Your data matches 159 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001530
(load all 6 compositions to match this statistic)
Mapping
St001530: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 1
[1,0,1,0]
 => 2
[1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => 3
[1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,1,0,0]
 => 3
[1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,0]
 => 3
[1,1,0,1,1,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => 2
[1,1,1,0,1,0,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => 2
Description
The depth of a Dyck path. That is the depth of the corresponding Nakayama algebra with a linear quiver.
Matching statistic: St000374
(load all 8 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
 => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => [3,1,2] => 2 = 3 - 1
[1,1,1,0,0,0]
 => [3,2,1] => [2,3,1] => [3,2,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,3,4,2] => [1,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 3 = 4 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,4,1,2] => [4,3,2,1] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,3,1,4] => [3,2,1,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,4,1,3] => [4,2,1,3] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [2,3,4,1] => [4,2,3,1] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [3,2,4,1] => [2,4,3,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,4,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,4,5,2,3] => [1,5,4,3,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,3,5,2,4] => [1,5,3,2,4] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [1,3,4,5,2] => [1,5,3,4,2] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,4,3,5,2] => [1,3,5,4,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,5,4,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,5,1,2,3] => [5,4,2,3,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,4,1,2,5] => [4,3,2,1,5] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,5,1,2,4] => [5,3,2,1,4] => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,4,5,1,2] => [5,4,3,2,1] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [4,3,5,1,2] => [4,5,3,2,1] => 1 = 2 - 1
Description
The number of exclusive right-to-left minima of a permutation. This is the number of right-to-left minima that are not left-to-right maxima. This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3. Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$. See also [[St000213]] and [[St000119]].
Matching statistic: St001418
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001418: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [1,2] => [1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0]
 => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,0]
 => [2,3,1] => [1,2,3] => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,5,6,4,3,2] => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,4,6,5,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,6,4,5,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,4,5,6,3,2] => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,3,6,5,4,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,3,5,6,4,2] => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,6,5,3,4,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,6,4,2,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,5,6,3,4,2] => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,3,4,6,5,2] => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,3,6,4,5,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,6,3,4,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,3,4,5,6,2] => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,2,4,6,5,3] => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,6,5,4,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,5,6,4,2,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,6,5,3,2,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,5,3,2,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,5,6,3,2,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,4,6,5,2,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,6,4,5,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,6,3,2,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,4,5,6,2,3] => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 3 = 4 - 1
Description
Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. The stable Auslander algebra is by definition the stable endomorphism ring of the direct sum of all indecomposable modules.
Matching statistic: St000068
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 1
[1,0,1,0]
 => [2,1] => [2,1] => ([],2)
 => 2
[1,1,0,0]
 => [1,2] => [1,2] => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => [2,3,1] => ([(1,2)],3)
 => 2
[1,0,1,1,0,0]
 => [2,3,1] => [3,2,1] => ([],3)
 => 3
[1,1,0,0,1,0]
 => [3,1,2] => [3,1,2] => ([(1,2)],3)
 => 2
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => 2
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 2
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [4,3,1,2] => ([(2,3)],4)
 => 3
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 2
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,2,3,1] => ([(2,3)],4)
 => 3
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,4,2,1] => ([(2,3)],4)
 => 3
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [4,3,2,1] => ([],4)
 => 4
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 2
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 2
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => 2
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [5,4,3,1,2] => ([(3,4)],5)
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [4,5,3,2,1] => ([(3,4)],5)
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [5,4,3,2,1] => ([],5)
 => 5
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,6,1] => [3,4,6,1,5,2] => ([(0,4),(1,3),(1,5),(4,2),(4,5)],6)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [4,5,3,2,6,1] => [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [6,3,4,2,5,1] => [3,5,4,1,6,2] => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,5)],6)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,6,1] => [4,6,3,1,5,2] => ([(0,5),(1,4),(1,5),(2,3),(2,5)],6)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [5,4,2,3,6,1] => [4,6,1,2,5,3] => ([(0,4),(1,3),(1,5),(4,2),(4,5)],6)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [4,5,2,3,6,1] => [6,4,1,2,5,3] => ([(1,5),(2,3),(3,4),(3,5)],6)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [5,3,2,4,6,1] => [3,4,1,6,5,2] => ([(0,3),(1,2),(1,4),(1,5),(3,4),(3,5)],6)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,6,1] => [3,6,1,4,5,2] => ([(0,4),(0,5),(1,3),(1,5),(5,2)],6)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [5,3,4,2,1,6] => [3,5,4,1,2,6] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,1,6] => [3,5,1,4,2,6] => ([(0,3),(0,4),(1,2),(1,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [6,4,2,1,3,5] => [4,6,1,3,2,5] => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,5)],6)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [5,4,2,1,3,6] => [4,5,1,3,2,6] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,2,2,3,3,3,3,4}
Description
The number of minimal elements in a poset.
Matching statistic: St000996
(load all 7 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[1,0,1,0]
 => [3,1,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,0]
 => [2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,3,4,2] => [1,3,4,2] => 2 = 3 - 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,5,4,3,2] => [1,5,2,3,4] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,4,5,3,2] => [1,4,5,2,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,3,5,4,2] => [1,3,5,2,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,5,3,4,2] => [1,5,2,3,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,3,4,5,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,2,5,4,3] => [1,2,5,3,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,2,4,5,3] => [1,2,4,5,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,5,4,2,3] => [1,5,2,3,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,5,3,2,4] => [1,5,2,4,3] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,4,5,2,3] => [1,4,5,2,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,2,5,3,4] => [1,2,5,3,4] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,5,2,3,4] => [1,5,2,3,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,6,2,3,4,5] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [1,5,6,4,3,2] => [1,5,6,2,3,4] => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,4,6,5,3,2] => [1,4,6,2,3,5] => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,6,4,5,3,2] => [1,6,2,3,4,5] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [1,4,5,6,3,2] => [1,4,5,6,2,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,3,6,5,4,2] => [1,3,6,2,4,5] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [1,3,5,6,4,2] => [1,3,5,6,2,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,6,5,3,4,2] => [1,6,2,3,4,5] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,6,4,2,3,5] => [1,6,2,3,5,4] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [1,5,6,3,4,2] => [1,5,6,2,3,4] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [1,3,4,6,5,2] => [1,3,4,6,2,5] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,3,6,4,5,2] => [1,3,6,2,4,5] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,6,3,4,5,2] => [1,6,2,3,4,5] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [1,3,4,5,6,2] => [1,3,4,5,6,2] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,2,5,6,4,3] => [1,2,5,6,3,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,2,4,6,5,3] => [1,2,4,6,3,5] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,2,6,4,5,3] => [1,2,6,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,2,4,5,6,3] => [1,2,4,5,6,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,6,5,4,2,3] => [1,6,2,3,4,5] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [1,5,6,4,2,3] => [1,5,6,2,3,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,6,5,3,2,4] => [1,6,2,4,3,5] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,5,3,2,6,4] => [1,5,2,6,3,4] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [1,5,6,3,2,4] => [1,5,6,2,4,3] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,4,6,5,2,3] => [1,4,6,2,3,5] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,6,4,5,2,3] => [1,6,2,3,4,5] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,6,3,2,4,5] => [1,6,2,4,5,3] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [1,4,5,6,2,3] => [1,4,5,6,2,3] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => [1,7,5,3,2,4,6] => [1,7,2,4,6,3,5] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => [1,7,6,4,2,3,5] => [1,7,2,3,5,4,6] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => [1,6,7,4,2,3,5] => [1,6,7,2,3,5,4] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [1,7,4,2,3,5,6] => [1,7,2,3,5,6,4] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [1,7,5,2,3,4,6] => [1,7,2,3,4,6,5] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [1,7,4,5,2,3,6] => [1,7,2,3,6,4,5] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,4,1,2,3,5,6] => [1,7,6,5,3,2,4] => [1,7,2,4,3,5,6] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,4,1,2,3,7,5] => [1,6,7,5,3,2,4] => [1,6,7,2,4,3,5] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => [1,7,5,3,2,6,4] => [1,7,2,6,3,4,5] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => [1,5,3,2,6,7,4] => [1,5,2,6,7,3,4] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => [1,5,7,6,3,2,4] => [1,5,7,2,4,3,6] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [1,7,5,6,3,2,4] => [1,7,2,4,3,5,6] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => [1,5,6,7,3,2,4] => [1,5,6,7,2,4,3] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [1,7,5,2,3,4,6] => [1,7,2,3,4,6,5] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [1,7,6,3,2,4,5] => [1,7,2,4,5,3,6] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [7,4,1,6,2,3,5] => [1,7,5,2,4,6,3] => [1,7,2,4,6,3,5] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [1,6,7,3,2,4,5] => [1,6,7,2,4,5,3] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => [1,7,3,2,4,5,6] => [1,7,2,4,5,6,3] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => [1,7,6,4,2,3,5] => [1,7,2,3,5,4,6] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => [1,6,7,4,2,3,5] => [1,6,7,2,3,5,4] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => [1,7,6,3,4,2,5] => [1,7,2,5,3,4,6] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [6,7,5,1,2,3,4] => [1,6,3,5,2,7,4] => [1,6,2,7,3,5,4] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => [1,6,7,3,4,2,5] => [1,6,7,2,5,3,4] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => [1,7,4,2,3,5,6] => [1,7,2,3,5,6,4] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => [1,7,3,4,2,5,6] => [1,7,2,5,6,3,4] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => [1,7,5,2,3,4,6] => [1,7,2,3,4,6,5] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => [1,7,4,5,2,3,6] => [1,7,2,3,6,4,5] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => [1,7,3,4,5,2,6] => [1,7,2,6,3,4,5] => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4} - 1
Description
The number of exclusive left-to-right maxima of a permutation. This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St001372
(load all 5 compositions to match this statistic)
Mapping
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 10 => 01 => 01 => 1
[1,0,1,0]
 => 1010 => 0101 => 1001 => 2
[1,1,0,0]
 => 1100 => 1001 => 0101 => 1
[1,0,1,0,1,0]
 => 101010 => 010101 => 110001 => 3
[1,0,1,1,0,0]
 => 101100 => 011001 => 010101 => 1
[1,1,0,0,1,0]
 => 110010 => 100101 => 101001 => 2
[1,1,0,1,0,0]
 => 110100 => 101001 => 011001 => 2
[1,1,1,0,0,0]
 => 111000 => 110001 => 001101 => 2
[1,0,1,0,1,0,1,0]
 => 10101010 => 01010101 => 11100001 => 4
[1,0,1,0,1,1,0,0]
 => 10101100 => 01011001 => 01100101 => 2
[1,0,1,1,0,0,1,0]
 => 10110010 => 01100101 => 10101001 => 2
[1,0,1,1,0,1,0,0]
 => 10110100 => 01101001 => 01101001 => 2
[1,0,1,1,1,0,0,0]
 => 10111000 => 01110001 => 00101101 => 2
[1,1,0,0,1,0,1,0]
 => 11001010 => 10010101 => 11010001 => 3
[1,1,0,0,1,1,0,0]
 => 11001100 => 10011001 => 01010101 => 1
[1,1,0,1,0,0,1,0]
 => 11010010 => 10100101 => 10110001 => 2
[1,1,0,1,0,1,0,0]
 => 11010100 => 10101001 => 01110001 => 3
[1,1,0,1,1,0,0,0]
 => 11011000 => 10110001 => 00110101 => 2
[1,1,1,0,0,0,1,0]
 => 11100010 => 11000101 => 10011001 => 2
[1,1,1,0,0,1,0,0]
 => 11100100 => 11001001 => 01011001 => 2
[1,1,1,0,1,0,0,0]
 => 11101000 => 11010001 => 00111001 => 3
[1,1,1,1,0,0,0,0]
 => 11110000 => 11100001 => 00011101 => 3
[1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 0101010101 => 1111000001 => 5
[1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 0101011001 => 0111000101 => 3
[1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 0101100101 => 1011001001 => 2
[1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 0101101001 => 0111001001 => 3
[1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 0101110001 => 0011001101 => 2
[1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 0110010101 => 1101010001 => 3
[1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 0110011001 => 0101010101 => 1
[1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 0110100101 => 1011010001 => 2
[1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 0110101001 => 0111010001 => 3
[1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 0110110001 => 0011010101 => 2
[1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 0111000101 => 1001011001 => 2
[1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 0111001001 => 0101011001 => 2
[1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 0111010001 => 0011011001 => ? ∊ {2,2,2,3,3,3,4,4}
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0111100001 => 0001011101 => 3
[1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 1001010101 => 1110100001 => 4
[1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 1001011001 => 0110100101 => 2
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 1001100101 => 1010101001 => 2
[1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 1001101001 => 0110101001 => 2
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 1001110001 => 0010101101 => 2
[1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 1010010101 => 1101100001 => 3
[1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 1010011001 => 0101100101 => 2
[1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 1010100101 => 1011100001 => 3
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 1010101001 => 0111100001 => 4
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 1010110001 => 0011100101 => ? ∊ {2,2,2,3,3,3,4,4}
[1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 1011000101 => 1001101001 => 2
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 1011001001 => 0101101001 => 2
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 1011010001 => 0011101001 => ? ∊ {2,2,2,3,3,3,4,4}
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 1011100001 => 0001101101 => 2
[1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 1100010101 => 1100110001 => 3
[1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 1100011001 => 0100110101 => 2
[1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 1100100101 => 1010110001 => 2
[1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => 1101010001 => 0011110001 => ? ∊ {2,2,2,3,3,3,4,4}
[1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 1110000101 => 1000111001 => ? ∊ {2,2,2,3,3,3,4,4}
[1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => 1110001001 => 0100111001 => ? ∊ {2,2,2,3,3,3,4,4}
[1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => 1110010001 => 0010111001 => ? ∊ {2,2,2,3,3,3,4,4}
[1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => 1110100001 => 0001111001 => ? ∊ {2,2,2,3,3,3,4,4}
[1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => 010101110001 => 001110001101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,0,1,0,1,1,0,1,1,0,0,0]
 => 101011011000 => 010110110001 => 001110010101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,0,1,0,1,1,1,0,1,0,0,0]
 => 101011101000 => 010111010001 => 001110011001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,0,1,1,0,1,0,1,1,0,0,0]
 => 101101011000 => 011010110001 => 001110100101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,0,1,1,0,1,1,0,1,0,0,0]
 => 101101101000 => 011011010001 => 001110101001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,0,1,1,1,0,1,0,1,0,0,0]
 => 101110101000 => 011101010001 => 001110110001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,0,1,1,1,1,0,0,0,0,1,0]
 => 101111000010 => 011110000101 => 100010111001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,0,1,1,1,1,0,0,0,1,0,0]
 => 101111000100 => 011110001001 => 010010111001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,0,1,1,1,1,0,0,1,0,0,0]
 => 101111001000 => 011110010001 => 001010111001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => 011110100001 => 000110111001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,0,0,1,1,1,0,1,0,0,0]
 => 110011101000 => 100111010001 => 001101011001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,0,1,0,1,0,1,1,0,0,0]
 => 110101011000 => 101010110001 => 001111000101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,0,1,0,1,1,0,1,0,0,0]
 => 110101101000 => 101011010001 => 001111001001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,0,1,1,0,1,0,1,0,0,0]
 => 110110101000 => 101101010001 => 001111010001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,0,1,1,1,0,0,0,0,1,0]
 => 110111000010 => 101110000101 => 100011011001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,0,1,1,1,0,0,0,1,0,0]
 => 110111000100 => 101110001001 => 010011011001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,0,1,1,1,0,0,1,0,0,0]
 => 110111001000 => 101110010001 => 001011011001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,0,1,1,1,0,1,0,0,0,0]
 => 110111010000 => 101110100001 => 000111011001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,0,0,1,0,1,1,0,0,0]
 => 111001011000 => 110010110001 => 001101100101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,0,0,1,1,0,1,0,0,0]
 => 111001101000 => 110011010001 => 001101101001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,0,1,0,0,0,1,1,0,0]
 => 111010001100 => 110100011001 => 010011100101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,0,1,0,0,1,1,0,0,0]
 => 111010011000 => 110100110001 => 001011100101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,0,1,0,1,0,1,0,0,0]
 => 111010101000 => 110101010001 => 001111100001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,0,1,0,1,1,0,0,0,0]
 => 111010110000 => 110101100001 => 000111100101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,0,1,1,0,0,0,0,1,0]
 => 111011000010 => 110110000101 => 100011101001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,0,1,1,0,0,0,1,0,0]
 => 111011000100 => 110110001001 => 010011101001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,0,1,1,0,0,1,0,0,0]
 => 111011001000 => 110110010001 => 001011101001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,0,1,1,0,1,0,0,0,0]
 => 111011010000 => 110110100001 => 000111101001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,1,0,0,1,0,1,0,0,0]
 => 111100101000 => 111001010001 => 001101110001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => 111010000101 => 100011110001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,1,0,1,0,0,0,1,0,0]
 => 111101000100 => 111010001001 => 010011110001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,1,0,1,0,0,1,0,0,0]
 => 111101001000 => 111010010001 => 001011110001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,1,0,1,0,1,0,0,0,0]
 => 111101010000 => 111010100001 => 000111110001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,1,1,0,0,0,0,0,1,0]
 => 111110000010 => 111100000101 => 100001111001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,1,1,0,0,0,0,1,0,0]
 => 111110000100 => 111100001001 => 010001111001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,1,1,0,0,0,1,0,0,0]
 => 111110001000 => 111100010001 => 001001111001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,1,1,0,0,1,0,0,0,0]
 => 111110010000 => 111100100001 => 000101111001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
[1,1,1,1,1,0,1,0,0,0,0,0]
 => 111110100000 => 111101000001 => 000011111001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5}
Description
The length of a longest cyclic run of ones of a binary word. Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
Matching statistic: St000392
(load all 8 compositions to match this statistic)
Mapping
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 10 => 01 => 10 => 1
[1,0,1,0]
 => 1010 => 0101 => 1010 => 1
[1,1,0,0]
 => 1100 => 0110 => 0011 => 2
[1,0,1,0,1,0]
 => 101010 => 010101 => 101010 => 1
[1,0,1,1,0,0]
 => 101100 => 010110 => 001011 => 2
[1,1,0,0,1,0]
 => 110010 => 011001 => 101100 => 2
[1,1,0,1,0,0]
 => 110100 => 011010 => 001101 => 2
[1,1,1,0,0,0]
 => 111000 => 011100 => 001110 => 3
[1,0,1,0,1,0,1,0]
 => 10101010 => 01010101 => 10101010 => 1
[1,0,1,0,1,1,0,0]
 => 10101100 => 01010110 => 00101011 => 2
[1,0,1,1,0,0,1,0]
 => 10110010 => 01011001 => 10101100 => 2
[1,0,1,1,0,1,0,0]
 => 10110100 => 01011010 => 00101101 => 2
[1,0,1,1,1,0,0,0]
 => 10111000 => 01011100 => 00101110 => 3
[1,1,0,0,1,0,1,0]
 => 11001010 => 01100101 => 10110010 => 2
[1,1,0,0,1,1,0,0]
 => 11001100 => 01100110 => 00110011 => 2
[1,1,0,1,0,0,1,0]
 => 11010010 => 01101001 => 10110100 => 2
[1,1,0,1,0,1,0,0]
 => 11010100 => 01101010 => 00110101 => 2
[1,1,0,1,1,0,0,0]
 => 11011000 => 01101100 => 00110110 => 2
[1,1,1,0,0,0,1,0]
 => 11100010 => 01110001 => 10111000 => 3
[1,1,1,0,0,1,0,0]
 => 11100100 => 01110010 => 00111001 => 3
[1,1,1,0,1,0,0,0]
 => 11101000 => 01110100 => 00111010 => 3
[1,1,1,1,0,0,0,0]
 => 11110000 => 01111000 => 00111100 => 4
[1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 0101010101 => 1010101010 => 1
[1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 0101010110 => 0010101011 => 2
[1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 0101011001 => 1010101100 => 2
[1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 0101011010 => 0010101101 => 2
[1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 0101011100 => 0010101110 => 3
[1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 0101100101 => 1010110010 => 2
[1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 0101100110 => 0010110011 => 2
[1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 0101101001 => 1010110100 => 2
[1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 0101101010 => 0010110101 => 2
[1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 0101101100 => 0010110110 => 2
[1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 0101110001 => 1010111000 => 3
[1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 0101110010 => 0010111001 => ? ∊ {2,2,2,2,3,3,3,3,3,4}
[1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 0101110100 => 0010111010 => 3
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0101111000 => 0010111100 => 4
[1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 0110010101 => 1011001010 => 2
[1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 0110010110 => 0011001011 => 2
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 0110011001 => 1011001100 => 2
[1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 0110011010 => 0011001101 => 2
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 0110011100 => 0011001110 => ? ∊ {2,2,2,2,3,3,3,3,3,4}
[1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 0110100101 => 1011010010 => 2
[1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 0110100110 => 0011010011 => 2
[1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 0110101001 => 1011010100 => 2
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 0110101010 => 0011010101 => 2
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 0110101100 => 0011010110 => ? ∊ {2,2,2,2,3,3,3,3,3,4}
[1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 0110110001 => 1011011000 => 2
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 0110110010 => 0011011001 => ? ∊ {2,2,2,2,3,3,3,3,3,4}
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 0110110100 => 0011011010 => ? ∊ {2,2,2,2,3,3,3,3,3,4}
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 0110111000 => 0011011100 => ? ∊ {2,2,2,2,3,3,3,3,3,4}
[1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 0111000101 => 1011100010 => 3
[1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 0111000110 => 0011100011 => ? ∊ {2,2,2,2,3,3,3,3,3,4}
[1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 0111001001 => 1011100100 => 3
[1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => 0111001010 => 0011100101 => ? ∊ {2,2,2,2,3,3,3,3,3,4}
[1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => 0111001100 => 0011100110 => 3
[1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => 0111010001 => 1011101000 => 3
[1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => 0111010010 => 0011101001 => ? ∊ {2,2,2,2,3,3,3,3,3,4}
[1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => 0111010100 => 0011101010 => 3
[1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => 0111011000 => 0011101100 => 3
[1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => 0111100010 => 0011110001 => ? ∊ {2,2,2,2,3,3,3,3,3,4}
[1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => 010101011100 => 001010101110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,0,1,0,1,1,0,1,1,0,0,0]
 => 101011011000 => 010101101100 => 001010110110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,0,1,0,1,1,1,0,0,1,0,0]
 => 101011100100 => 010101110010 => 001010111001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,0,1,0,1,1,1,0,1,0,0,0]
 => 101011101000 => 010101110100 => 001010111010 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,0,1,0,1,1,1,1,0,0,0,0]
 => 101011110000 => 010101111000 => 001010111100 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,0,1,1,0,0,1,1,1,0,0,0]
 => 101100111000 => 010110011100 => 001011001110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,0,1,1,0,1,1,0,0,1,0,0]
 => 101101100100 => 010110110010 => 001011011001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,0,1,1,1,0,0,0,1,1,0,0]
 => 101110001100 => 010111000110 => 001011100011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,0,1,1,1,0,0,1,0,1,0,0]
 => 101110010100 => 010111001010 => 001011100101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,0,1,1,1,0,1,0,0,1,0,0]
 => 101110100100 => 010111010010 => 001011101001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,0,1,1,1,1,0,0,0,1,0,0]
 => 101111000100 => 010111100010 => 001011110001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,0,0,1,0,1,1,1,0,0,0]
 => 110010111000 => 011001011100 => 001100101110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,0,0,1,1,0,1,1,0,0,0]
 => 110011011000 => 011001101100 => 001100110110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,0,0,1,1,1,0,0,1,0,0]
 => 110011100100 => 011001110010 => 001100111001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,0,0,1,1,1,0,1,0,0,0]
 => 110011101000 => 011001110100 => 001100111010 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,0,0,1,1,1,1,0,0,0,0]
 => 110011110000 => 011001111000 => 001100111100 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,0,1,0,0,1,1,1,0,0,0]
 => 110100111000 => 011010011100 => 001101001110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,0,1,0,1,0,1,1,0,0,0]
 => 110101011000 => 011010101100 => 001101010110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,0,1,0,1,1,0,0,1,0,0]
 => 110101100100 => 011010110010 => 001101011001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,0,1,0,1,1,0,1,0,0,0]
 => 110101101000 => 011010110100 => 001101011010 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,0,1,0,1,1,1,0,0,0,0]
 => 110101110000 => 011010111000 => 001101011100 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,0,1,1,0,0,0,1,1,0,0]
 => 110110001100 => 011011000110 => 001101100011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,0,1,1,0,0,1,0,1,0,0]
 => 110110010100 => 011011001010 => 001101100101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,0,1,1,0,1,0,0,1,0,0]
 => 110110100100 => 011011010010 => 001101101001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,0,1,1,1,0,0,0,1,0,0]
 => 110111000100 => 011011100010 => 001101110001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,0,0,0,1,0,1,1,0,0]
 => 111000101100 => 011100010110 => 001110001011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,0,0,0,1,1,0,1,0,0]
 => 111000110100 => 011100011010 => 001110001101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,0,0,0,1,1,1,0,0,0]
 => 111000111000 => 011100011100 => 001110001110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,0,0,1,0,0,1,1,0,0]
 => 111001001100 => 011100100110 => 001110010011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,0,0,1,0,1,0,1,0,0]
 => 111001010100 => 011100101010 => 001110010101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,0,0,1,0,1,1,0,0,0]
 => 111001011000 => 011100101100 => 001110010110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,0,0,1,1,0,0,1,0,0]
 => 111001100100 => 011100110010 => 001110011001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,0,0,1,1,0,1,0,0,0]
 => 111001101000 => 011100110100 => 001110011010 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,0,0,1,1,1,0,0,0,0]
 => 111001110000 => 011100111000 => 001110011100 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,0,1,0,0,0,1,1,0,0]
 => 111010001100 => 011101000110 => 001110100011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,0,1,0,0,1,0,1,0,0]
 => 111010010100 => 011101001010 => 001110100101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,0,1,0,0,1,1,0,0,0]
 => 111010011000 => 011101001100 => 001110100110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,0,1,0,1,0,0,1,0,0]
 => 111010100100 => 011101010010 => 001110101001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,0,1,0,1,0,1,0,0,0]
 => 111010101000 => 011101010100 => 001110101010 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,0,1,0,1,1,0,0,0,0]
 => 111010110000 => 011101011000 => 001110101100 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000982
(load all 10 compositions to match this statistic)
Mapping
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 10 => 01 => 10 => 1
[1,0,1,0]
 => 1010 => 0101 => 1010 => 1
[1,1,0,0]
 => 1100 => 0110 => 0011 => 2
[1,0,1,0,1,0]
 => 101010 => 010101 => 101010 => 1
[1,0,1,1,0,0]
 => 101100 => 010110 => 001011 => 2
[1,1,0,0,1,0]
 => 110010 => 011001 => 101100 => 2
[1,1,0,1,0,0]
 => 110100 => 011010 => 001101 => 2
[1,1,1,0,0,0]
 => 111000 => 011100 => 001110 => 3
[1,0,1,0,1,0,1,0]
 => 10101010 => 01010101 => 10101010 => 1
[1,0,1,0,1,1,0,0]
 => 10101100 => 01010110 => 00101011 => 2
[1,0,1,1,0,0,1,0]
 => 10110010 => 01011001 => 10101100 => 2
[1,0,1,1,0,1,0,0]
 => 10110100 => 01011010 => 00101101 => 2
[1,0,1,1,1,0,0,0]
 => 10111000 => 01011100 => 00101110 => 3
[1,1,0,0,1,0,1,0]
 => 11001010 => 01100101 => 10110010 => 2
[1,1,0,0,1,1,0,0]
 => 11001100 => 01100110 => 00110011 => 2
[1,1,0,1,0,0,1,0]
 => 11010010 => 01101001 => 10110100 => 2
[1,1,0,1,0,1,0,0]
 => 11010100 => 01101010 => 00110101 => 2
[1,1,0,1,1,0,0,0]
 => 11011000 => 01101100 => 00110110 => 2
[1,1,1,0,0,0,1,0]
 => 11100010 => 01110001 => 10111000 => 3
[1,1,1,0,0,1,0,0]
 => 11100100 => 01110010 => 00111001 => 3
[1,1,1,0,1,0,0,0]
 => 11101000 => 01110100 => 00111010 => 3
[1,1,1,1,0,0,0,0]
 => 11110000 => 01111000 => 00111100 => 4
[1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 0101010101 => 1010101010 => 1
[1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 0101010110 => 0010101011 => 2
[1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 0101011001 => 1010101100 => 2
[1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 0101011010 => 0010101101 => 2
[1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 0101011100 => 0010101110 => 3
[1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 0101100101 => 1010110010 => 2
[1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 0101100110 => 0010110011 => 2
[1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 0101101001 => 1010110100 => 2
[1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 0101101010 => 0010110101 => 2
[1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 0101101100 => 0010110110 => 2
[1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 0101110001 => 1010111000 => 3
[1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 0101110010 => 0010111001 => ? ∊ {2,2,2,2,2,3,3,3,3,4}
[1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 0101110100 => 0010111010 => 3
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0101111000 => 0010111100 => 4
[1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 0110010101 => 1011001010 => 2
[1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 0110010110 => 0011001011 => 2
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 0110011001 => 1011001100 => 2
[1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 0110011010 => 0011001101 => 2
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 0110011100 => 0011001110 => ? ∊ {2,2,2,2,2,3,3,3,3,4}
[1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 0110100101 => 1011010010 => 2
[1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 0110100110 => 0011010011 => 2
[1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 0110101001 => 1011010100 => 2
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 0110101010 => 0011010101 => 2
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 0110101100 => 0011010110 => ? ∊ {2,2,2,2,2,3,3,3,3,4}
[1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 0110110001 => 1011011000 => 3
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 0110110010 => 0011011001 => ? ∊ {2,2,2,2,2,3,3,3,3,4}
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 0110110100 => 0011011010 => ? ∊ {2,2,2,2,2,3,3,3,3,4}
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 0110111000 => 0011011100 => ? ∊ {2,2,2,2,2,3,3,3,3,4}
[1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 0111000101 => 1011100010 => 3
[1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 0111000110 => 0011100011 => ? ∊ {2,2,2,2,2,3,3,3,3,4}
[1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 0111001001 => 1011100100 => 3
[1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => 0111001010 => 0011100101 => ? ∊ {2,2,2,2,2,3,3,3,3,4}
[1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => 0111001100 => 0011100110 => 3
[1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => 0111010001 => 1011101000 => 3
[1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => 0111010010 => 0011101001 => ? ∊ {2,2,2,2,2,3,3,3,3,4}
[1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => 0111010100 => 0011101010 => 3
[1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => 0111011000 => 0011101100 => 3
[1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => 0111100010 => 0011110001 => ? ∊ {2,2,2,2,2,3,3,3,3,4}
[1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => 010101011100 => 001010101110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,0,1,0,1,1,0,1,1,0,0,0]
 => 101011011000 => 010101101100 => 001010110110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,0,1,0,1,1,1,0,0,1,0,0]
 => 101011100100 => 010101110010 => 001010111001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,0,1,0,1,1,1,0,1,0,0,0]
 => 101011101000 => 010101110100 => 001010111010 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,0,1,0,1,1,1,1,0,0,0,0]
 => 101011110000 => 010101111000 => 001010111100 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,0,1,1,0,0,1,1,1,0,0,0]
 => 101100111000 => 010110011100 => 001011001110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,0,1,1,0,1,1,0,0,1,0,0]
 => 101101100100 => 010110110010 => 001011011001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,0,1,1,1,0,0,0,1,1,0,0]
 => 101110001100 => 010111000110 => 001011100011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,0,1,1,1,0,0,1,0,1,0,0]
 => 101110010100 => 010111001010 => 001011100101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,0,1,1,1,0,1,0,0,1,0,0]
 => 101110100100 => 010111010010 => 001011101001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,0,1,1,1,1,0,0,0,1,0,0]
 => 101111000100 => 010111100010 => 001011110001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,0,0,1,0,1,1,1,0,0,0]
 => 110010111000 => 011001011100 => 001100101110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,0,0,1,1,0,1,1,0,0,0]
 => 110011011000 => 011001101100 => 001100110110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,0,0,1,1,1,0,0,1,0,0]
 => 110011100100 => 011001110010 => 001100111001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,0,0,1,1,1,0,1,0,0,0]
 => 110011101000 => 011001110100 => 001100111010 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,0,0,1,1,1,1,0,0,0,0]
 => 110011110000 => 011001111000 => 001100111100 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,0,1,0,0,1,1,1,0,0,0]
 => 110100111000 => 011010011100 => 001101001110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,0,1,0,1,0,1,1,0,0,0]
 => 110101011000 => 011010101100 => 001101010110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,0,1,0,1,1,0,0,1,0,0]
 => 110101100100 => 011010110010 => 001101011001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,0,1,0,1,1,0,1,0,0,0]
 => 110101101000 => 011010110100 => 001101011010 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,0,1,0,1,1,1,0,0,0,0]
 => 110101110000 => 011010111000 => 001101011100 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,0,1,1,0,0,0,1,1,0,0]
 => 110110001100 => 011011000110 => 001101100011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,0,1,1,0,0,1,0,1,0,0]
 => 110110010100 => 011011001010 => 001101100101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,0,1,1,0,1,0,0,1,0,0]
 => 110110100100 => 011011010010 => 001101101001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,0,1,1,1,0,0,0,1,0,0]
 => 110111000100 => 011011100010 => 001101110001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,1,0,0,0,1,0,1,1,0,0]
 => 111000101100 => 011100010110 => 001110001011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,1,0,0,0,1,1,0,1,0,0]
 => 111000110100 => 011100011010 => 001110001101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,1,0,0,0,1,1,1,0,0,0]
 => 111000111000 => 011100011100 => 001110001110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,1,0,0,1,0,0,1,1,0,0]
 => 111001001100 => 011100100110 => 001110010011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,1,0,0,1,0,1,0,1,0,0]
 => 111001010100 => 011100101010 => 001110010101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,1,0,0,1,0,1,1,0,0,0]
 => 111001011000 => 011100101100 => 001110010110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,1,0,0,1,1,0,0,1,0,0]
 => 111001100100 => 011100110010 => 001110011001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,1,0,0,1,1,0,1,0,0,0]
 => 111001101000 => 011100110100 => 001110011010 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,1,0,0,1,1,1,0,0,0,0]
 => 111001110000 => 011100111000 => 001110011100 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,1,0,1,0,0,0,1,1,0,0]
 => 111010001100 => 011101000110 => 001110100011 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,1,0,1,0,0,1,0,1,0,0]
 => 111010010100 => 011101001010 => 001110100101 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,1,0,1,0,0,1,1,0,0,0]
 => 111010011000 => 011101001100 => 001110100110 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,1,0,1,0,1,0,0,1,0,0]
 => 111010100100 => 011101010010 => 001110101001 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,1,0,1,0,1,0,1,0,0,0]
 => 111010101000 => 011101010100 => 001110101010 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
[1,1,1,0,1,0,1,1,0,0,0,0]
 => 111010110000 => 011101011000 => 001110101100 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5}
Description
The length of the longest constant subword.
Matching statistic: St000969
Mapping
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000969: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [[1],[]]
 => []
 => []
 => ? = 1
[1,0,1,0]
 => [[1,1],[]]
 => []
 => []
 => ? ∊ {1,2}
[1,1,0,0]
 => [[2],[]]
 => []
 => []
 => ? ∊ {1,2}
[1,0,1,0,1,0]
 => [[1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,3}
[1,0,1,1,0,0]
 => [[2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,3}
[1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,0,1,0,0]
 => [[3],[]]
 => []
 => []
 => ? ∊ {1,2,2,3}
[1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,3}
[1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,1,1,1,0,0,0,1,0,0]
 => [[4,3],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,1,1,0,1,0,0,0,0]
 => [[4,4],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [[4,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 3
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [[4,4,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 4
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [[5,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [[4,4,1],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [[4,3,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [[4,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [[4,3,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [[4,4,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [[6],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [[5,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [[4,2,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
Description
We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. Then we calculate the global dimension of that CNakayama algebra.
Matching statistic: St001028
Mapping
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001028: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [[1],[]]
 => []
 => []
 => ? = 1
[1,0,1,0]
 => [[1,1],[]]
 => []
 => []
 => ? ∊ {1,2}
[1,1,0,0]
 => [[2],[]]
 => []
 => []
 => ? ∊ {1,2}
[1,0,1,0,1,0]
 => [[1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,3}
[1,0,1,1,0,0]
 => [[2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,3}
[1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,0,1,0,0]
 => [[3],[]]
 => []
 => []
 => ? ∊ {1,2,2,3}
[1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,3}
[1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,3,3,3,4}
[1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,1,1,1,0,0,0,1,0,0]
 => [[4,3],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3],[1]]
 => [1]
 => [1,0]
 => 2
[1,1,1,1,0,1,0,0,0,0]
 => [[4,4],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [[4,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 3
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [[4,4,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 4
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [[5,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [[4,4,1],[2]]
 => [2]
 => [1,0,1,0]
 => 3
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [[4,3,1],[1]]
 => [1]
 => [1,0]
 => 2
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [[4,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [[4,3,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [[4,4,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3,1],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [[6],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [[5,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [[4,2,2],[]]
 => []
 => []
 => ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6}
Description
Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
The following 149 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001180Number of indecomposable injective modules with projective dimension at most 1. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001638The book thickness of a graph. St001060The distinguishing index of a graph. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000028The number of stack-sorts needed to sort a permutation. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001644The dimension of a graph. St000308The height of the tree associated to a permutation. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000893The number of distinct diagonal sums of an alternating sign matrix. St000007The number of saliances of the permutation. St000264The girth of a graph, which is not a tree. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001875The number of simple modules with projective dimension at most 1. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000035The number of left outer peaks of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000153The number of adjacent cycles of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000991The number of right-to-left minima of a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000455The second largest eigenvalue of a graph if it is integral. St000528The height of a poset. St000907The number of maximal antichains of minimal length in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000080The rank of the poset. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001545The second Elser number of a connected graph. St000474Dyson's crank of a partition. St000993The multiplicity of the largest part of an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001498The normalised height of a Nakayama algebra with magnitude 1. St001571The Cartan determinant of the integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001432The order dimension of the partition. St001488The number of corners of a skew partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000477The weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001279The sum of the parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St000983The length of the longest alternating subword. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St000381The largest part of an integer composition. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001864The number of excedances of a signed permutation. St000451The length of the longest pattern of the form k 1 2. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000662The staircase size of the code of a permutation. St000075The orbit size of a standard tableau under promotion. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001730The number of times the path corresponding to a binary word crosses the base line. St000120The number of left tunnels of a Dyck path. St000225Difference between largest and smallest parts in a partition. St000328The maximum number of child nodes in a tree. St000670The reversal length of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000808The number of up steps of the associated bargraph. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001570The minimal number of edges to add to make a graph Hamiltonian. St001589The nesting number of a perfect matching. St001742The difference of the maximal and the minimal degree in a graph. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001557The number of inversions of the second entry of a permutation. St001566The length of the longest arithmetic progression in a permutation. St001578The minimal number of edges to add or remove to make a graph a line graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001624The breadth of a lattice. St000386The number of factors DDU in a Dyck path. St000884The number of isolated descents of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001435The number of missing boxes in the first row. St001877Number of indecomposable injective modules with projective dimension 2. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000703The number of deficiencies of a permutation. St000022The number of fixed points of a permutation. St000390The number of runs of ones in a binary word. St000731The number of double exceedences of a permutation.