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Your data matches 457 different statistics following compositions of up to 3 maps.
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Matching statistic: St000441
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [1,3,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [3,1,4,2] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,3,1,5,2] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,5,2,1,4] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [5,4,3,1,6,2] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,4,2] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,2,4,3] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [4,6,3,2,1,5] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,4,3,6,5,2] => 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [4,1,5,3,2] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,5,1,4,2] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,5,4,1,3] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,2,5,4] => 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [5,1,3,6,4,2] => 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [3,4,6,1,5,2] => 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [5,4,2,6,1,3] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [4,6,1,2,5,3] => 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [5,3,6,2,1,4] => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [2,6,3,5,1,4] => 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [5,4,1,6,3,2] => 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,3,6,5,4,2] => 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [4,6,3,1,5,2] => 0
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [2,4,6,5,1,3] => 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [5,1,6,2,4,3] => 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,2,5,4,3] => 0
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [3,6,5,2,1,4] => 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [1,3,4,7,6,5,2] => 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,4,6,5,3,2] => 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [5,3,6,1,4,2] => 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,3,5,4,2] => 0
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [5,2,6,4,1,3] => 0
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,6,2,5,1,3] => 0
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,5,2,4,3] => 0
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [5,1,6,4,3,2] => 0
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,6,1,5,3,2] => 0
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [3,6,5,1,4,2] => 0
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [2,6,5,4,1,3] => 0
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => 0
[]
=> []
=> [1] => [1] => 0
Description
The number of successions of a permutation.
A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St000664
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000664: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000664: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [1,3,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [3,1,4,2] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,3,1,5,2] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,5,2,1,4] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [5,4,3,1,6,2] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,4,2] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,2,4,3] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [4,6,3,2,1,5] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,4,3,6,5,2] => 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [4,1,5,3,2] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,5,1,4,2] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,5,4,1,3] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,2,5,4] => 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [5,1,3,6,4,2] => 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [3,4,6,1,5,2] => 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [5,4,2,6,1,3] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [4,6,1,2,5,3] => 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [5,3,6,2,1,4] => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [2,6,3,5,1,4] => 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [5,4,1,6,3,2] => 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,3,6,5,4,2] => 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [4,6,3,1,5,2] => 0
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [2,4,6,5,1,3] => 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [5,1,6,2,4,3] => 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,2,5,4,3] => 0
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [3,6,5,2,1,4] => 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [1,3,4,7,6,5,2] => 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,4,6,5,3,2] => 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [5,3,6,1,4,2] => 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,3,5,4,2] => 0
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [5,2,6,4,1,3] => 0
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,6,2,5,1,3] => 0
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,5,2,4,3] => 0
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [5,1,6,4,3,2] => 0
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,6,1,5,3,2] => 0
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [3,6,5,1,4,2] => 0
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [2,6,5,4,1,3] => 0
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => 0
[]
=> []
=> [1] => [1] => 0
Description
The number of right ropes of a permutation.
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right rope is a large ascent after a raft of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
Matching statistic: St000665
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [1,3,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [3,1,4,2] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,3,1,5,2] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,5,2,1,4] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [5,4,3,1,6,2] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,4,2] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,2,4,3] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [4,6,3,2,1,5] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,4,3,6,5,2] => 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [4,1,5,3,2] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,5,1,4,2] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,5,4,1,3] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,2,5,4] => 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [5,1,3,6,4,2] => 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [3,4,6,1,5,2] => 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [5,4,2,6,1,3] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [4,6,1,2,5,3] => 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [5,3,6,2,1,4] => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [2,6,3,5,1,4] => 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [5,4,1,6,3,2] => 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,3,6,5,4,2] => 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [4,6,3,1,5,2] => 0
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [2,4,6,5,1,3] => 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [5,1,6,2,4,3] => 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,2,5,4,3] => 0
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [3,6,5,2,1,4] => 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [1,3,4,7,6,5,2] => 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,4,6,5,3,2] => 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [5,3,6,1,4,2] => 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,3,5,4,2] => 0
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [5,2,6,4,1,3] => 0
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,6,2,5,1,3] => 0
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,5,2,4,3] => 0
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [5,1,6,4,3,2] => 0
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,6,1,5,3,2] => 0
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [3,6,5,1,4,2] => 0
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [2,6,5,4,1,3] => 0
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => 0
[]
=> []
=> [1] => [1] => 0
Description
The number of rafts of a permutation.
Let $\pi$ be a permutation of length $n$. A small ascent of $\pi$ is an index $i$ such that $\pi(i+1)= \pi(i)+1$, see [[St000441]], and a raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents.
Matching statistic: St001652
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001652: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001652: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [1,3,2] => 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [3,1,4,2] => 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,3,1,5,2] => 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,5,2,1,4] => 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [5,4,3,1,6,2] => 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,4,2] => 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,2,4,3] => 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [4,6,3,2,1,5] => 1 = 0 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,4,3,6,5,2] => 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [4,1,5,3,2] => 1 = 0 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,5,1,4,2] => 1 = 0 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,5,4,1,3] => 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,2,5,4] => 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [5,1,3,6,4,2] => 1 = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [3,4,6,1,5,2] => 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [5,4,2,6,1,3] => 1 = 0 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => 1 = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [4,6,1,2,5,3] => 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [5,3,6,2,1,4] => 1 = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [2,6,3,5,1,4] => 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [5,4,1,6,3,2] => 1 = 0 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,3,6,5,4,2] => 1 = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [4,6,3,1,5,2] => 1 = 0 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [2,4,6,5,1,3] => 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [5,1,6,2,4,3] => 1 = 0 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,2,5,4,3] => 1 = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [3,6,5,2,1,4] => 1 = 0 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [1,3,4,7,6,5,2] => 2 = 1 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,4,6,5,3,2] => 1 = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [5,3,6,1,4,2] => 1 = 0 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,3,5,4,2] => 1 = 0 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [5,2,6,4,1,3] => 1 = 0 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,6,2,5,1,3] => 1 = 0 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,5,2,4,3] => 1 = 0 + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [5,1,6,4,3,2] => 1 = 0 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,6,1,5,3,2] => 1 = 0 + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [3,6,5,1,4,2] => 1 = 0 + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [2,6,5,4,1,3] => 1 = 0 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => 1 = 0 + 1
[]
=> []
=> [1] => [1] => 1 = 0 + 1
Description
The length of a longest interval of consecutive numbers.
For a permutation $\pi=\pi_1,\dots,\pi_n$, this statistic returns the length of a longest subsequence $\pi_k,\dots,\pi_\ell$ such that $\pi_{i+1} = \pi_i + 1$ for $i\in\{k,\dots,\ell-1\}$.
Matching statistic: St001662
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [1,3,2] => 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [3,1,4,2] => 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,3,1,5,2] => 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,5,2,1,4] => 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [5,4,3,1,6,2] => 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,4,2] => 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,2,4,3] => 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [4,6,3,2,1,5] => 1 = 0 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,4,3,6,5,2] => 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [4,1,5,3,2] => 1 = 0 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,5,1,4,2] => 1 = 0 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,5,4,1,3] => 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,2,5,4] => 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [5,1,3,6,4,2] => 1 = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [3,4,6,1,5,2] => 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [5,4,2,6,1,3] => 1 = 0 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => 1 = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [4,6,1,2,5,3] => 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [5,3,6,2,1,4] => 1 = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [2,6,3,5,1,4] => 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [5,4,1,6,3,2] => 1 = 0 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,3,6,5,4,2] => 1 = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [4,6,3,1,5,2] => 1 = 0 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [2,4,6,5,1,3] => 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [5,1,6,2,4,3] => 1 = 0 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,2,5,4,3] => 1 = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [3,6,5,2,1,4] => 1 = 0 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [1,3,4,7,6,5,2] => 2 = 1 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,4,6,5,3,2] => 1 = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [5,3,6,1,4,2] => 1 = 0 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,3,5,4,2] => 1 = 0 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [5,2,6,4,1,3] => 1 = 0 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,6,2,5,1,3] => 1 = 0 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,5,2,4,3] => 1 = 0 + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [5,1,6,4,3,2] => 1 = 0 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,6,1,5,3,2] => 1 = 0 + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [3,6,5,1,4,2] => 1 = 0 + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [2,6,5,4,1,3] => 1 = 0 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => 1 = 0 + 1
[]
=> []
=> [1] => [1] => 1 = 0 + 1
Description
The length of the longest factor of consecutive numbers in a permutation.
Matching statistic: St000214
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 0
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => ? = 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => 0
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 0
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 0
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 0
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 0
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 0
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 0
[]
=> []
=> [1] => 0
Description
The number of adjacencies of a permutation.
An adjacency of a permutation $\pi$ is an index $i$ such that $\pi(i)-1 = \pi(i+1)$. Adjacencies are also known as ''small descents''.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St001137
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001137: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001137: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[]
=> []
=> []
=> []
=> ? = 0
Description
Number of simple modules that are 3-regular in the corresponding Nakayama algebra.
Matching statistic: St001663
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001663: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001663: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [2,3,1] => 0
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [3,4,1,2] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,3,1] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,5,1,2,3] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,2,3,1,4] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [5,6,1,2,3,4] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [2,4,5,3,1] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,3,4,1,2] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [3,4,2,5,1] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,2,3,1,4,5] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [2,4,1,5,6,3] => 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [3,4,5,1,2] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,5,2,3,1] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,5,3,4,1] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [3,5,2,4,6,1] => 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [3,5,6,4,1,2] => 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [5,6,2,4,3,1] => 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6,4,5,1,2,3] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [4,6,5,2,3,1] => 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6,3,4,1,2,5] => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,3,4,2,5,1] => 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [4,5,6,1,2,3] => 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [2,3,5,6,4,1] => 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [5,6,2,3,1,4] => 0
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [2,6,4,5,3,1] => 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [4,5,3,6,1,2] => 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [3,4,5,2,6,1] => 0
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [2,6,3,4,1,5] => 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [2,3,6,7,5,4,1] => ? = 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [2,4,5,6,3,1] => 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [5,6,3,4,1,2] => 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [3,4,2,5,6,1] => 0
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [3,6,4,5,1,2] => 0
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6,4,5,2,3,1] => 0
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [2,4,5,3,6,1] => 0
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [3,4,5,6,1,2] => 0
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,5,6,2,3,1] => 0
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [2,5,6,3,4,1] => 0
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [2,3,6,4,5,1] => 0
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => 0
[]
=> []
=> [1] => [1] => 0
Description
The number of occurrences of the Hertzsprung pattern 132 in a permutation.
A Hertzsprung occurrence of the pattern $\tau=(\tau_1,\dots,\tau_k)$ in a permutation $\pi$ is a factor $\pi_i, \pi_{i+1}, \dots,\pi_{i+k-1}$ of $\pi$ such that $\pi_{i+j-1} - \tau_j$ is constant for $1\leq j\leq k$.
Matching statistic: St001063
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001063: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001063: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[]
=> []
=> []
=> []
=> ? = 0 + 1
Description
Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra.
Matching statistic: St001064
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001064: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001064: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[]
=> []
=> []
=> []
=> ? = 0 + 1
Description
Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules.
The following 447 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St000322The skewness of a graph. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001715The number of non-records in a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000879The number of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001434The number of negative sum pairs of a signed permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000210Minimum over maximum difference of elements in cycles. St000223The number of nestings in the permutation. St000237The number of small exceedances. St000352The Elizalde-Pak rank of a permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000546The number of global descents of a permutation. St000666The number of right tethers of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St001468The smallest fixpoint of a permutation. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000516The number of stretching pairs 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. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000487The length of the shortest cycle of a permutation. St000990The first ascent of a permutation. St000842The breadth of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St000002The number of occurrences of the pattern 123 in a permutation. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000405The number of occurrences of the pattern 1324 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000629The defect of a binary word. St000787The number of flips required to make a perfect matching noncrossing. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001394The genus of a permutation. St001513The number of nested exceedences of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000717The number of ordinal summands of a poset. St000788The number of nesting-similar perfect matchings of a perfect matching. St001260The permanent of an alternating sign matrix. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001549The number of restricted non-inversions between exceedances. St001810The number of fixed points of a permutation smaller than its largest moved point. St000056The decomposition (or block) number of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000042The number of crossings of a perfect matching. St000051The size of the left subtree of a binary tree. St000052The number of valleys of a Dyck path not on the x-axis. St000091The descent variation of a composition. St000095The number of triangles of a graph. St000133The "bounce" of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000234The number of global ascents of a permutation. St000296The length of the symmetric border of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000317The cycle descent number of a permutation. St000351The determinant of the adjacency matrix of a graph. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000368The Altshuler-Steinberg determinant of a graph. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000674The number of hills of a Dyck path. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000731The number of double exceedences of a permutation. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001119The length of a shortest maximal path in a graph. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001271The competition number of a graph. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001395The number of strictly unfriendly partitions of a graph. St001411The number of patterns 321 or 3412 in a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001536The number of cyclic misalignments of a permutation. St001537The number of cyclic crossings of a permutation. 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. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001705The number of occurrences of the pattern 2413 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. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001871The number of triconnected components of a graph. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000054The first entry of the permutation. St000069The number of maximal elements of a poset. St000084The number of subtrees. St000096The number of spanning trees of a graph. St000287The number of connected components of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000450The number of edges minus the number of vertices plus 2 of a graph. St000627The exponent of a binary word. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000740The last entry of a permutation. St000775The multiplicity of the largest eigenvalue in a graph. St000843The decomposition number of a perfect matching. St000876The number of factors in the Catalan decomposition of a binary word. St000889The number of alternating sign matrices with the same antidiagonal sums. St000991The number of right-to-left minima of a permutation. St001048The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. 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. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001272The number of graphs with the same degree sequence. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001316The domatic number of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001597The Frobenius rank of a skew partition. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001828The Euler characteristic of a graph. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000733The row containing the largest entry of a standard tableau. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. 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. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St001204Call 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. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001552The number of inversions between excedances and fixed points of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St000694The number of affine bounded permutations that project to a given permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001344The neighbouring number of a permutation. St001590The crossing number of a perfect matching. St001722The number of minimal chains with small intervals between a binary word and the top element. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000022The number of fixed points of a permutation. St000090The variation of a composition. St000221The number of strong fixed points of a permutation. St000225Difference between largest and smallest parts in a partition. St000295The length of the border of a binary word. St000379The number of Hamiltonian cycles in a graph. St000461The rix statistic of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000699The toughness times the least common multiple of 1,. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000873The aix statistic of a permutation. St000989The number of final rises of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001281The normalized isoperimetric number of a graph. St001301The first Betti number of the order complex associated with the poset. St001381The fertility of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001429The number of negative entries in a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001577The minimal number of edges to add or remove to make a graph a cograph. St001836The number of occurrences of a 213 pattern in the restricted growth word of a perfect matching. St001856The number of edges in the reduced word graph of a permutation. St000061The number of nodes on the left branch of a binary tree. St000570The Edelman-Greene number of a permutation. St000654The first descent of a permutation. St000781The number of proper colouring schemes of a Ferrers diagram. St000908The length of the shortest maximal antichain in a poset. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001081The number of minimal length factorizations of a permutation into star transpositions. St001162The minimum jump of a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001487The number of inner corners of a skew partition. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001589The nesting number of a perfect matching. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000485The length of the longest cycle of a permutation. St001042The size of the automorphism group of the leaf labelled binary unordered tree associated with the perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. 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 $eA$. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001811The Castelnuovo-Mumford regularity of a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001520The number of strict 3-descents. St001557The number of inversions of the second entry of a permutation. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000914The sum of the values of the Möbius function of a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000534The number of 2-rises of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000264The girth of a graph, which is not a tree. St000297The number of leading ones in a binary word. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001960The number of descents of a permutation minus one if its first entry is not one. St000768The number of peaks in an integer composition. St000764The number of strong records in an integer composition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000310The minimal degree of a vertex of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000650The number of 3-rises of a permutation. St000286The number of connected components of the complement of a graph. St000657The smallest part of an integer composition. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001518The number of graphs with the same ordinary spectrum as the given graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000807The sum of the heights of the valleys of the associated bargraph. St000894The trace of an alternating sign matrix. St000943The number of spots the most unlucky car had to go further in a parking function. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001524The degree of symmetry of a binary word. St001556The number of inversions of the third entry of a permutation. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001845The number of join irreducibles minus the rank of a lattice. St001867The number of alignments of type EN of a signed permutation. St000763The sum of the positions of the strong records of an integer composition. St000805The number of peaks of the associated bargraph. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St001884The number of borders of a binary word. St000900The minimal number of repetitions of a part in an integer composition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000902 The minimal number of repetitions of an integer composition. St001413Half the length of the longest even length palindromic prefix of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001889The size of the connectivity set of a signed permutation. 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 $eA$. St000455The second largest eigenvalue of a graph if it is integral. St001621The number of atoms of a lattice. St000417The size of the automorphism group of the ordered tree. St001058The breadth of the ordered tree. St000068The number of minimal elements in a poset. St001410The minimal entry of a semistandard tableau. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000039The number of crossings of a permutation. St000247The number of singleton blocks of a set partition. St000355The number of occurrences of the pattern 21-3. St000462The major index minus the number of excedences of a permutation. St000496The rcs statistic of a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000962The 3-shifted major index of a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001781The interlacing number of a set partition. St001857The number of edges in the reduced word graph of a signed permutation. St000021The number of descents of a permutation. St000154The sum of the descent bottoms of a permutation. St000253The crossing number of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000729The minimal arc length of a set partition. St000864The number of circled entries of the shifted recording tableau of a permutation. 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. St001806The upper middle entry of a permutation. St001928The number of non-overlapping descents in a permutation. St000105The number of blocks in the set partition. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000504The cardinality of the first block of a set partition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000823The number of unsplittable factors of the set partition. St000862The number of parts of the shifted shape of a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St000219The number of occurrences of the pattern 231 in a permutation. St000338The number of pixed points of a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000488The number of cycles of a permutation of length at most 2. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000649The number of 3-excedences of a permutation. St000779The tier of a permutation. St000850The number of 1/2-balanced pairs in a poset. St000872The number of very big descents of a permutation. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001850The number of Hecke atoms of a permutation. St000023The number of inner peaks of a permutation. St000353The number of inner valleys of a permutation. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) 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. St000633The size of the automorphism group of a poset. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000886The number of permutations with the same antidiagonal sums. St000925The number of topologically connected components of a set partition. St001399The distinguishing number of a poset. St001472The permanent of the Coxeter matrix of the poset. St001839The number of excedances of a set partition. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000251The number of nonsingleton blocks of a set partition. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000619The number of cyclic descents of a permutation. St000679The pruning number of an ordered tree. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000836The number of descents of distance 2 of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St000495The number of inversions of distance at most 2 of a permutation. St000638The number of up-down runs of a permutation. St000831The number of indices that are either descents or recoils. St001060The distinguishing index of a graph.
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