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Your data matches 106 different statistics following compositions of up to 3 maps.
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Matching statistic: St000054
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => 1
[1,2] => [.,[.,.]]
=> [2,1] => 2
[2,1] => [[.,.],.]
=> [1,2] => 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 3
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 2
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 3
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => 3
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 4
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 4
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 4
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 4
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 4
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 3
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 3
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 4
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 4
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 4
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 4
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 4
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 4
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 3
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 5
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 5
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 5
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 4
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 5
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 5
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 4
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 4
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 5
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 5
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 5
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 5
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 5
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000740
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 1
[1,2] => [1,0,1,0]
=> [2,1] => 1
[2,1] => [1,1,0,0]
=> [1,2] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [2,3,1] => 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,1,3] => 3
[2,1,3] => [1,1,0,0,1,0]
=> [1,3,2] => 2
[2,3,1] => [1,1,0,1,0,0]
=> [3,1,2] => 2
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 3
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 4
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 4
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 4
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 4
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 5
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 4
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 4
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 5
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 5
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 5
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 5
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 5
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 4
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 4
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 4
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 4
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 4
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St001291
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 4
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 4
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St001725
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001725: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001725: 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] => [1,1] => ([(0,1)],2)
=> 2
[1,2,3] => [3] => ([],3)
=> 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3,4] => [4] => ([],4)
=> 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 4
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 4
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 4
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
Description
The harmonious chromatic number of a graph.
A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices.
Matching statistic: St000141
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => 0 = 1 - 1
[1,2] => [.,[.,.]]
=> [2,1] => 1 = 2 - 1
[2,1] => [[.,.],.]
=> [1,2] => 0 = 1 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 2 = 3 - 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1 = 2 - 1
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 2 = 3 - 1
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => 2 = 3 - 1
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => 1 = 2 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 3 = 4 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2 = 3 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 3 = 4 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 3 = 4 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2 = 3 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1 = 2 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 3 = 4 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 3 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 3 = 4 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 3 = 4 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 3 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 3 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 3 = 4 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 3 = 4 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3 = 4 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 3 = 4 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3 = 4 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2 = 3 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1 = 2 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 2 = 3 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 2 = 3 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1 = 2 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 4 = 5 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 3 = 4 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 4 = 5 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 4 = 5 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 3 = 4 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 2 = 3 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 4 = 5 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 3 = 4 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 4 = 5 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 4 = 5 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 3 = 4 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 3 = 4 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 4 = 5 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 4 = 5 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 4 = 5 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 4 = 5 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 4 = 5 - 1
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000171
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0 = 1 - 1
[1,2] => [2] => ([],2)
=> 0 = 1 - 1
[2,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3] => [3] => ([],3)
=> 0 = 1 - 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,1,3] => [1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,2,3,4] => [4] => ([],4)
=> 0 = 1 - 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,3,4,5] => [5] => ([],5)
=> 0 = 1 - 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
Description
The degree of the graph.
This is the maximal vertex degree of a graph.
Matching statistic: St000316
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => 0 = 1 - 1
[1,2] => [.,[.,.]]
=> [2,1] => 1 = 2 - 1
[2,1] => [[.,.],.]
=> [1,2] => 0 = 1 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 2 = 3 - 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1 = 2 - 1
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 2 = 3 - 1
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => 2 = 3 - 1
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => 1 = 2 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 3 = 4 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2 = 3 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 3 = 4 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 3 = 4 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2 = 3 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1 = 2 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 3 = 4 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 3 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 3 = 4 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 3 = 4 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 3 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 3 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 3 = 4 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 3 = 4 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3 = 4 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 3 = 4 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3 = 4 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2 = 3 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1 = 2 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 2 = 3 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 2 = 3 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1 = 2 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 4 = 5 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 3 = 4 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 4 = 5 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 4 = 5 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 3 = 4 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 2 = 3 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 4 = 5 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 3 = 4 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 4 = 5 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 4 = 5 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 3 = 4 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 3 = 4 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 4 = 5 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 4 = 5 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 4 = 5 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 4 = 5 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 4 = 5 - 1
Description
The number of non-left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a **non-left-to-right-maximum** if there exists a $j < i$ such that $\sigma_j > \sigma_i$.
Matching statistic: St000987
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0 = 1 - 1
[1,2] => [2] => ([],2)
=> 0 = 1 - 1
[2,1] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3] => [3] => ([],3)
=> 0 = 1 - 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,1,3] => [1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,2,3,4] => [4] => ([],4)
=> 0 = 1 - 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,2,3,4,5] => [5] => ([],5)
=> 0 = 1 - 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
Description
The number of positive eigenvalues of the Laplacian matrix of the graph.
This is the number of vertices minus the number of connected components of the graph.
Matching statistic: St001225
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001225: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001225: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
Description
The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St001227
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 0 = 1 - 1
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0 = 1 - 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
The following 96 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000240The number of indices that are not small excedances. St000363The number of minimal vertex covers of a graph. St000443The number of long tunnels of a Dyck path. St000501The size of the first part in the decomposition of a permutation. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000822The Hadwiger number of the graph. St000839The largest opener of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001330The hat guessing number of a graph. St001389The number of partitions of the same length below the given integer partition. St001494The Alon-Tarsi number of a graph. St001497The position of the largest weak excedence of a permutation. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000010The length of the partition. St000019The cardinality of the support of a permutation. St000024The number of double up and double down steps of a Dyck path. St000051The size of the left subtree of a binary tree. St000067The inversion number of the alternating sign matrix. St000133The "bounce" of a permutation. St000272The treewidth of a graph. St000304The load of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000362The size of a minimal vertex cover of a graph. St000439The position of the first down step of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000536The pathwidth of a graph. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St001120The length of a longest path in a graph. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001358The largest degree of a regular subgraph 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. St001644The dimension of a graph. St001721The degree of a binary word. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001962The proper pathwidth of a graph. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St000444The length of the maximal rise of a Dyck path. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000216The absolute length of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000730The maximal arc length of a set partition. St000809The reduced reflection length of the permutation. St000874The position of the last double rise in a Dyck path. St000957The number of Bruhat lower covers of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001812The biclique partition number of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000840The number of closers smaller than the largest opener in a perfect matching. St001645The pebbling number of a connected graph. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001742The difference of the maximal and the minimal degree in a graph. St000264The girth of a graph, which is not a tree. St001875The number of simple modules with projective dimension at most 1. St001769The reflection length of a signed permutation. St000173The segment statistic of a semistandard tableau. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001877Number of indecomposable injective modules with projective dimension 2.
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