Your data matches 40 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000381
(load all 15 compositions to match this statistic)
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => 1
[1,0,1,0]
 => [1,1] => [2] => 2
[1,1,0,0]
 => [2] => [1,1] => 1
[1,0,1,0,1,0]
 => [1,1,1] => [3] => 3
[1,0,1,1,0,0]
 => [1,2] => [1,2] => 2
[1,1,0,0,1,0]
 => [2,1] => [2,1] => 2
[1,1,0,1,0,0]
 => [2,1] => [2,1] => 2
[1,1,1,0,0,0]
 => [3] => [1,1,1] => 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => 4
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,3] => 3
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,2] => 2
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [2,2] => 2
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,1,2] => 2
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [3,1] => 3
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,2,1] => 2
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [3,1] => 3
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [3,1] => 3
[1,1,0,1,1,0,0,0]
 => [2,2] => [1,2,1] => 2
[1,1,1,0,0,0,1,0]
 => [3,1] => [2,1,1] => 2
[1,1,1,0,0,1,0,0]
 => [3,1] => [2,1,1] => 2
[1,1,1,0,1,0,0,0]
 => [3,1] => [2,1,1] => 2
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1] => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,4] => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,3] => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [2,3] => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,1,3] => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [3,2] => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,2,2] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [3,2] => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [3,2] => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [1,2,2] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [2,1,2] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [2,1,2] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [2,1,2] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,1,1,2] => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [4,1] => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [2,2,1] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,2,1] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [4,1] => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [4,1] => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [4,1] => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,3,1] => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [2,2,1] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [2,2,1] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [2,2,1] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1,2,1] => 2
Description
The largest part of an integer composition.
Matching statistic: St000684
(load all 3 compositions to match this statistic)
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
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%
Values
[1,0]
 => [1] => [1,0]
 => 1
[1,0,1,0]
 => [1,1] => [1,0,1,0]
 => 2
[1,1,0,0]
 => [2] => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,0,1,1,0,0]
 => [1,2] => [1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [2,1] => [1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => [2,1] => [1,1,0,0,1,0]
 => 2
[1,1,1,0,0,0]
 => [3] => [1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
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 3 compositions to match this statistic)
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
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%
Values
[1,0]
 => [1] => [1,0]
 => 1
[1,0,1,0]
 => [1,1] => [1,0,1,0]
 => 2
[1,1,0,0]
 => [2] => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,0,1,1,0,0]
 => [1,2] => [1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [2,1] => [1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => [2,1] => [1,1,0,0,1,0]
 => 2
[1,1,1,0,0,0]
 => [3] => [1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
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: St000013
(load all 2 compositions to match this statistic)
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
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%
Values
[1,0]
 => [1] => [1] => [1,0]
 => 1
[1,0,1,0]
 => [1,1] => [2] => [1,1,0,0]
 => 2
[1,1,0,0]
 => [2] => [1,1] => [1,0,1,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 3
[1,0,1,1,0,0]
 => [1,2] => [1,2] => [1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [2,1] => [2,1] => [1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => [2,1] => [2,1] => [1,1,0,0,1,0]
 => 2
[1,1,1,0,0,0]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 3
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 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
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
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%
Values
[1,0]
 => [1] => [1] => [1]
 => 1
[1,0,1,0]
 => [1,1] => [2] => [2]
 => 2
[1,1,0,0]
 => [2] => [1,1] => [1,1]
 => 1
[1,0,1,0,1,0]
 => [1,1,1] => [3] => [3]
 => 3
[1,0,1,1,0,0]
 => [1,2] => [2,1] => [2,1]
 => 2
[1,1,0,0,1,0]
 => [2,1] => [1,2] => [2,1]
 => 2
[1,1,0,1,0,0]
 => [2,1] => [1,2] => [2,1]
 => 2
[1,1,1,0,0,0]
 => [3] => [1,1,1] => [1,1,1]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => [4]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [3,1] => [3,1]
 => 3
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,2] => [2,2]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [2,2] => [2,2]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,3] => [2,1,1] => [2,1,1]
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,3] => [3,1]
 => 3
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,2,1] => [2,1,1]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [1,3] => [3,1]
 => 3
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,3] => [3,1]
 => 3
[1,1,0,1,1,0,0,0]
 => [2,2] => [1,2,1] => [2,1,1]
 => 2
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,1,2] => [2,1,1]
 => 2
[1,1,1,0,0,1,0,0]
 => [3,1] => [1,1,2] => [2,1,1]
 => 2
[1,1,1,0,1,0,0,0]
 => [3,1] => [1,1,2] => [2,1,1]
 => 2
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1] => [1,1,1,1]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => [5]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [4,1] => [4,1]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [3,2] => [3,2]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [3,2] => [3,2]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [3,1,1] => [3,1,1]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [2,3] => [3,2]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1] => [2,2,1]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [2,3] => [3,2]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [2,3] => [3,2]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [2,2,1] => [2,2,1]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [2,1,2] => [2,2,1]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [2,1,2] => [2,2,1]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [2,1,2] => [2,2,1]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [2,1,1,1] => [2,1,1,1]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,4] => [4,1]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => [3,1,1]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [1,2,2] => [2,2,1]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [1,2,2] => [2,2,1]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,2,1,1] => [2,1,1,1]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [1,4] => [4,1]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => [3,1,1]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [1,4] => [4,1]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,4] => [4,1]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,3,1] => [3,1,1]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [1,2,2] => [2,2,1]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [1,2,2] => [2,2,1]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [1,2,2] => [2,2,1]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,2,1,1] => [2,1,1,1]
 => 2
Description
The largest part of an integer partition.
Matching statistic: St000982
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => 1 => 1 => 1
[1,0,1,0]
 => [1,1] => 11 => 11 => 2
[1,1,0,0]
 => [2] => 10 => 01 => 1
[1,0,1,0,1,0]
 => [1,1,1] => 111 => 111 => 3
[1,0,1,1,0,0]
 => [1,2] => 110 => 011 => 2
[1,1,0,0,1,0]
 => [2,1] => 101 => 001 => 2
[1,1,0,1,0,0]
 => [2,1] => 101 => 001 => 2
[1,1,1,0,0,0]
 => [3] => 100 => 101 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1111 => 1111 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,2] => 1110 => 0111 => 3
[1,0,1,1,0,0,1,0]
 => [1,2,1] => 1101 => 0011 => 2
[1,0,1,1,0,1,0,0]
 => [1,2,1] => 1101 => 0011 => 2
[1,0,1,1,1,0,0,0]
 => [1,3] => 1100 => 1011 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,1] => 1011 => 0001 => 3
[1,1,0,0,1,1,0,0]
 => [2,2] => 1010 => 1001 => 2
[1,1,0,1,0,0,1,0]
 => [2,1,1] => 1011 => 0001 => 3
[1,1,0,1,0,1,0,0]
 => [2,1,1] => 1011 => 0001 => 3
[1,1,0,1,1,0,0,0]
 => [2,2] => 1010 => 1001 => 2
[1,1,1,0,0,0,1,0]
 => [3,1] => 1001 => 1101 => 2
[1,1,1,0,0,1,0,0]
 => [3,1] => 1001 => 1101 => 2
[1,1,1,0,1,0,0,0]
 => [3,1] => 1001 => 1101 => 2
[1,1,1,1,0,0,0,0]
 => [4] => 1000 => 0101 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 11111 => 11111 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 11110 => 01111 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 11101 => 00111 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => 11101 => 00111 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 11100 => 10111 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 11011 => 00011 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 11010 => 10011 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => 11011 => 00011 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => 11011 => 00011 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => 11010 => 10011 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 11001 => 11011 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => 11001 => 11011 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => 11001 => 11011 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 11000 => 01011 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => 10111 => 00001 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => 10110 => 10001 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 10101 => 11001 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => 10101 => 11001 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => 10100 => 01001 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => 10111 => 00001 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => 10110 => 10001 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => 10111 => 00001 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => 10111 => 00001 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => 10110 => 10001 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => 10101 => 11001 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => 10101 => 11001 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => 10101 => 11001 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => 10100 => 01001 => 2
Description
The length of the longest constant subword.
Matching statistic: St000444
(load all 15 compositions to match this statistic)
Mapping
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1,0]
 => ? = 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St001062
(load all 10 compositions to match this statistic)
Mapping
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => {{1}}
 => ? = 1
[1,0,1,0]
 => [1,1,0,0]
 => {{1,2}}
 => 2
[1,1,0,0]
 => [1,0,1,0]
 => {{1},{2}}
 => 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 3
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 2
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 2
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 2
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 3
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 2
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 3
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 2
[1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 3
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 3
[1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 2
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 2
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 2
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 2
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => {{1,4,5},{2,3}}
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => {{1,2,5},{3,4}}
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => {{1,4},{2,3},{5}}
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => {{1,2,3,5},{4}}
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => {{1,2,4},{3},{5}}
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => {{1,5},{2,4},{3}}
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 3
Description
The maximal size of a block of a set partition.
Matching statistic: St000392
(load all 6 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ 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%
Values
[1,0]
 => [1] => => ? = 1 - 1
[1,0,1,0]
 => [2,1] => 1 => 1 = 2 - 1
[1,1,0,0]
 => [1,2] => 0 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [3,2,1] => 11 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [2,3,1] => 01 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [3,1,2] => 10 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,1,3] => 10 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,2,3] => 00 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 111 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 011 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 101 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 101 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 001 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 110 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 010 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 110 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 110 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => 010 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 100 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 100 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 100 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 000 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 1111 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => 0111 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => 1011 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => 1011 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 0011 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => 1101 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 0101 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => 1101 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => 1101 => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => 0101 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 1001 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => 1001 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 1001 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0001 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => 1110 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => 0110 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 1010 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => 1010 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 0010 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => 1110 => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => 0110 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => 1110 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => 1110 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => 0110 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => 1010 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => 1010 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => 1010 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => 0010 => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 1100 => 2 = 3 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001039
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [.,.]
 => [1,0]
 => ? = 1
[1,0,1,0]
 => [1,2] => [.,[.,.]]
 => [1,0,1,0]
 => 2
[1,1,0,0]
 => [2,1] => [[.,.],.]
 => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 3
[1,0,1,1,0,0]
 => [1,3,2] => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 2
[1,1,1,0,0,0]
 => [3,1,2] => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000442The maximal area to the right of an up step of a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000451The length of the longest pattern of the form k 1 2. St001090The number of pop-stack-sorts needed to sort a permutation. St000662The staircase size of the code of a permutation. St000209Maximum difference of elements in cycles. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000956The maximal displacement of a permutation. St000308The height of the tree associated to a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000141The maximum drop size of a permutation. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001372The length of a longest cyclic run of ones of a binary word. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. 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. St000328The maximum number of child nodes in a tree. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. 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. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000983The length of the longest alternating subword. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001624The breadth of a lattice.