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Your data matches 284 different statistics following compositions of up to 3 maps.
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Matching statistic: St000381
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 1
[2] => 2
[1,1,1] => 1
[1,2] => 2
[2,1] => 2
[3] => 3
[1,1,1,1] => 1
[1,1,2] => 2
[1,2,1] => 2
[1,3] => 3
[2,1,1] => 2
[2,2] => 2
[3,1] => 3
[4] => 4
[1,1,1,1,1] => 1
[1,1,1,2] => 2
[1,1,2,1] => 2
[1,1,3] => 3
[1,2,1,1] => 2
[1,2,2] => 2
[1,3,1] => 3
[1,4] => 4
[2,1,1,1] => 2
[2,1,2] => 2
[2,2,1] => 2
[2,3] => 3
[3,1,1] => 3
[3,2] => 3
[4,1] => 4
[5] => 5
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 2
[1,1,1,2,1] => 2
[1,1,1,3] => 3
[1,1,2,1,1] => 2
[1,1,2,2] => 2
[1,1,3,1] => 3
[1,1,4] => 4
[1,2,1,1,1] => 2
[1,2,1,2] => 2
[1,2,2,1] => 2
[1,2,3] => 3
[1,3,1,1] => 3
[1,3,2] => 3
[1,4,1] => 4
[1,5] => 5
[2,1,1,1,1] => 2
[2,1,1,2] => 2
[2,1,2,1] => 2
Description
The largest part of an integer composition.
Matching statistic: St001235
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St001235: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 2
[2] => 1
[1,1,1] => 3
[1,2] => 2
[2,1] => 2
[3] => 1
[1,1,1,1] => 4
[1,1,2] => 3
[1,2,1] => 2
[1,3] => 2
[2,1,1] => 3
[2,2] => 2
[3,1] => 2
[4] => 1
[1,1,1,1,1] => 5
[1,1,1,2] => 4
[1,1,2,1] => 3
[1,1,3] => 3
[1,2,1,1] => 3
[1,2,2] => 2
[1,3,1] => 2
[1,4] => 2
[2,1,1,1] => 4
[2,1,2] => 3
[2,2,1] => 2
[2,3] => 2
[3,1,1] => 3
[3,2] => 2
[4,1] => 2
[5] => 1
[1,1,1,1,1,1] => 6
[1,1,1,1,2] => 5
[1,1,1,2,1] => 4
[1,1,1,3] => 4
[1,1,2,1,1] => 3
[1,1,2,2] => 3
[1,1,3,1] => 3
[1,1,4] => 3
[1,2,1,1,1] => 4
[1,2,1,2] => 3
[1,2,2,1] => 2
[1,2,3] => 2
[1,3,1,1] => 3
[1,3,2] => 2
[1,4,1] => 2
[1,5] => 2
[2,1,1,1,1] => 5
[2,1,1,2] => 4
[2,1,2,1] => 3
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: St000013
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,1] => [1,0,1,0]
=> 1
[2] => [1,1,0,0]
=> 2
[1,1,1] => [1,0,1,0,1,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> 2
[2,1] => [1,1,0,0,1,0]
=> 2
[3] => [1,1,1,0,0,0]
=> 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> 3
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 3
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 4
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1
[1,1] => [1,1]
=> 1
[2] => [2]
=> 2
[1,1,1] => [1,1,1]
=> 1
[1,2] => [2,1]
=> 2
[2,1] => [2,1]
=> 2
[3] => [3]
=> 3
[1,1,1,1] => [1,1,1,1]
=> 1
[1,1,2] => [2,1,1]
=> 2
[1,2,1] => [2,1,1]
=> 2
[1,3] => [3,1]
=> 3
[2,1,1] => [2,1,1]
=> 2
[2,2] => [2,2]
=> 2
[3,1] => [3,1]
=> 3
[4] => [4]
=> 4
[1,1,1,1,1] => [1,1,1,1,1]
=> 1
[1,1,1,2] => [2,1,1,1]
=> 2
[1,1,2,1] => [2,1,1,1]
=> 2
[1,1,3] => [3,1,1]
=> 3
[1,2,1,1] => [2,1,1,1]
=> 2
[1,2,2] => [2,2,1]
=> 2
[1,3,1] => [3,1,1]
=> 3
[1,4] => [4,1]
=> 4
[2,1,1,1] => [2,1,1,1]
=> 2
[2,1,2] => [2,2,1]
=> 2
[2,2,1] => [2,2,1]
=> 2
[2,3] => [3,2]
=> 3
[3,1,1] => [3,1,1]
=> 3
[3,2] => [3,2]
=> 3
[4,1] => [4,1]
=> 4
[5] => [5]
=> 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> 2
[1,1,1,2,1] => [2,1,1,1,1]
=> 2
[1,1,1,3] => [3,1,1,1]
=> 3
[1,1,2,1,1] => [2,1,1,1,1]
=> 2
[1,1,2,2] => [2,2,1,1]
=> 2
[1,1,3,1] => [3,1,1,1]
=> 3
[1,1,4] => [4,1,1]
=> 4
[1,2,1,1,1] => [2,1,1,1,1]
=> 2
[1,2,1,2] => [2,2,1,1]
=> 2
[1,2,2,1] => [2,2,1,1]
=> 2
[1,2,3] => [3,2,1]
=> 3
[1,3,1,1] => [3,1,1,1]
=> 3
[1,3,2] => [3,2,1]
=> 3
[1,4,1] => [4,1,1]
=> 4
[1,5] => [5,1]
=> 5
[2,1,1,1,1] => [2,1,1,1,1]
=> 2
[2,1,1,2] => [2,2,1,1]
=> 2
[2,1,2,1] => [2,2,1,1]
=> 2
Description
The largest part of an integer partition.
Matching statistic: St000684
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000684: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000684: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,1] => [1,0,1,0]
=> 2
[2] => [1,1,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2] => [1,0,1,1,0,0]
=> 2
[2,1] => [1,1,0,0,1,0]
=> 2
[3] => [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 4
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3
Description
The global dimension of the LNakayama algebra associated to a Dyck path.
An n-LNakayama algebra is a quiver algebra with a directed line as a connected quiver with $n$ points for $n \geq 2$. Number those points from the left to the right by $0,1,\ldots,n-1$.
The algebra is then uniquely determined by the dimension $c_i$ of the projective indecomposable modules at point $i$. Such algebras are then uniquely determined by lists of the form $[c_0,c_1,...,c_{n-1}]$ with the conditions: $c_{n-1}=1$ and $c_i -1 \leq c_{i+1}$ for all $i$. The number of such algebras is then the $n-1$-st Catalan number $C_{n-1}$.
One can get also an interpretation with Dyck paths by associating the top boundary of the Auslander-Reiten quiver (which is a Dyck path) to those algebras. Example: [3,4,3,3,2,1] corresponds to the Dyck path [1,1,0,1,1,0,0,1,0,0].
Conjecture: that there is an explicit bijection between $n$-LNakayama algebras with global dimension bounded by $m$ and Dyck paths with height at most $m$.
Examples:
* For $m=2$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1,2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192.
* For $m=3$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418.
Matching statistic: St000686
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000686: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000686: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,1] => [1,0,1,0]
=> 2
[2] => [1,1,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2] => [1,0,1,1,0,0]
=> 2
[2,1] => [1,1,0,0,1,0]
=> 2
[3] => [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 4
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3
Description
The finitistic dominant dimension of a Dyck path.
To every LNakayama algebra there is a corresponding Dyck path, see also [[St000684]]. We associate the finitistic dominant dimension of the algebra to the corresponding Dyck path.
Matching statistic: St000774
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000774: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000774: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 1
[1,1] => ([(0,1)],2)
=> 1
[2] => ([],2)
=> 2
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2] => ([(1,2)],3)
=> 2
[2,1] => ([(0,2),(1,2)],3)
=> 1
[3] => ([],3)
=> 3
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,3] => ([(2,3)],4)
=> 3
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4] => ([],4)
=> 4
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4] => ([(3,4)],5)
=> 4
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,3] => ([(2,4),(3,4)],5)
=> 3
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[5] => ([],5)
=> 5
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 4
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,5] => ([(4,5)],6)
=> 5
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The maximal multiplicity of a Laplacian eigenvalue in a graph.
Matching statistic: St000930
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000930: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000930: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,1] => [1,0,1,0]
=> 2
[2] => [1,1,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2] => [1,0,1,1,0,0]
=> 2
[2,1] => [1,1,0,0,1,0]
=> 2
[3] => [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 4
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3
Description
The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver.
The $k$-Gorenstein degree is the maximal number $k$ such that the algebra is $k$-Gorenstein. We apply the convention that the value is equal to the global dimension of the algebra in case the $k$-Gorenstein degree is greater than or equal to the global dimension.
Matching statistic: St000982
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 1
[1,1] => 11 => 2
[2] => 10 => 1
[1,1,1] => 111 => 3
[1,2] => 110 => 2
[2,1] => 101 => 1
[3] => 100 => 2
[1,1,1,1] => 1111 => 4
[1,1,2] => 1110 => 3
[1,2,1] => 1101 => 2
[1,3] => 1100 => 2
[2,1,1] => 1011 => 2
[2,2] => 1010 => 1
[3,1] => 1001 => 2
[4] => 1000 => 3
[1,1,1,1,1] => 11111 => 5
[1,1,1,2] => 11110 => 4
[1,1,2,1] => 11101 => 3
[1,1,3] => 11100 => 3
[1,2,1,1] => 11011 => 2
[1,2,2] => 11010 => 2
[1,3,1] => 11001 => 2
[1,4] => 11000 => 3
[2,1,1,1] => 10111 => 3
[2,1,2] => 10110 => 2
[2,2,1] => 10101 => 1
[2,3] => 10100 => 2
[3,1,1] => 10011 => 2
[3,2] => 10010 => 2
[4,1] => 10001 => 3
[5] => 10000 => 4
[1,1,1,1,1,1] => 111111 => 6
[1,1,1,1,2] => 111110 => 5
[1,1,1,2,1] => 111101 => 4
[1,1,1,3] => 111100 => 4
[1,1,2,1,1] => 111011 => 3
[1,1,2,2] => 111010 => 3
[1,1,3,1] => 111001 => 3
[1,1,4] => 111000 => 3
[1,2,1,1,1] => 110111 => 3
[1,2,1,2] => 110110 => 2
[1,2,2,1] => 110101 => 2
[1,2,3] => 110100 => 2
[1,3,1,1] => 110011 => 2
[1,3,2] => 110010 => 2
[1,4,1] => 110001 => 3
[1,5] => 110000 => 4
[2,1,1,1,1] => 101111 => 4
[2,1,1,2] => 101110 => 3
[2,1,2,1] => 101101 => 2
Description
The length of the longest constant subword.
Matching statistic: St000983
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 1
[1,1] => 11 => 1
[2] => 10 => 2
[1,1,1] => 111 => 1
[1,2] => 110 => 2
[2,1] => 101 => 3
[3] => 100 => 2
[1,1,1,1] => 1111 => 1
[1,1,2] => 1110 => 2
[1,2,1] => 1101 => 3
[1,3] => 1100 => 2
[2,1,1] => 1011 => 3
[2,2] => 1010 => 4
[3,1] => 1001 => 2
[4] => 1000 => 2
[1,1,1,1,1] => 11111 => 1
[1,1,1,2] => 11110 => 2
[1,1,2,1] => 11101 => 3
[1,1,3] => 11100 => 2
[1,2,1,1] => 11011 => 3
[1,2,2] => 11010 => 4
[1,3,1] => 11001 => 2
[1,4] => 11000 => 2
[2,1,1,1] => 10111 => 3
[2,1,2] => 10110 => 3
[2,2,1] => 10101 => 5
[2,3] => 10100 => 4
[3,1,1] => 10011 => 2
[3,2] => 10010 => 3
[4,1] => 10001 => 2
[5] => 10000 => 2
[1,1,1,1,1,1] => 111111 => 1
[1,1,1,1,2] => 111110 => 2
[1,1,1,2,1] => 111101 => 3
[1,1,1,3] => 111100 => 2
[1,1,2,1,1] => 111011 => 3
[1,1,2,2] => 111010 => 4
[1,1,3,1] => 111001 => 2
[1,1,4] => 111000 => 2
[1,2,1,1,1] => 110111 => 3
[1,2,1,2] => 110110 => 3
[1,2,2,1] => 110101 => 5
[1,2,3] => 110100 => 4
[1,3,1,1] => 110011 => 2
[1,3,2] => 110010 => 3
[1,4,1] => 110001 => 2
[1,5] => 110000 => 2
[2,1,1,1,1] => 101111 => 3
[2,1,1,2] => 101110 => 3
[2,1,2,1] => 101101 => 3
Description
The length of the longest alternating subword.
This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.
The following 274 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001530The depth of a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000010The length of the partition. St000025The number of initial rises of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000308The height of the tree associated to a permutation. St000392The length of the longest run of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000527The width of the poset. St000628The balance of a binary word. St000676The number of odd rises of a Dyck path. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000734The last entry in the first row of a standard tableau. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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:
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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001268The size of the largest ordinal summand in the poset. St001372The length of a longest cyclic run of ones of a binary word. St001399The distinguishing number of a poset. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001652The length of a longest interval of consecutive numbers. St001809The index of the step at the first peak of maximal height in a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000094The depth of an ordered tree. St000141The maximum drop size of a permutation. St000209Maximum difference of elements in cycles. St000306The bounce count of a Dyck path. St000439The position of the first down step of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000662The staircase size of the code of a permutation. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001090The number of pop-stack-sorts needed to sort a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000054The first entry of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000328The maximum number of child nodes in a tree. St000378The diagonal inversion number of an integer partition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000470The number of runs in a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000505The biggest entry in the block containing the 1. St000528The height of a poset. St000542The number of left-to-right-minima of a permutation. St000651The maximal size of a rise in a permutation. St000691The number of changes of a binary word. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St000822The Hadwiger number of the graph. St000877The depth of the binary word interpreted as a path. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001058The breadth of the ordered tree. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call 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 special CNakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001494The Alon-Tarsi number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001580The acyclic chromatic number of a graph. St001662The length of the longest factor of consecutive numbers in a permutation. St001674The number of vertices of the largest induced star graph in the graph. St001717The largest size of an interval in a poset. St001733The number of weak left to right maxima of a Dyck path. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000080The rank of the poset. St000157The number of descents of a standard tableau. St000171The degree of the graph. St000225Difference between largest and smallest parts in a partition. St000245The number of ascents of a permutation. St000272The treewidth of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000996The number of exclusive left-to-right maxima of a permutation. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. 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. St001180Number of indecomposable injective modules with projective dimension at most 1. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001644The dimension of a graph. St001826The maximal number of leaves on a vertex of a graph. St001962The proper pathwidth of a graph. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St000956The maximal displacement of a permutation. St000061The number of nodes on the left branch of a binary tree. St000504The cardinality of the first block of a set partition. St000653The last descent of a permutation. St000702The number of weak deficiencies of a permutation. St000823The number of unsplittable factors of the set partition. St000083The number of left oriented leafs of a binary tree except the first one. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000730The maximal arc length of a set partition. St000874The position of the last double rise in a Dyck path. St000989The number of final rises of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001497The position of the largest weak excedence of a permutation. St000652The maximal difference between successive positions of a permutation. St000746The number of pairs with odd minimum in a perfect matching. St000840The number of closers smaller than the largest opener in a perfect matching. 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. 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. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000075The orbit size of a standard tableau under promotion. St001654The monophonic hull number of a graph. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. 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 $eA$. St001589The nesting number of a perfect matching. St000155The number of exceedances (also excedences) of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. 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 $eA$. St001590The crossing number of a perfect matching. St000317The cycle descent number of a permutation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001060The distinguishing index of a graph. St000260The radius of a connected graph. St001118The acyclic chromatic index of a graph. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000455The second largest eigenvalue of a graph if it is integral. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree. St001863The number of weak excedances of a signed permutation. St001568The smallest positive integer that does not appear twice in the partition. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001267The length of the Lyndon factorization of the binary word. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001712The number of natural descents of a standard Young tableau. St001948The number of augmented double ascents of a permutation. St001875The number of simple modules with projective dimension at most 1. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St000035The number of left outer peaks of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000441The number of successions of a permutation. St000871The number of very big ascents of a permutation. St000307The number of rowmotion orbits of a poset. St000632The jump number of the poset. St000056The decomposition (or block) number of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000239The number of small weak excedances. St000254The nesting number of a set partition. St000354The number of recoils of a permutation. St000502The number of successions of a set partitions. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000864The number of circled entries of the shifted recording tableau of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001864The number of excedances of a signed permutation. St000023The number of inner peaks of a permutation. St000091The descent variation of a composition. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000234The number of global ascents of a permutation. St000353The number of inner valleys of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000486The number of cycles of length at least 3 of a permutation. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St001388The number of non-attacking neighbors of a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001728The number of invisible descents of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. 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.
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