searching the database
Your data matches 473 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: St001389
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001389: Integer partitions ⟶ 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]
=> [1,2] => [2]
=> 2
[1,1,0,0]
=> [2,1] => [1,1]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [3]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,1,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,1,1]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,1,1]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [2,2,1]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [2,1,1,1]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [2,1,1,1]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [2,1,1,1]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [2,1,1,1]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [2,1,1,1]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [6]
=> 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [5,1]
=> 5
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [4,1,1]
=> 4
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [3,1,1,1]
=> 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [2,1,1,1,1]
=> 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [5,1]
=> 5
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => [2,2,1,1]
=> 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => [2,1,1,1,1]
=> 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => [4,1,1]
=> 4
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => [2,1,1,1,1]
=> 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => [3,1,1,1]
=> 3
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => [2,2,1,1]
=> 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [2,1,1,1,1]
=> 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [2,1,1,1,1]
=> 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => [2,1,1,1,1]
=> 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => [2,1,1,1,1]
=> 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [2,1,1,1,1]
=> 2
Description
The number of partitions of the same length below the given integer partition.
For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St000141
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,2,1,4] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,4,2] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,3,4,1] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,3,2,1,5] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,4,3,2] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,4,1,2,5] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,3,1,4,5] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,4,5,2,3] => 2 = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,3,4,1,5] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [1,2,4,5,3] => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [1,3,4,5,2] => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,3,4,5,1] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [5,4,3,2,1,6] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [4,3,2,1,5,6] => 3 = 4 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [3,2,1,4,5,6] => 2 = 3 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,1,3,4,5,6] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [1,6,5,4,3,2] => 4 = 5 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [3,4,1,2,5,6] => 2 = 3 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => [2,3,1,4,5,6] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,2,3] => [1,2,6,5,4,3] => 3 = 4 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => [2,3,4,1,5,6] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,2,3,4] => [1,2,3,6,5,4] => 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [1,2,5,6,3,4] => 2 = 3 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6] => [2,3,4,5,1,6] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => [1,2,3,5,6,4] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [4,1,2,3,5,6] => [1,2,4,5,6,3] => 1 = 2 - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,2,4,5,6] => [1,3,4,5,6,2] => 1 = 2 - 1
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000245
(load all 52 compositions to match this statistic)
(load all 52 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,1,3,2] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,4,3,2] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,1,4,2,3] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,1,3,2] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [2,1,5,3,4] => 2 = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [5,1,4,3,2] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [3,2,1,5,4] => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [2,1,5,4,3] => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [1,5,4,3,2] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [6,1,2,3,4,5] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [6,5,1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [6,5,4,1,2,3] => 2 = 3 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [6,5,4,3,1,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 4 = 5 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => [6,5,1,4,2,3] => 2 = 3 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => [6,5,4,1,3,2] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => 3 = 4 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,6,5,4,2,1] => [6,5,1,4,3,2] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => [3,2,1,6,4,5] => 2 = 3 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,6,5,3,2,1] => [6,1,5,4,3,2] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => [4,3,2,1,6,5] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,4,3,6,2,1] => [3,2,1,6,5,4] => 1 = 2 - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,4,6,3,2,1] => [2,1,6,5,4,3] => 1 = 2 - 1
Description
The number of ascents of a permutation.
Matching statistic: St000394
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000662
(load all 57 compositions to match this statistic)
(load all 57 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,2,1,4] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,4,2] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,3,4,1] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,3,2,1,5] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,4,3,2] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,4,1,2,5] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,3,1,4,5] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,4,5,2,3] => 2 = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,3,4,1,5] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [1,2,4,5,3] => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [1,3,4,5,2] => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,3,4,5,1] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [5,4,3,2,1,6] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [4,3,2,1,5,6] => 3 = 4 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [3,2,1,4,5,6] => 2 = 3 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,1,3,4,5,6] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [1,6,5,4,3,2] => 4 = 5 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [3,4,1,2,5,6] => 2 = 3 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => [2,3,1,4,5,6] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,2,3] => [1,2,6,5,4,3] => 3 = 4 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => [2,3,4,1,5,6] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,2,3,4] => [1,2,3,6,5,4] => 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [1,2,5,6,3,4] => 2 = 3 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6] => [2,3,4,5,1,6] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => [1,2,3,5,6,4] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [4,1,2,3,5,6] => [1,2,4,5,6,3] => 1 = 2 - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,2,4,5,6] => [1,3,4,5,6,2] => 1 = 2 - 1
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St000672
(load all 61 compositions to match this statistic)
(load all 61 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,1,3,2] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,4,3,2] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,1,4,2,3] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,1,3,2] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [2,1,5,3,4] => 2 = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [5,1,4,3,2] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [3,2,1,5,4] => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [2,1,5,4,3] => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [1,5,4,3,2] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [6,1,2,3,4,5] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [6,5,1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [6,5,4,1,2,3] => 2 = 3 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [6,5,4,3,1,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 4 = 5 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => [6,5,1,4,2,3] => 2 = 3 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => [6,5,4,1,3,2] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => 3 = 4 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,6,5,4,2,1] => [6,5,1,4,3,2] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => [3,2,1,6,4,5] => 2 = 3 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,6,5,3,2,1] => [6,1,5,4,3,2] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => [4,3,2,1,6,5] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,4,3,6,2,1] => [3,2,1,6,5,4] => 1 = 2 - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,4,6,3,2,1] => [2,1,6,5,4,3] => 1 = 2 - 1
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St001176
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,1]
=> 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [2]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,1,1]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2]
=> 2 = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [4,1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [4,1]
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [4,1]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [2,1,1,1,1]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [3,1,1,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [4,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [5,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [2,1,1,1,1]
=> 4 = 5 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => [4,2]
=> 2 = 3 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => [5,1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => [3,1,1,1]
=> 3 = 4 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => [5,1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => [4,1,1]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => [4,2]
=> 2 = 3 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [5,1]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [5,1]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => [5,1]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => [5,1]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [5,1]
=> 1 = 2 - 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000010
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: 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]
=> [1,2] => [1,2] => [1,1]
=> 2
[1,1,0,0]
=> [2,1] => [2,1] => [2]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [2,1]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [2,1,1]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [3,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,4,1,3] => [2,1,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [3,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [3,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [3,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [2,1,1,1]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [3,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [4,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,2,4,1,3] => [3,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,2,3,1] => [4,1]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,1,1]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,5,1,4] => [3,1,1]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [5,3,4,2,1] => [4,1]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [4,1]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => [4,1]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [4,3,5,2,1] => [4,1]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => [4,1]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [5]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [6,1,2,3,4,5] => [2,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [6,5,1,2,3,4] => [3,1,1,1]
=> 4
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [6,5,4,1,2,3] => [4,1,1]
=> 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [6,5,4,3,1,2] => [5,1]
=> 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,1,1,1]
=> 5
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => [6,5,2,4,1,3] => [4,1,1]
=> 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => [6,5,4,2,3,1] => [5,1]
=> 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => [3,1,1,1]
=> 4
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,6,5,4,2,1] => [6,5,3,4,2,1] => [5,1]
=> 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => [4,1,1]
=> 3
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => [4,3,2,6,1,5] => [4,1,1]
=> 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,6,5,3,2,1] => [6,4,5,3,2,1] => [5,1]
=> 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => [5,1]
=> 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => [5,4,3,2,6,1] => [5,1]
=> 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,4,3,6,2,1] => [5,4,3,6,2,1] => [5,1]
=> 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,4,6,3,2,1] => [5,4,6,3,2,1] => [5,1]
=> 2
Description
The length of the partition.
Matching statistic: St000011
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 4
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000013
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 4
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 2
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
The following 463 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000063The number of linear extensions of a certain poset defined for an integer partition. St000071The number of maximal chains in a poset. St000108The number of partitions contained in the given partition. St000147The largest part of an integer partition. St000507The number of ascents of a standard tableau. St000532The total number of rook placements on a Ferrers board. St000676The number of odd rises of a Dyck path. St000738The first entry in the last row of a standard tableau. St001330The hat guessing number of a graph. St001400The total number of Littlewood-Richardson tableaux of given shape. St001462The number of factors of a standard tableaux under concatenation. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000157The number of descents of a standard tableau. St000214The number of adjacencies of a permutation. St000228The size of a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000377The dinv defect of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000439The position of the first down step of a Dyck path. St000441The number of successions of a permutation. St000459The hook length of the base cell of a partition. St000703The number of deficiencies of a permutation. St000784The maximum of the length and the largest part of the integer partition. St000996The number of exclusive left-to-right maxima of a permutation. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000007The number of saliances of the permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001581The achromatic number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000288The number of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000632The jump number of the poset. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001644The dimension of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000024The number of double up and double down steps of a Dyck path. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000058The order of a permutation. St000153The number of adjacent cycles of a permutation. St000172The Grundy number of a graph. St000444The length of the maximal rise of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000053The number of valleys of the Dyck path. St000211The rank of the set partition. St000234The number of global ascents of a permutation. St000306The bounce count of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000502The number of successions of a set partitions. St000728The dimension of a set partition. St000730The maximal arc length of a set partition. St000874The position of the last double rise in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000308The height of the tree associated to a permutation. St000528The height of a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St000028The number of stack-sorts needed to sort a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000651The maximal size of a rise in a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001717The largest size of an interval in a poset. St000067The inversion number of the alternating sign matrix. St000332The positive inversions of an alternating sign matrix. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000470The number of runs in a permutation. St000702The number of weak deficiencies of a permutation. St001304The number of maximally independent sets of vertices of a graph. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St001270The bandwidth of a graph. St001427The number of descents of a signed permutation. St001489The maximum of the number of descents and the number of inverse descents. St001962The proper pathwidth of a graph. St000331The number of upper interactions of a Dyck path. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000062The length of the longest increasing subsequence of the permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000822The Hadwiger number of the graph. St000991The number of right-to-left minima of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000021The number of descents of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000336The leg major index of a standard tableau. St000829The Ulam distance of a permutation to the identity permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. 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. St000015The number of peaks of a Dyck path. St000056The decomposition (or block) number of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000240The number of indices that are not small excedances. St000314The number of left-to-right-maxima of a permutation. St000485The length of the longest cycle of a permutation. St000542The number of left-to-right-minima of a permutation. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001530The depth of a Dyck path. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000080The rank of the poset. St000120The number of left tunnels of a Dyck path. St000154The sum of the descent bottoms of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000460The hook length of the last cell along the main diagonal of an integer partition. St000809The reduced reflection length of the permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001175The size of a partition minus the hook length of the base cell. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001360The number of covering relations in Young's lattice below a partition. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001480The number of simple summands of the module J^2/J^3. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000472The sum of the ascent bottoms 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. St000653The last descent of a permutation. St000795The mad of a permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001812The biclique partition number of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001331The size of the minimal feedback vertex set. St000144The pyramid weight of the Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001420Half the length of a longest factor which is its own reverse-complement 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. St001432The order dimension of the partition. St000291The number of descents of a binary word. St000731The number of double exceedences of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000527The width of the poset. St000652The maximal difference between successive positions of a permutation. St000862The number of parts of the shifted shape of a permutation. St000758The length of the longest staircase fitting into an integer composition. St001321The number of vertices of the largest induced subforest of a graph. St000619The number of cyclic descents of a permutation. St000836The number of descents of distance 2 of a permutation. St000741The Colin de Verdière graph invariant. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000837The number of ascents of distance 2 of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001645The pebbling number of a connected graph. St000185The weighted size of a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001741The largest integer such that all patterns of this size are contained in the permutation. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001726The number of visible inversions of a permutation. St000039The number of crossings of a permutation. St000307The number of rowmotion orbits of a poset. St001180Number of indecomposable injective modules with projective dimension at most 1. St001214The aft of an integer partition. St000145The Dyson rank of a partition. St001668The number of points of the poset minus the width of the poset. St000806The semiperimeter of the associated bargraph. St000681The Grundy value of Chomp on Ferrers diagrams. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001863The number of weak excedances of a signed permutation. St001651The Frankl number of a lattice. St001875The number of simple modules with projective dimension at most 1. St001624The breadth of a lattice. St001820The size of the image of the pop stack sorting operator. St001896The number of right descents of a signed permutations. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000781The number of proper colouring schemes of a Ferrers diagram. St000808The number of up steps of the associated bargraph. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000264The girth of a graph, which is not a tree. St000493The los statistic of a set partition. St000497The lcb statistic of a set partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000455The second largest eigenvalue of a graph if it is integral. St000699The toughness times the least common multiple of 1,. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000456The monochromatic index of a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001626The number of maximal proper sublattices of a lattice. St000352The Elizalde-Pak rank of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001621The number of atoms of a lattice. St000260The radius of a connected graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000451The length of the longest pattern of the form k 1 2. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000834The number of right outer peaks of a permutation. St000907The number of maximal antichains of minimal length in a poset. St001128The exponens consonantiae of a partition. St001435The number of missing boxes in the first row. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000035The number of left outer peaks of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001488The number of corners of a skew partition. St000742The number of big ascents of a permutation after prepending zero. St000884The number of isolated descents of a permutation. St001060The distinguishing index of a graph. St000366The number of double descents of a permutation. St000871The number of very big ascents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001867The number of alignments of type EN of a signed permutation. St000201The number of leaf nodes in a binary tree. St000390The number of runs of ones in a binary word. St000568The hook number of a binary tree. St000659The number of rises of length at least 2 of a Dyck path. St000919The number of maximal left branches of a binary tree. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St000098The chromatic number of a graph. St000356The number of occurrences of the pattern 13-2. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001115The number of even descents of a permutation. St001394The genus of a permutation. St000850The number of 1/2-balanced pairs in a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001118The acyclic chromatic index of a graph. St000243The number of cyclic valleys and cyclic peaks of a permutation. 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 $eA$. 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 $eA$. 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 $eA$. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001864The number of excedances of a signed permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000239The number of small weak excedances. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001737The number of descents of type 2 in a permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St000023The number of inner peaks of a permutation. St000353The number of inner valleys of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001712The number of natural descents of a standard Young tableau. St001840The number of descents of a set partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000254The nesting number of a set partition. St000292The number of ascents of a binary word. St000340The number of non-final maximal constant sub-paths of length greater than one. St000387The matching number of a graph. St000710The number of big deficiencies of a permutation. St000925The number of topologically connected components of a set partition. St000942The number of critical left to right maxima of the parking functions. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001220The width of a permutation. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001405The number of bonds in a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001512The minimum rank of a graph. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001732The number of peaks visible from the left. St001734The lettericity of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001874Lusztig's a-function for the symmetric group. St001928The number of non-overlapping descents in a permutation. St000004The major index of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000174The flush statistic of a semistandard tableau. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000247The number of singleton blocks of a set partition. St000317The cycle descent number of a permutation. St000338The number of pixed points of a permutation. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000386The number of factors DDU in a Dyck path. St000504The cardinality of the first block of a set partition. St000640The rank of the largest boolean interval in a poset. St000650The number of 3-rises of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000794The mak of a permutation. St000820The number of compositions obtained by rotating the composition. St000872The number of very big descents of a permutation. St000918The 2-limited packing number of a graph. St000961The shifted major index of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000989The number of final rises of a permutation. St001029The size of the core of a graph. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001062The maximal size of a block of a set partition. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001388The number of non-attacking neighbors of a permutation. St001423The number of distinct cubes in a binary word. St001471The magnitude of a Dyck path. St001494The Alon-Tarsi number of a graph. St001520The number of strict 3-descents. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule 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. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001781The interlacing number of a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001823The Stasinski-Voll length of a signed permutation. St001839The number of excedances of a set partition. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001902The number of potential covers of a poset. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000735The last entry on the main diagonal of a standard tableau. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 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. St000634The number of endomorphisms of a poset. St000639The number of relations in a poset. St000641The number of non-empty boolean intervals in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. 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. St000100The number of linear extensions of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000327The number of cover relations in a poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000635The number of strictly order preserving maps of a poset into itself. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St000993The multiplicity of the largest part of an integer partition. St001637The number of (upper) dissectors of a poset. St001890The maximum magnitude of the Möbius function 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. St001095The number of non-isomorphic posets with precisely one further covering relation.
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!