searching the database
Your data matches 40 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000381
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
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%
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)
(load all 3 compositions to match this statistic)
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%
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: St000147
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%
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
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%
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: St000013
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●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] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 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] => [3,1] => [1,1,1,0,0,0,1,0]
=> 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] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 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] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 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] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 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] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 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] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 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] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 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] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 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] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,2,2] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,2,2] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,3,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,3,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,3,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
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: St000392
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●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
[1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,2,3,6,1] => ? => ? = 5 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [7,6,4,3,2,5,8,1] => ? => ? = 5 - 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [8,7,6,4,3,5,1,2] => ? => ? = 5 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,3,6,1,2] => ? => ? = 5 - 1
[1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,2,3,1,6] => ? => ? = 5 - 1
[1,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,3,1,2,5] => ? => ? = 5 - 1
[1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,2,1,3,6] => ? => ? = 6 - 1
[1,1,1,0,1,0,0,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,2,1,3,8] => ? => ? = 5 - 1
[1,1,1,0,1,0,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,2,1,4,8] => ? => ? = 5 - 1
[1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> [8,6,5,3,1,2,4,7] => ? => ? = 5 - 1
[1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [7,6,5,3,1,2,4,8] => ? => ? = 5 - 1
[1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [8,7,4,3,1,2,5,6] => ? => ? = 5 - 1
[1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> [8,5,4,3,1,2,6,7] => ? => ? = 5 - 1
[1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> [7,5,4,2,1,3,6,8] => ? => ? = 5 - 1
[1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [8,6,3,2,1,4,5,7] => ? => ? = 5 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000444
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 97%●distinct values known / distinct values provided: 88%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 97%●distinct values known / distinct values provided: 88%
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] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 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] => [3,1] => [1,1,1,0,0,0,1,0]
=> 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] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 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] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 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] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 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] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 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] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 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] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 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] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 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] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,2,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,2,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,3] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,2,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,2,2] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,2,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,2,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,2,2] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,3,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,3,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,3,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,4] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,2,1,1,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St000442
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 97%●distinct values known / distinct values provided: 88%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 97%●distinct values known / distinct values provided: 88%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 1 - 1
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 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]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[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]
=> 1 = 2 - 1
[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]
=> 1 = 2 - 1
[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]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[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]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[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]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[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]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,2,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,2,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,3] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,2,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,2,2] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,2,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,2,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,2,2] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,3,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,3,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,3,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,4] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,2,1,1,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St001039
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: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
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: 94% ●values known / values provided: 94%●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
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6,8,7] => ?
=> ?
=> ? = 5
[1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,4,2,5,6,3,7,8] => ?
=> ?
=> ? = 5
[1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,6,3,7,8] => ?
=> ?
=> ? = 5
[1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,7,3,8] => ?
=> ?
=> ? = 5
[1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,4,2,5,6,8,7] => ?
=> ?
=> ? = 5
[1,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0]
=> [3,1,4,5,2,6,8,7] => ?
=> ?
=> ? = 5
[1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,6,2,7,8] => ?
=> ?
=> ? = 6
[1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
=> [3,4,1,5,2,6,8,7] => ?
=> ?
=> ? = 5
[1,1,1,0,1,0,1,0,0,1,0,1,1,0,0,0]
=> [3,4,5,1,6,8,2,7] => ?
=> ?
=> ? = 5
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [4,1,2,5,3,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [4,1,2,5,6,3,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> [4,1,2,5,6,7,3,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,8,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [4,1,5,2,3,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> [4,1,5,2,6,3,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> [4,1,5,2,6,7,3,8] => ?
=> ?
=> ? = 5
[1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [4,1,5,2,6,7,8,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [4,1,5,6,2,3,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [4,1,5,6,2,7,3,8] => ?
=> ?
=> ? = 5
[1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,7,8,3] => ?
=> ?
=> ? = 5
[1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> [4,1,5,6,7,2,3,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> [4,1,5,6,7,2,8,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [4,1,5,6,7,8,2,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [4,5,1,2,3,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [4,5,1,2,6,3,7,8] => ?
=> ?
=> ? = 5
[1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> [4,5,1,2,6,7,3,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [4,5,1,2,6,7,8,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [4,5,1,6,2,3,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [4,5,1,6,2,7,3,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
=> [4,5,1,6,2,7,8,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [4,5,1,6,7,2,3,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> [4,5,1,6,7,2,8,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> [4,5,1,6,7,8,2,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [4,5,6,1,2,3,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [4,5,6,1,2,7,3,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,6,1,2,7,8,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [4,5,6,1,7,2,3,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> [4,5,6,1,7,2,8,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> [4,5,6,1,7,8,2,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [4,5,6,7,1,2,3,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,7,1,2,8,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [4,5,6,7,1,8,2,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,8,1,2,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St001062
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●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
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> {{1,7,8},{2,3,4,5,6}}
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,5,6,7}}
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> {{1,7},{2,3,4,5,6},{8}}
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> {{1,2,3,4,5},{6},{7,8}}
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> {{1,2,3,4,5},{6,8},{7}}
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> {{1,8},{2,3,4,5,6},{7}}
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 5
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> {{1,5,6,7,8},{2,3,4}}
=> ? = 5
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> {{1,2,6,7,8},{3,4,5}}
=> ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 5
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2},{3,4,5,6,7},{8}}
=> ? = 5
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,4,5,6,7},{2,3},{8}}
=> ? = 5
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,5,6,7},{3,4},{8}}
=> ? = 5
[1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,6,7},{4,5},{8}}
=> ? = 5
[1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,7},{5,6},{8}}
=> ? = 5
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2},{3},{4,5,6,7,8}}
=> ? = 5
[1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> {{1,2},{3,5,6,7,8},{4}}
=> ? = 5
[1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> {{1,2},{3,4,6,7,8},{5}}
=> ? = 5
[1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> {{1,2},{3,4,5,7,8},{6}}
=> ? = 5
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1,2},{3,4,5,6,8},{7}}
=> ? = 5
[1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> {{1,5,6,7,8},{2,3},{4}}
=> ? = 5
[1,0,1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> {{1,4,6,7,8},{2,3},{5}}
=> ? = 5
[1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
=> {{1,4,5,7,8},{2,3},{6}}
=> ? = 5
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> {{1,4,5,6,8},{2,3},{7}}
=> ? = 5
[1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> {{1,2,6,7,8},{3,4},{5}}
=> ? = 5
[1,0,1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> {{1,2,5,7,8},{3,4},{6}}
=> ? = 5
[1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
=> {{1,2,5,6,8},{3,4},{7}}
=> ? = 5
[1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,5},{6}}
=> ? = 5
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> {{1,2,3,6,8},{4,5},{7}}
=> ? = 5
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6},{7}}
=> ? = 5
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> {{1},{2,3,4,5,6},{7,8}}
=> ? = 5
[1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> {{1},{2,8},{3,4,5,6,7}}
=> ? = 5
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> {{1},{2,3,4,5,6},{7},{8}}
=> ? = 5
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 5
[1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> {{1},{2,5,6,7,8},{3,4}}
=> ? = 5
[1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> {{1},{2,3,6,7,8},{4,5}}
=> ? = 5
[1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> {{1},{2,3,4,7,8},{5,6}}
=> ? = 5
[1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,8},{6,7}}
=> ? = 5
[1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> {{1,3,4,5,6,7},{2},{8}}
=> ? = 6
[1,1,0,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> {{1,3,4,5,6},{2},{7,8}}
=> ? = 5
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> {{1,8},{2},{3,4,5,6,7}}
=> ? = 5
[1,1,0,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> {{1,3,4,5,6},{2},{7},{8}}
=> ? = 5
[1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,4,5,6,7},{3},{8}}
=> ? = 6
[1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> {{1,2,4,5,6},{3},{7,8}}
=> ? = 5
Description
The maximal size of a block of a set partition.
The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000686The finitistic dominant dimension 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.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!