2,3,4] => 000 => 0
[3,4,2,1] => [1,2,3,4] => 000 => 0
[4,1,2,3] => [1,2,3,4] => 000 => 0
[4,1,3,2] => [1,3,2,4] => 010 => 1
[4,2,1,3] => [1,3,2,4] => 010 => 1
[4,2,3,1] => [1,2,3,4] => 000 => 0
[4,3,1,2] => [1,2,3,4] => 000 => 0
[4,3,2,1] => [1,2,3,4] => 000 => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,2,4,5,3] => [1,2,4,5,3] => 0001 => 1
[1,2,5,3,4] => [1,2,5,3,4] => 0010 => 1
[1,2,5,4,3] => [1,2,5,3,4] => 0010 => 1
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 1
[1,3,4,2,5] => [1,3,4,2,5] => 0010 => 1
[1,3,4,5,2] => [1,3,4,5,2] => 0001 => 1
[1,3,5,2,4] => [1,3,5,2,4] => 0010 => 1
[1,3,5,4,2] => [1,3,5,2,4] => 0010 => 1
[1,4,2,3,5] => [1,4,2,3,5] => 0100 => 1
[1,4,2,5,3] => [1,4,2,5,3] => 0101 => 1
[1,4,3,2,5] => [1,4,2,5,3] => 0101 => 1
[1,4,3,5,2] => [1,4,2,3,5] => 0100 => 1
[1,4,5,2,3] => [1,4,5,2,3] => 0010 => 1
[1,4,5,3,2] => [1,4,5,2,3] => 0010 => 1
Description
The balance of a binary word.
The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1].
A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.
Matching statistic: St001124
(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
St001124: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,1]
=> 0
[2,1] => [1,2] => [1,1]
=> 0
[1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> 1
[2,1,3] => [1,3,2] => [2,1]
=> 1
[2,3,1] => [1,2,3] => [1,1,1]
=> 0
[3,1,2] => [1,2,3] => [1,1,1]
=> 0
[3,2,1] => [1,2,3] => [1,1,1]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[1,3,4,2] => [1,3,4,2] => [2,1,1]
=> 1
[1,4,2,3] => [1,4,2,3] => [2,1,1]
=> 1
[1,4,3,2] => [1,4,2,3] => [2,1,1]
=> 1
[2,1,3,4] => [1,3,4,2] => [2,1,1]
=> 1
[2,1,4,3] => [1,4,2,3] => [2,1,1]
=> 1
[2,3,1,4] => [1,4,2,3] => [2,1,1]
=> 1
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,4,1,3] => [1,3,2,4] => [2,1,1]
=> 1
[2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 1
[3,1,2,4] => [1,2,4,3] => [2,1,1]
=> 1
[3,1,4,2] => [1,4,2,3] => [2,1,1]
=> 1
[3,2,1,4] => [1,4,2,3] => [2,1,1]
=> 1
[3,2,4,1] => [1,2,4,3] => [2,1,1]
=> 1
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> 0
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> 0
[4,1,3,2] => [1,3,2,4] => [2,1,1]
=> 1
[4,2,1,3] => [1,3,2,4] => [2,1,1]
=> 1
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> 0
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,1,1,1]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,1,1,1]
=> 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,1,1,1]
=> 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,1,1]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1]
=> 1
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1]
=> 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,1,1]
=> 1
[1,4,5,3,2] => [1,4,5,2,3] => [2,1,1,1]
=> 1
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St001192
(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
St001192: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [2] => [1,1,0,0]
=> 0
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [3] => [1,1,1,0,0,0]
=> 0
[2,3,1] => [3] => [1,1,1,0,0,0]
=> 0
[3,1,2] => [3] => [1,1,1,0,0,0]
=> 0
[3,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,5,3,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
Description
The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$.
Matching statistic: St001333
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => ([],2)
=> 0
[2,1] => [2] => ([],2)
=> 0
[1,2,3] => [3] => ([],3)
=> 0
[1,3,2] => [1,2] => ([(1,2)],3)
=> 1
[2,1,3] => [3] => ([],3)
=> 0
[2,3,1] => [3] => ([],3)
=> 0
[3,1,2] => [3] => ([],3)
=> 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [4] => ([],4)
=> 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [4] => ([],4)
=> 0
[2,3,4,1] => [4] => ([],4)
=> 0
[2,4,1,3] => [4] => ([],4)
=> 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [4] => ([],4)
=> 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [4] => ([],4)
=> 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [4] => ([],4)
=> 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> 1
[1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The cardinality of a minimal edge-isolating set of a graph.
Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$.
This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains only the graph with one edge.
The following 209 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001393The induced matching number of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000058The order of a permutation. St000159The number of distinct parts of the integer partition. St000298The order dimension or Dushnik-Miller dimension of a poset. St000308The height of the tree associated to a permutation. St000485The length of the longest cycle of a permutation. St000783The side length of the largest staircase partition fitting into a partition. St000846The maximal number of elements covering an element of a poset. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. 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. St000028The number of stack-sorts needed to sort a permutation. St000183The side length of the Durfee square of an integer partition. St000225Difference between largest and smallest parts in a partition. St000253The crossing number of a set partition. St000254The nesting number of a set 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. St000442The maximal area to the right of an up step of a Dyck path. St000481The number of upper covers of a partition in dominance order. 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. St000552The number of cut vertices of a graph. St000659The number of rises of length at least 2 of a Dyck path. 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. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with 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. 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. 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. 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. 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. 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. St000444The length of the maximal rise of a Dyck path. 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. St000630The length of the shortest palindromic decomposition of a binary word. St000668The least common multiple of the parts of the partition. 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. St001062The maximal size of a block of a set partition. St001093The detour number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001128The exponens consonantiae of a partition. 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. St000094The depth of an ordered tree. St000759The smallest missing part in an integer partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001112The 3-weak dynamic number of a graph. 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:
St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001110The 3-dynamic chromatic number of a graph. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000640The rank of the largest boolean interval in a poset. 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. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations 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. St001924The number of cells in an integer partition whose arm and leg length coincide. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. 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. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001877Number of indecomposable injective modules with projective dimension 2. St000456The monochromatic index of a connected graph. 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. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St001488The number of corners of a skew partition. St000454The largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000632The jump number of the poset. St001820The size of the image of the pop stack sorting operator. 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. 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. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001860The number of factors of the Stanley symmetric function associated with a 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. St001569The maximal modular displacement of a permutation. St001773The number of minimal elements in Bruhat order not less than the 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. 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. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001570The minimal number of edges to add to make a graph Hamiltonian.
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