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Your data matches 361 different statistics following compositions of up to 3 maps.
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Matching statistic: St000473
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(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000473: 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
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 0 = -1 + 1
[1,2] => [1,2] => [2]
=> 1 = 0 + 1
[2,1] => [1,2] => [2]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [3]
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => [3]
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => [3]
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => [3]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [4]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [4]
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [4]
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [4]
=> 1 = 0 + 1
[2,1,3,4] => [1,2,3,4] => [4]
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [4]
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [4]
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [4]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[2,1,3,4,5] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[2,1,3,5,4] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[2,1,4,3,5] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[2,3,1,4,5] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[2,3,1,5,4] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[2,3,4,1,5] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[3,5,1,2,4] => [1,3,2,5,4] => [3,2]
=> 2 = 1 + 1
[3,5,1,4,2] => [1,3,2,5,4] => [3,2]
=> 2 = 1 + 1
[3,5,2,1,4] => [1,3,2,5,4] => [3,2]
=> 2 = 1 + 1
[3,5,2,4,1] => [1,3,2,5,4] => [3,2]
=> 2 = 1 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
Description
The number of parts of a partition that are strictly bigger than the number of ones.
This is part of the definition of Dyson's crank of a partition, see [[St000474]].
Matching statistic: St000662
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0 = -1 + 1
[1,2] => [1,2] => [2,1] => 1 = 0 + 1
[2,1] => [1,2] => [2,1] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[1,3,2] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[2,1,3] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[2,3,1] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[2,1,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[2,1,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[2,1,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[2,3,1,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[2,3,1,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[2,3,4,1,5] => [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[3,5,1,2,4] => [1,3,2,5,4] => [2,4,3,1,5] => 2 = 1 + 1
[3,5,1,4,2] => [1,3,2,5,4] => [2,4,3,1,5] => 2 = 1 + 1
[3,5,2,1,4] => [1,3,2,5,4] => [2,4,3,1,5] => 2 = 1 + 1
[3,5,2,4,1] => [1,3,2,5,4] => [2,4,3,1,5] => 2 = 1 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1 = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1 = 0 + 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1 = 0 + 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1 = 0 + 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1 = 0 + 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1 = 0 + 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1 = 0 + 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1 = 0 + 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1 = 0 + 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1 = 0 + 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1 = 0 + 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1 = 0 + 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1 = 0 + 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1 = 0 + 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => 1 = 0 + 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: St000846
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0 = -1 + 1
[1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,5,1,2,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[3,5,1,4,2] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[3,5,2,1,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[3,5,2,4,1] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St000920
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [] => []
=> 0 = -1 + 1
[1,2] => [1] => [1,0]
=> 1 = 0 + 1
[2,1] => [1] => [1,0]
=> 1 = 0 + 1
[1,2,3] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[2,3,1] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> 1 = 0 + 1
[2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,3,4,5] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,3,5,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,4,3,5] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,4,5,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,3,1,4,5] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[2,3,1,5,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[2,3,4,1,5] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,3,4,5,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,5,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,5,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,5,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,5,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,3,4,5,6] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,5,4,6] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,3,5,6,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,4,3,5,6] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,4,3,6,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,4,5,3,6] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,4,5,6,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,2,4,5,6] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2,4,6,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2,5,4,6] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,2,5,6,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,4,2,5,6] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,3,4,2,6,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,3,4,5,2,6] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
Description
The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:
$$
\lfloor\log_2(1+height(D))\rfloor
$$
Matching statistic: St001280
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001280: 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
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 0 = -1 + 1
[1,2] => [1,2] => [2]
=> 1 = 0 + 1
[2,1] => [1,2] => [2]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [3]
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => [3]
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => [3]
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => [3]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [4]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [4]
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [4]
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [4]
=> 1 = 0 + 1
[2,1,3,4] => [1,2,3,4] => [4]
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [4]
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [4]
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [4]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[2,1,3,4,5] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[2,1,3,5,4] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[2,1,4,3,5] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[2,3,1,4,5] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[2,3,1,5,4] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[2,3,4,1,5] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [5]
=> 1 = 0 + 1
[3,5,1,2,4] => [1,3,2,5,4] => [3,2]
=> 2 = 1 + 1
[3,5,1,4,2] => [1,3,2,5,4] => [3,2]
=> 2 = 1 + 1
[3,5,2,1,4] => [1,3,2,5,4] => [3,2]
=> 2 = 1 + 1
[3,5,2,4,1] => [1,3,2,5,4] => [3,2]
=> 2 = 1 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000451
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 = -1 + 2
[1,2] => [1,2] => [2,1] => 2 = 0 + 2
[2,1] => [1,2] => [2,1] => 2 = 0 + 2
[1,2,3] => [1,2,3] => [3,2,1] => 2 = 0 + 2
[1,3,2] => [1,2,3] => [3,2,1] => 2 = 0 + 2
[2,1,3] => [1,2,3] => [3,2,1] => 2 = 0 + 2
[2,3,1] => [1,2,3] => [3,2,1] => 2 = 0 + 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 2 = 0 + 2
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 2 = 0 + 2
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 2 = 0 + 2
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 2 = 0 + 2
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 2 = 0 + 2
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 2 = 0 + 2
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 2 = 0 + 2
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 2 = 0 + 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 2 = 0 + 2
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 2 = 0 + 2
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 2 = 0 + 2
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 2 = 0 + 2
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 2 = 0 + 2
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 2 = 0 + 2
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => 2 = 0 + 2
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => 2 = 0 + 2
[2,1,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 2 = 0 + 2
[2,1,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 2 = 0 + 2
[2,1,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 2 = 0 + 2
[2,1,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 2 = 0 + 2
[2,3,1,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 2 = 0 + 2
[2,3,1,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 2 = 0 + 2
[2,3,4,1,5] => [1,2,3,4,5] => [5,4,3,2,1] => 2 = 0 + 2
[2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => 2 = 0 + 2
[3,5,1,2,4] => [1,3,2,5,4] => [4,5,2,3,1] => 3 = 1 + 2
[3,5,1,4,2] => [1,3,2,5,4] => [4,5,2,3,1] => 3 = 1 + 2
[3,5,2,1,4] => [1,3,2,5,4] => [4,5,2,3,1] => 3 = 1 + 2
[3,5,2,4,1] => [1,3,2,5,4] => [4,5,2,3,1] => 3 = 1 + 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 2 = 0 + 2
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 2 = 0 + 2
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 2 = 0 + 2
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 2 = 0 + 2
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 2 = 0 + 2
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 2 = 0 + 2
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 2 = 0 + 2
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 2 = 0 + 2
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 2 = 0 + 2
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 2 = 0 + 2
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 2 = 0 + 2
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 2 = 0 + 2
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 2 = 0 + 2
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 2 = 0 + 2
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 2 = 0 + 2
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St000891
(load all 70 compositions to match this statistic)
(load all 70 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000891: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000891: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 = -1 + 2
[1,2] => [1,2] => [1,2] => 2 = 0 + 2
[2,1] => [1,2] => [1,2] => 2 = 0 + 2
[1,2,3] => [1,2,3] => [1,2,3] => 2 = 0 + 2
[1,3,2] => [1,2,3] => [1,2,3] => 2 = 0 + 2
[2,1,3] => [1,2,3] => [1,2,3] => 2 = 0 + 2
[2,3,1] => [1,2,3] => [1,2,3] => 2 = 0 + 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 2 = 0 + 2
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 2 = 0 + 2
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 2 = 0 + 2
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 2 = 0 + 2
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 2 = 0 + 2
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 2 = 0 + 2
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 2 = 0 + 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 2 = 0 + 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[3,5,1,2,4] => [1,3,2,5,4] => [3,4,2,5,1] => 3 = 1 + 2
[3,5,1,4,2] => [1,3,2,5,4] => [3,4,2,5,1] => 3 = 1 + 2
[3,5,2,1,4] => [1,3,2,5,4] => [3,4,2,5,1] => 3 = 1 + 2
[3,5,2,4,1] => [1,3,2,5,4] => [3,4,2,5,1] => 3 = 1 + 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 2 = 0 + 2
Description
The number of distinct diagonal sums of a permutation matrix.
For example, the sums of the diagonals of the matrix $$\left(\begin{array}{rrrr}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0
\end{array}\right)$$
are $(1,0,1,0,2,0)$, so the statistic is $3$.
Matching statistic: St001642
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001642: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001642: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1 = -1 + 2
[1,2] => [1,2] => ([],2)
=> 2 = 0 + 2
[2,1] => [1,2] => ([],2)
=> 2 = 0 + 2
[1,2,3] => [1,2,3] => ([],3)
=> 2 = 0 + 2
[1,3,2] => [1,2,3] => ([],3)
=> 2 = 0 + 2
[2,1,3] => [1,2,3] => ([],3)
=> 2 = 0 + 2
[2,3,1] => [1,2,3] => ([],3)
=> 2 = 0 + 2
[1,2,3,4] => [1,2,3,4] => ([],4)
=> 2 = 0 + 2
[1,2,4,3] => [1,2,3,4] => ([],4)
=> 2 = 0 + 2
[1,3,2,4] => [1,2,3,4] => ([],4)
=> 2 = 0 + 2
[1,3,4,2] => [1,2,3,4] => ([],4)
=> 2 = 0 + 2
[2,1,3,4] => [1,2,3,4] => ([],4)
=> 2 = 0 + 2
[2,1,4,3] => [1,2,3,4] => ([],4)
=> 2 = 0 + 2
[2,3,1,4] => [1,2,3,4] => ([],4)
=> 2 = 0 + 2
[2,3,4,1] => [1,2,3,4] => ([],4)
=> 2 = 0 + 2
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 2 = 0 + 2
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> 2 = 0 + 2
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> 2 = 0 + 2
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> 2 = 0 + 2
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> 2 = 0 + 2
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> 2 = 0 + 2
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> 2 = 0 + 2
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> 2 = 0 + 2
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> 2 = 0 + 2
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> 2 = 0 + 2
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> 2 = 0 + 2
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> 2 = 0 + 2
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> 2 = 0 + 2
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> 2 = 0 + 2
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> 2 = 0 + 2
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> 2 = 0 + 2
[3,5,1,2,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 3 = 1 + 2
[3,5,1,4,2] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 3 = 1 + 2
[3,5,2,1,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 3 = 1 + 2
[3,5,2,4,1] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 3 = 1 + 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> 2 = 0 + 2
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> 2 = 0 + 2
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> 2 = 0 + 2
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> 2 = 0 + 2
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> 2 = 0 + 2
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> 2 = 0 + 2
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> 2 = 0 + 2
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> 2 = 0 + 2
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> 2 = 0 + 2
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> 2 = 0 + 2
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> 2 = 0 + 2
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> 2 = 0 + 2
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> 2 = 0 + 2
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> 2 = 0 + 2
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> 2 = 0 + 2
Description
The Prague dimension of a graph.
This is the least number of complete graphs such that the graph is an induced subgraph of their (categorical) product.
Put differently, this is the least number $n$ such that the graph can be embedded into $\mathbb N^n$, where two points are connected by an edge if and only if they differ in all coordinates.
Matching statistic: St000455
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> ([(0,1)],2)
=> -1
[1,2] => [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 0
[2,1] => [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[3,5,1,2,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1
[3,5,1,4,2] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1
[3,5,2,1,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1
[3,5,2,4,1] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000021
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0 = -1 + 1
[1,2] => [1,2] => [2,1] => [2,1] => 1 = 0 + 1
[2,1] => [1,2] => [2,1] => [2,1] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [2,3,1] => [1,3,2] => 1 = 0 + 1
[1,3,2] => [1,2,3] => [2,3,1] => [1,3,2] => 1 = 0 + 1
[2,1,3] => [1,2,3] => [2,3,1] => [1,3,2] => 1 = 0 + 1
[2,3,1] => [1,2,3] => [2,3,1] => [1,3,2] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [1,2,4,3] => 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [1,2,4,3] => 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [1,2,4,3] => 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [1,2,4,3] => 1 = 0 + 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [1,2,4,3] => 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [1,2,4,3] => 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [1,2,4,3] => 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [1,2,4,3] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,5,4] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,5,4] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,5,4] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,5,4] => 1 = 0 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,5,4] => 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,5,4] => 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,5,4] => 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,5,4] => 1 = 0 + 1
[2,1,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,5,4] => 1 = 0 + 1
[2,1,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,5,4] => 1 = 0 + 1
[2,1,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,5,4] => 1 = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,5,4] => 1 = 0 + 1
[2,3,1,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,5,4] => 1 = 0 + 1
[2,3,1,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,5,4] => 1 = 0 + 1
[2,3,4,1,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,5,4] => 1 = 0 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,5,4] => 1 = 0 + 1
[3,5,1,2,4] => [1,3,2,5,4] => [2,4,3,1,5] => [4,1,3,2,5] => 2 = 1 + 1
[3,5,1,4,2] => [1,3,2,5,4] => [2,4,3,1,5] => [4,1,3,2,5] => 2 = 1 + 1
[3,5,2,1,4] => [1,3,2,5,4] => [2,4,3,1,5] => [4,1,3,2,5] => 2 = 1 + 1
[3,5,2,4,1] => [1,3,2,5,4] => [2,4,3,1,5] => [4,1,3,2,5] => 2 = 1 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,6,5] => 1 = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,6,5] => 1 = 0 + 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,6,5] => 1 = 0 + 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,6,5] => 1 = 0 + 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,6,5] => 1 = 0 + 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,6,5] => 1 = 0 + 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,6,5] => 1 = 0 + 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,6,5] => 1 = 0 + 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,6,5] => 1 = 0 + 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,6,5] => 1 = 0 + 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,6,5] => 1 = 0 + 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,6,5] => 1 = 0 + 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,6,5] => 1 = 0 + 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,6,5] => 1 = 0 + 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,6,5] => 1 = 0 + 1
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
The following 351 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000024The number of double up and double down steps of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000143The largest repeated part of a partition. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000245The number of ascents of a permutation. St000259The diameter of a connected graph. St000272The treewidth of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000352The Elizalde-Pak rank of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000778The metric dimension of a graph. St000783The side length of the largest staircase partition fitting into a partition. St000834The number of right outer peaks of a permutation. St000845The maximal number of elements covered by an element in a poset. St000864The number of circled entries of the shifted recording tableau of a permutation. St000884The number of isolated descents of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001096The size of the overlap set of a permutation. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001644The dimension of a graph. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001792The arboricity of a graph. St001928The number of non-overlapping descents in a permutation. St001962The proper pathwidth of a graph. St000010The length of the partition. St000013The height of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000308The height of the tree associated to a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000527The width of the poset. St000759The smallest missing part in an integer partition. St000822The Hadwiger number of the graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001116The game chromatic number of a graph. 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. St001261The Castelnuovo-Mumford regularity of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000570The Edelman-Greene number of a permutation. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000486The number of cycles of length at least 3 of a permutation. St000628The balance of a binary word. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001470The cyclic holeyness of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001731The factorization defect of a permutation. St001840The number of descents of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000092The number of outer peaks of a permutation. St000485The length of the longest cycle of a permutation. St000640The rank of the largest boolean interval in a poset. St000862The number of parts of the shifted shape of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001741The largest integer such that all patterns of this size are contained in the permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000225Difference between largest and smallest parts in a partition. St000251The number of nonsingleton blocks of a set partition. St000252The number of nodes of degree 3 of a binary tree. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000347The inversion sum of a binary word. St000353The number of inner valleys of a permutation. St000358The number of occurrences of the pattern 31-2. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000516The number of stretching pairs of a permutation. St000535The rank-width of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000563The number of overlapping pairs of blocks of a set partition. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000659The number of rises of length at least 2 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000710The number of big deficiencies of a permutation. St000730The maximal arc length of a set partition. St000732The number of double deficiencies of a permutation. St000779The tier of a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000872The number of very big descents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001114The number of odd descents of a permutation. St001394The genus of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. 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. St001423The number of distinct cubes in a binary word. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001469The holeyness of a permutation. St001587Half of the largest even part of an integer partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001657The number of twos in an integer partition. St001712The number of natural descents of a standard Young tableau. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001743The discrepancy of a graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000058The order of a permutation. St000099The number of valleys of a permutation, including the boundary. St000201The number of leaf nodes in a binary tree. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension of a poset. St000396The register function (or Horton-Strahler number) of a binary tree. St000442The maximal area to the right of an up step of a Dyck path. St000504The cardinality of the first block of a set partition. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000668The least common multiple of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000758The length of the longest staircase fitting into an integer composition. St000886The number of permutations with the same antidiagonal sums. St000983The length of the longest alternating subword. St001062The maximal size of a block of a set partition. St001128The exponens consonantiae of a partition. St001220The width of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001372The length of a longest cyclic run of ones of a binary word. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001568The smallest positive integer that does not appear twice in the partition. St001732The number of peaks visible from the left. St001735The number of permutations with the same set of runs. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St000444The length of the maximal rise of a Dyck path. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000624The normalized sum of the minimal distances to a greater element. St000711The number of big exceedences of a permutation. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001960The number of descents of a permutation minus one if its first entry is not one. St000354The number of recoils of a permutation. St000652The maximal difference between successive positions of a permutation. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000829The Ulam distance of a permutation to the identity permutation. St000933The number of multipartitions of sizes given by an integer partition. St001330The hat guessing number of a graph. St001569The maximal modular displacement of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000454The largest eigenvalue of a graph if it is integral. St000881The number of short braid edges in the graph of braid moves of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000741The Colin de Verdière graph invariant. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000893The number of distinct diagonal sums of an alternating sign matrix. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001877Number of indecomposable injective modules with projective dimension 2. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001625The Möbius invariant of a lattice. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000100The number of linear extensions of 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. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001868The number of alignments of type NE of a signed permutation. St000706The product of the factorials of the multiplicities of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001933The largest multiplicity of a part in an integer partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St001060The distinguishing index of a graph. St001175The size of a partition minus the hook length of the base cell. 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 $eA$. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001281The normalized isoperimetric number of a graph. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001498The normalised height of a Nakayama algebra with magnitude 1. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000264The girth of a graph, which is not a tree. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000260The radius of a connected graph. St000618The number of self-evacuating tableaux of given shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001890The maximum magnitude of the Möbius function of a poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000495The number of inversions of distance at most 2 of a permutation. St001488The number of corners of a skew partition. St001734The lettericity of a graph. St001557The number of inversions of the second entry of a permutation. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000478Another weight of a partition according to Alladi. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St000635The number of strictly order preserving maps of a poset into itself. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000068The number of minimal elements in a poset. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001613The binary logarithm of the size of the center of a lattice. St000456The monochromatic index of a connected graph. St001870The number of positive entries followed by a negative entry in a signed permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001395The number of strictly unfriendly partitions of a graph. St001621The number of atoms of a lattice. St000636The hull number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001654The monophonic hull number of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000258The burning number of a graph. St000273The domination number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001829The common independence number of a graph. St001896The number of right descents of a signed permutations. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St000679The pruning number of an ordered tree. St000905The number of different multiplicities of parts of an integer composition. St000907The number of maximal antichains of minimal length in a poset. St001429The number of negative entries in a signed permutation. St001555The order of a signed permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001863The number of weak excedances of a signed permutation. St001893The flag descent of a signed permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001857The number of edges in the reduced word graph of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St001854The size of the left Kazhdan-Lusztig cell, St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001858The number of covering elements of a signed permutation in absolute order. St000936The number of even values of the symmetric group character corresponding to the partition. St001052The length of the exterior of a permutation. St001410The minimal entry of a semistandard tableau. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000084The number of subtrees. St000328The maximum number of child nodes in a tree. St000842The breadth of a permutation. St001271The competition number of a graph. St001624The breadth of a lattice. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau.
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