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Your data matches 333 different statistics following compositions of up to 3 maps.
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Matching statistic: St000206
St000206: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 0
[1,1]
=> 0
[3]
=> 0
[2,1]
=> 0
[1,1,1]
=> 0
[4]
=> 0
[3,1]
=> 0
[2,2]
=> 0
[2,1,1]
=> 0
[1,1,1,1]
=> 0
[5]
=> 0
[4,1]
=> 0
[3,2]
=> 1
[3,1,1]
=> 0
[2,2,1]
=> 1
[2,1,1,1]
=> 0
[1,1,1,1,1]
=> 0
[6]
=> 0
[5,1]
=> 0
[4,2]
=> 3
[4,1,1]
=> 0
[3,3]
=> 1
[3,2,1]
=> 6
[3,1,1,1]
=> 0
[2,2,2]
=> 1
[2,2,1,1]
=> 1
[2,1,1,1,1]
=> 0
[1,1,1,1,1,1]
=> 0
[7]
=> 0
[6,1]
=> 0
[5,2]
=> 7
[5,1,1]
=> 0
[4,3]
=> 11
[4,2,1]
=> 16
[4,1,1,1]
=> 0
[3,3,1]
=> 17
[3,2,2]
=> 8
[3,2,1,1]
=> 8
[3,1,1,1,1]
=> 0
[2,2,2,1]
=> 7
[2,2,1,1,1]
=> 1
[2,1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> 0
[8]
=> 0
[7,1]
=> 0
[6,2]
=> 15
[6,1,1]
=> 0
[5,3]
=> 31
[5,2,1]
=> 43
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also [[St000205]].
Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St001344
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001344: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 59%●distinct values known / distinct values provided: 11%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001344: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 59%●distinct values known / distinct values provided: 11%
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] => [4,2,1,3] => 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [6,2,3,4,1,5] => 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,2,5,4] => 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] => [1,3,4,5,6,2] => 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [7,2,3,4,5,1,6] => ? = 1 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [3,1,4,2,6,5] => 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,1,5,3,4] => 1 = 0 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => 1 = 0 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,2,4,5,3] => 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,1,4,2,5,3] => 2 = 1 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [1,3,4,5,6,7,2] => 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [8,2,3,4,5,6,1,7] => ? ∊ {0,1,3,6} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => [3,4,1,5,2,7,6] => ? ∊ {0,1,3,6} + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6,2,1,4,3,5] => 1 = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [4,3,1,6,2,5] => 1 = 0 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [5,2,3,6,1,4] => 1 = 0 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 1 = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6,3,1,5,2,4] => 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [4,2,5,6,1,3] => 1 = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,5,2,4,6,3] => 2 = 1 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [1,7,4,5,2,6,3] => ? ∊ {0,1,3,6} + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [1,3,4,5,6,7,8,2] => ? ∊ {0,1,3,6} + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => [9,2,3,4,5,6,7,1,8] => ? ∊ {1,7,7,8,8,11,16,17} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => [3,4,5,1,6,2,8,7] => ? ∊ {1,7,7,8,8,11,16,17} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [2,7,4,1,5,3,6] => ? ∊ {1,7,7,8,8,11,16,17} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [3,5,4,1,7,2,6] => ? ∊ {1,7,7,8,8,11,16,17} + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,6,3,1,4,5] => 1 = 0 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,2,4,3,6,5] => 1 = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6,1,3,4,2,5] => 1 = 0 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,3,2,5,6,4] => 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6,2,1,5,3,4] => 1 = 0 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,2,6,3,5,4] => 1 = 0 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [3,7,5,1,6,2,4] => ? ∊ {1,7,7,8,8,11,16,17} + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,2,4,5,6,3] => 1 = 0 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [6,1,4,2,5,7,3] => ? ∊ {1,7,7,8,8,11,16,17} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [1,4,8,5,6,2,7,3] => ? ∊ {1,7,7,8,8,11,16,17} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [1,3,4,5,6,7,8,9,2] => ? ∊ {1,7,7,8,8,11,16,17} + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => [10,2,3,4,5,6,7,8,1,9] => ? ∊ {1,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,1,2,8] => [3,4,5,6,1,7,2,9,8] => ? ∊ {1,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,8,4,5,6,1,3,7] => [2,4,8,5,1,6,3,7] => ? ∊ {1,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => [3,4,6,5,1,8,2,7] => ? ∊ {1,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [7,2,5,3,1,4,6] => ? ∊ {1,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [1,2,5,4,3,7,6] => 1 = 0 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [1,4,7,3,5,2,6] => ? ∊ {1,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [6,2,3,4,7,1,5] => ? ∊ {1,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,3,2,4,6,5] => 1 = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [4,6,2,1,3,5] => 1 = 0 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,4,2,3,6,5] => 1 = 0 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [7,6,1,3,4,2,5] => ? ∊ {1,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,1,5,6,3,4] => 1 = 0 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,6,2,4] => 1 = 0 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,2,3,5,4] => 1 = 0 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [1,2,7,6,3,5,4] => 1 = 0 + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [3,5,8,6,1,7,2,4] => ? ∊ {1,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [4,2,5,6,7,1,3] => ? ∊ {1,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [1,5,2,4,6,7,3] => ? ∊ {1,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [1,7,4,5,2,6,8,3] => ? ∊ {1,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [1,4,5,9,6,7,2,8,3] => ? ∊ {1,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => [1,3,4,5,6,7,8,9,10,2] => ? ∊ {1,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
Description
The neighbouring number of a permutation.
For a permutation $\pi$, this is
$$\min \big(\big\{|\pi(k)-\pi(k+1)|:k\in\{1,\ldots,n-1\}\big\}\cup \big\{|\pi(1) - \pi(n)|\big\}\big).$$
Matching statistic: St000862
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 59%●distinct values known / distinct values provided: 11%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 59%●distinct values known / distinct values provided: 11%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> [3,1,2] => 2 = 0 + 2
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2 = 0 + 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 2 = 0 + 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 2 = 0 + 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 2 = 0 + 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 0 + 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2 = 0 + 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 2 = 0 + 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 2 = 0 + 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 2 = 0 + 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2 = 0 + 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2 = 0 + 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 2 = 0 + 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 3 = 1 + 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ? = 1 + 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => 2 = 0 + 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? ∊ {1,3,6} + 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 2 = 0 + 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 2 = 0 + 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 2 = 0 + 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 2 = 0 + 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 3 = 1 + 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => 2 = 0 + 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 3 = 1 + 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {1,3,6} + 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,2] => ? ∊ {1,3,6} + 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => 2 = 0 + 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [3,1,4,5,6,8,2,7] => ? ∊ {7,7,8,8,11,16,17} + 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? ∊ {7,7,8,8,11,16,17} + 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? ∊ {7,7,8,8,11,16,17} + 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 2 = 0 + 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 0 + 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 3 = 1 + 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 2 = 0 + 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 2 = 0 + 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? ∊ {7,7,8,8,11,16,17} + 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 2 = 0 + 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ? ∊ {7,7,8,8,11,16,17} + 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? ∊ {7,7,8,8,11,16,17} + 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,2] => ? ∊ {7,7,8,8,11,16,17} + 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => ? ∊ {0,0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 2
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [3,1,4,5,6,7,9,2,8] => ? ∊ {0,0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,4,1,5,6,8,3,7] => ? ∊ {0,0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => ? ∊ {0,0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? ∊ {0,0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? ∊ {0,0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ? ∊ {0,0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => ? ∊ {0,0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 2 = 0 + 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => 2 = 0 + 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => 2 = 0 + 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? ∊ {0,0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 2 = 0 + 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 3 = 1 + 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 2 = 0 + 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => ? ∊ {0,0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 2
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => ? ∊ {0,0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ? ∊ {0,0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 2
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ? ∊ {0,0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 2
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [8,4,1,5,6,7,2,3] => ? ∊ {0,0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => ? ∊ {0,0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 2
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [10,3,4,5,6,7,8,9,1,2] => ? ∊ {0,0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 2
Description
The number of parts of the shifted shape of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of parts of the shifted shape.
Matching statistic: St001513
(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
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001513: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 58%●distinct values known / distinct values provided: 11%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001513: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 58%●distinct values known / distinct values provided: 11%
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] => [4,2,1,3] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [3,4,1,5,2] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [1,3,4,2] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [4,2,5,1,3] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [3,4,5,1,6,2] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,5,3,4,2] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [3,5,2,1,4] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,3,2,5,4] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [4,2,5,6,1,3] => 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [3,4,5,6,1,7,2] => ? ∊ {1,1}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,5,6,3,4,2] => 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,2,1,5,3] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,2,3,1,4] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,3,6,2,5,4] => 0
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [4,2,5,6,7,1,3] => ? ∊ {1,1}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [3,4,5,6,7,1,8,2] => ? ∊ {1,3,6}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => [1,5,6,7,3,4,2] => ? ∊ {1,3,6}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [3,1,6,4,5,2] => 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [5,4,2,1,6,3] => 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [3,4,6,2,1,5] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,3,4,5,2] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [4,2,1,3,6,5] => 0
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [3,5,2,6,1,4] => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,2,5,3,1,4] => 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [1,3,6,7,2,5,4] => 0
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [4,2,5,6,7,8,1,3] => ? ∊ {1,3,6}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => [3,4,5,6,7,8,1,9,2] => ? ∊ {0,7,7,8,8,11,16,17}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => [1,5,6,7,8,3,4,2] => ? ∊ {0,7,7,8,8,11,16,17}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [3,1,6,7,4,5,2] => ? ∊ {0,7,7,8,8,11,16,17}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [6,4,5,2,1,7,3] => ? ∊ {0,7,7,8,8,11,16,17}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [3,4,1,5,6,2] => 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,6,3,4,5,2] => 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [4,2,5,1,6,3] => 0
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6,4,2,3,1,5] => 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [3,1,4,2,6,5] => 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,3,2,5,6,4] => 0
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [4,2,1,7,3,6,5] => ? ∊ {0,7,7,8,8,11,16,17}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [5,2,3,6,1,4] => 0
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [7,2,5,6,3,1,4] => ? ∊ {0,7,7,8,8,11,16,17}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [1,3,6,7,8,2,5,4] => ? ∊ {0,7,7,8,8,11,16,17}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [4,2,5,6,7,8,9,1,3] => ? ∊ {0,7,7,8,8,11,16,17}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => [3,4,5,6,7,8,9,1,10,2] => ? ∊ {0,1,8,8,11,14,15,24,28,30,31,31,39,43,53}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,1,2,8] => [1,5,6,7,8,9,3,4,2] => ? ∊ {0,1,8,8,11,14,15,24,28,30,31,31,39,43,53}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,8,4,5,6,1,3,7] => [3,1,6,7,8,4,5,2] => ? ∊ {0,1,8,8,11,14,15,24,28,30,31,31,39,43,53}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => [7,4,5,6,2,1,8,3] => ? ∊ {0,1,8,8,11,14,15,24,28,30,31,31,39,43,53}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [3,4,1,7,5,6,2] => ? ∊ {0,1,8,8,11,14,15,24,28,30,31,31,39,43,53}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [1,7,6,3,4,5,2] => ? ∊ {0,1,8,8,11,14,15,24,28,30,31,31,39,43,53}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [5,4,2,6,1,7,3] => ? ∊ {0,1,8,8,11,14,15,24,28,30,31,31,39,43,53}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [3,4,5,7,2,1,6] => ? ∊ {0,1,8,8,11,14,15,24,28,30,31,31,39,43,53}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,5,3,4,6,2] => 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [3,5,2,1,6,4] => 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,3,6,4,5,2] => 0
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [4,2,5,1,3,7,6] => ? ∊ {0,1,8,8,11,14,15,24,28,30,31,31,39,43,53}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [3,6,2,4,1,5] => 0
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,2,6,3,1,5] => 0
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,3,4,2,6,5] => 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [1,3,2,4,6,7,5] => 0
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [4,2,1,7,8,3,6,5] => ? ∊ {0,1,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [3,5,2,6,7,1,4] => ? ∊ {0,1,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [6,2,5,3,7,1,4] => ? ∊ {0,1,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [8,2,5,6,7,3,1,4] => ? ∊ {0,1,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [1,3,6,7,8,9,2,5,4] => ? ∊ {0,1,8,8,11,14,15,24,28,30,31,31,39,43,53}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => [4,2,5,6,7,8,9,10,1,3] => ? ∊ {0,1,8,8,11,14,15,24,28,30,31,31,39,43,53}
Description
The number of nested exceedences of a permutation.
For a permutation $\pi$, this is the number of pairs $i,j$ such that $i < j < \pi(j) < \pi(i)$. For exceedences, see [[St000155]].
Matching statistic: St001550
(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
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001550: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 58%●distinct values known / distinct values provided: 11%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001550: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 58%●distinct values known / distinct values provided: 11%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [1,3,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [2,4,1,3] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,4,2] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,5,1,3,4] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,4,5,2] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,1,3,4,5] => 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] => [2,1,4,3,5] => 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] => [3,1,4,5,6,2] => 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,7,1,3,4,5,6] => ? ∊ {1,1}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [3,4,6,1,5,2] => 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,3,5,1,4] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,2,5,4] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [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] => [4,6,1,2,5,3] => 0
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,4,5,6,7,2] => ? ∊ {1,1}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [2,8,1,3,4,5,6,7] => ? ∊ {1,1,3,6}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => [3,4,7,1,5,6,2] => ? ∊ {1,1,3,6}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [2,4,6,5,1,3] => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,4,3,6,5,2] => 0
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,1,5,3,4,6] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,6,3,2,5,4] => 0
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,1,4,5,3,6] => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,6,2,4,3] => 0
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [4,7,1,2,5,6,3] => ? ∊ {1,1,3,6}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [3,1,4,5,6,7,8,2] => ? ∊ {1,1,3,6}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => [2,9,1,3,4,5,6,7,8] => ? ∊ {1,7,7,8,8,11,16,17}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => [3,4,8,1,5,6,7,2] => ? ∊ {1,7,7,8,8,11,16,17}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [2,4,7,1,5,3,6] => ? ∊ {1,7,7,8,8,11,16,17}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [3,5,1,7,4,6,2] => ? ∊ {1,7,7,8,8,11,16,17}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [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] => [1,2,4,6,5,3] => 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,4,1,2,6,5] => 0
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [5,1,3,6,4,2] => 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,3,5,1,4] => 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,2,6,3,5,4] => 0
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [3,7,1,5,2,6,4] => ? ∊ {1,7,7,8,8,11,16,17}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,2,5,6,3] => 0
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [4,1,6,2,5,7,3] => ? ∊ {1,7,7,8,8,11,16,17}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [4,8,1,2,5,6,7,3] => ? ∊ {1,7,7,8,8,11,16,17}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [3,1,4,5,6,7,8,9,2] => ? ∊ {1,7,7,8,8,11,16,17}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => [2,10,1,3,4,5,6,7,8,9] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,1,2,8] => [3,4,9,1,5,6,7,8,2] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,8,4,5,6,1,3,7] => [2,4,8,1,5,6,3,7] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => [3,6,1,8,4,5,7,2] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [2,3,7,5,1,4,6] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [1,2,5,7,6,4,3] => 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,5,1,3,7,6,2] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [2,1,6,3,4,5,7] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,3,4,6,5,2] => 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [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] => [1,2,6,4,5,3] => 0
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,7,4,1,2,6,5] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,1,5,6,3,4] => 0
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [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] => [1,6,2,3,5,4] => 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [1,2,7,6,3,5,4] => 0
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [3,8,1,5,6,2,7,4] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,1,4,5,6,3,7] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,2,6,4,7,3] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [4,1,7,2,5,6,8,3] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [4,9,1,2,5,6,7,8,3] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => [3,1,4,5,6,7,8,9,10,2] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
Description
The number of inversions between exceedances where the greater exceedance is linked.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{ile}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(j) < \sigma(i) \wedge \sigma^{-1}(j) < j \}.$$
Matching statistic: St001906
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001906: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 58%●distinct values known / distinct values provided: 11%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001906: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 58%●distinct values known / distinct values provided: 11%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [1,3,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [2,4,1,3] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,4,2] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,5,1,3,4] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,4,5,2] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,1,3,4,5] => 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] => [2,1,4,3,5] => 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] => [3,1,4,5,6,2] => 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,7,1,3,4,5,6] => ? ∊ {0,0}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [3,4,6,1,5,2] => 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,3,5,1,4] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,2,5,4] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [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] => [4,6,1,2,5,3] => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,4,5,6,7,2] => ? ∊ {0,0}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [2,8,1,3,4,5,6,7] => ? ∊ {0,1,3,6}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => [3,4,7,1,5,6,2] => ? ∊ {0,1,3,6}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [2,4,6,5,1,3] => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,4,3,6,5,2] => 0
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,1,5,3,4,6] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,6,3,2,5,4] => 0
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,1,4,5,3,6] => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,6,2,4,3] => 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [4,7,1,2,5,6,3] => ? ∊ {0,1,3,6}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [3,1,4,5,6,7,8,2] => ? ∊ {0,1,3,6}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => [2,9,1,3,4,5,6,7,8] => ? ∊ {0,7,7,8,8,11,16,17}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => [3,4,8,1,5,6,7,2] => ? ∊ {0,7,7,8,8,11,16,17}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [2,4,7,1,5,3,6] => ? ∊ {0,7,7,8,8,11,16,17}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [3,5,1,7,4,6,2] => ? ∊ {0,7,7,8,8,11,16,17}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [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] => [1,2,4,6,5,3] => 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,4,1,2,6,5] => 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [5,1,3,6,4,2] => 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,3,5,1,4] => 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,2,6,3,5,4] => 0
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [3,7,1,5,2,6,4] => ? ∊ {0,7,7,8,8,11,16,17}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,2,5,6,3] => 0
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [4,1,6,2,5,7,3] => ? ∊ {0,7,7,8,8,11,16,17}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [4,8,1,2,5,6,7,3] => ? ∊ {0,7,7,8,8,11,16,17}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [3,1,4,5,6,7,8,9,2] => ? ∊ {0,7,7,8,8,11,16,17}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => [2,10,1,3,4,5,6,7,8,9] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,1,2,8] => [3,4,9,1,5,6,7,8,2] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,8,4,5,6,1,3,7] => [2,4,8,1,5,6,3,7] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => [3,6,1,8,4,5,7,2] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [2,3,7,5,1,4,6] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [1,2,5,7,6,4,3] => 0
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,5,1,3,7,6,2] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [2,1,6,3,4,5,7] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,3,4,6,5,2] => 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [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] => [1,2,6,4,5,3] => 0
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,7,4,1,2,6,5] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,1,5,6,3,4] => 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [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] => [1,6,2,3,5,4] => 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [1,2,7,6,3,5,4] => 0
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [3,8,1,5,6,2,7,4] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,1,4,5,6,3,7] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,2,6,4,7,3] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [4,1,7,2,5,6,8,3] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [4,9,1,2,5,6,7,8,3] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => [3,1,4,5,6,7,8,9,10,2] => ? ∊ {0,8,8,11,14,15,24,28,30,31,31,39,43,53}
Description
Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation.
Let $\pi$ be a permutation. Its total displacement [[St000830]] is $D(\pi) = \sum_i |\pi(i) - i|$, and its absolute length [[St000216]] is the minimal number $T(\pi)$ of transpositions whose product is $\pi$. Finally, let $I(\pi)$ be the number of inversions [[St000018]] of $\pi$.
This statistic equals $\left(D(\pi)-T(\pi)-I(\pi)\right)/2$.
Diaconis and Graham [1] proved that this statistic is always nonnegative.
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: 11% ●values known / values provided: 56%●distinct values known / distinct values provided: 11%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000664: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 56%●distinct values known / distinct values provided: 11%
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
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [6,5,4,3,1,7,2] => ? ∊ {1,1}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,4,3,6,5,2] => 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
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [5,7,4,3,2,1,6] => ? ∊ {1,1}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [7,6,5,4,3,1,8,2] => ? ∊ {0,1,3,6}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => [1,5,4,3,7,6,2] => ? ∊ {0,1,3,6}
[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
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [1,7,4,3,2,6,5] => ? ∊ {0,1,3,6}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [6,8,5,4,3,2,1,7] => ? ∊ {0,1,3,6}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => [8,7,6,5,4,3,1,9,2] => ? ∊ {1,7,7,8,8,11,16,17}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => [1,6,5,4,3,8,7,2] => ? ∊ {1,7,7,8,8,11,16,17}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [6,1,4,3,7,5,2] => ? ∊ {1,7,7,8,8,11,16,17}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [3,5,4,7,1,6,2] => ? ∊ {1,7,7,8,8,11,16,17}
[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
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [5,7,1,3,2,6,4] => ? ∊ {1,7,7,8,8,11,16,17}
[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
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [2,7,4,3,6,1,5] => ? ∊ {1,7,7,8,8,11,16,17}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [1,8,5,4,3,2,7,6] => ? ∊ {1,7,7,8,8,11,16,17}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [7,9,6,5,4,3,2,1,8] => ? ∊ {1,7,7,8,8,11,16,17}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => [9,8,7,6,5,4,3,1,10,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,1,2,8] => [1,7,6,5,4,3,9,8,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,8,4,5,6,1,3,7] => [7,1,5,4,3,8,6,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => [3,6,5,4,8,1,7,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [6,5,1,3,7,4,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[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
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,5,7,3,1,6,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [6,5,4,2,7,1,3] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[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
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [5,7,4,1,2,6,3] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[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
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [1,7,2,3,6,5,4] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [6,8,1,4,3,2,7,5] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [6,4,7,3,2,1,5] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [3,7,4,6,2,1,5] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [2,8,5,4,3,7,1,6] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [1,9,6,5,4,3,2,8,7] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => [8,10,7,6,5,4,3,2,1,9] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
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: St001551
(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
St001551: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 56%●distinct values known / distinct values provided: 11%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001551: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 56%●distinct values known / distinct values provided: 11%
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
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [6,5,4,3,1,7,2] => ? ∊ {1,1}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,4,3,6,5,2] => 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
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [5,7,4,3,2,1,6] => ? ∊ {1,1}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [7,6,5,4,3,1,8,2] => ? ∊ {0,0,3,6}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => [1,5,4,3,7,6,2] => ? ∊ {0,0,3,6}
[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] => 1
[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] => 0
[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
[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
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [1,7,4,3,2,6,5] => ? ∊ {0,0,3,6}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [6,8,5,4,3,2,1,7] => ? ∊ {0,0,3,6}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => [8,7,6,5,4,3,1,9,2] => ? ∊ {0,7,7,8,8,11,16,17}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => [1,6,5,4,3,8,7,2] => ? ∊ {0,7,7,8,8,11,16,17}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [6,1,4,3,7,5,2] => ? ∊ {0,7,7,8,8,11,16,17}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [3,5,4,7,1,6,2] => ? ∊ {0,7,7,8,8,11,16,17}
[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
[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
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [5,7,1,3,2,6,4] => ? ∊ {0,7,7,8,8,11,16,17}
[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
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [2,7,4,3,6,1,5] => ? ∊ {0,7,7,8,8,11,16,17}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [1,8,5,4,3,2,7,6] => ? ∊ {0,7,7,8,8,11,16,17}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [7,9,6,5,4,3,2,1,8] => ? ∊ {0,7,7,8,8,11,16,17}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => [9,8,7,6,5,4,3,1,10,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,1,2,8] => [1,7,6,5,4,3,9,8,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,8,4,5,6,1,3,7] => [7,1,5,4,3,8,6,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => [3,6,5,4,8,1,7,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [6,5,1,3,7,4,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[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] => 0
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,5,7,3,1,6,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [6,5,4,2,7,1,3] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[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] => 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] => 0
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [5,7,4,1,2,6,3] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[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
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [1,7,2,3,6,5,4] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [6,8,1,4,3,2,7,5] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [6,4,7,3,2,1,5] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [3,7,4,6,2,1,5] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [2,8,5,4,3,7,1,6] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [1,9,6,5,4,3,2,8,7] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => [8,10,7,6,5,4,3,2,1,9] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
Description
The number of restricted non-inversions between exceedances where the rightmost exceedance is linked.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \wedge \sigma^{-1}(j) < j \}.$$
Matching statistic: St001715
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001715: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 56%●distinct values known / distinct values provided: 11%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001715: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 56%●distinct values known / distinct values provided: 11%
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
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [6,5,4,3,1,7,2] => ? ∊ {0,0}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,4,3,6,5,2] => 1
[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] => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [5,7,4,3,2,1,6] => ? ∊ {0,0}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [7,6,5,4,3,1,8,2] => ? ∊ {0,1,3,6}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => [1,5,4,3,7,6,2] => ? ∊ {0,1,3,6}
[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
[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] => 0
[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] => 0
[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] => 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [1,7,4,3,2,6,5] => ? ∊ {0,1,3,6}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [6,8,5,4,3,2,1,7] => ? ∊ {0,1,3,6}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => [8,7,6,5,4,3,1,9,2] => ? ∊ {1,7,7,8,8,11,16,17}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => [1,6,5,4,3,8,7,2] => ? ∊ {1,7,7,8,8,11,16,17}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [6,1,4,3,7,5,2] => ? ∊ {1,7,7,8,8,11,16,17}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [3,5,4,7,1,6,2] => ? ∊ {1,7,7,8,8,11,16,17}
[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
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [5,7,1,3,2,6,4] => ? ∊ {1,7,7,8,8,11,16,17}
[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
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [2,7,4,3,6,1,5] => ? ∊ {1,7,7,8,8,11,16,17}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [1,8,5,4,3,2,7,6] => ? ∊ {1,7,7,8,8,11,16,17}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [7,9,6,5,4,3,2,1,8] => ? ∊ {1,7,7,8,8,11,16,17}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => [9,8,7,6,5,4,3,1,10,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,1,2,8] => [1,7,6,5,4,3,9,8,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,8,4,5,6,1,3,7] => [7,1,5,4,3,8,6,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => [3,6,5,4,8,1,7,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [6,5,1,3,7,4,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[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] => 0
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,5,7,3,1,6,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [6,5,4,2,7,1,3] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[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] => 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [5,7,4,1,2,6,3] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[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
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [1,7,2,3,6,5,4] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [6,8,1,4,3,2,7,5] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [6,4,7,3,2,1,5] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [3,7,4,6,2,1,5] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [2,8,5,4,3,7,1,6] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [1,9,6,5,4,3,2,8,7] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => [8,10,7,6,5,4,3,2,1,9] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53}
Description
The number of non-records in a permutation.
A record in a permutation $\pi$ is a value $\pi(j)$ which is a left-to-right minimum, a left-to-right maximum, a right-to-left minimum, or a right-to-left maximum.
For example, in the permutation $\pi = [1, 4, 3, 2, 5]$, the values $1$ is a left-to-right minimum, $1, 4, 5$ are left-to-right maxima, $5, 2, 1$ are right-to-left minima and $5$ is a right-to-left maximum. Hence, $3$ is the unique non-record.
Permutations without non-records are called square [1].
Matching statistic: St001652
(load all 3 compositions to match this statistic)
(load all 3 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: 11% ●values known / values provided: 56%●distinct values known / distinct values provided: 11%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001652: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 56%●distinct values known / distinct values provided: 11%
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
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [6,5,4,3,1,7,2] => ? ∊ {1,1} + 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
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [5,7,4,3,2,1,6] => ? ∊ {1,1} + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [7,6,5,4,3,1,8,2] => ? ∊ {0,1,3,6} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => [1,5,4,3,7,6,2] => ? ∊ {0,1,3,6} + 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
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [1,7,4,3,2,6,5] => ? ∊ {0,1,3,6} + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [6,8,5,4,3,2,1,7] => ? ∊ {0,1,3,6} + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => [8,7,6,5,4,3,1,9,2] => ? ∊ {1,7,7,8,8,11,16,17} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => [1,6,5,4,3,8,7,2] => ? ∊ {1,7,7,8,8,11,16,17} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [6,1,4,3,7,5,2] => ? ∊ {1,7,7,8,8,11,16,17} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [3,5,4,7,1,6,2] => ? ∊ {1,7,7,8,8,11,16,17} + 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
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [5,7,1,3,2,6,4] => ? ∊ {1,7,7,8,8,11,16,17} + 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
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [2,7,4,3,6,1,5] => ? ∊ {1,7,7,8,8,11,16,17} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [1,8,5,4,3,2,7,6] => ? ∊ {1,7,7,8,8,11,16,17} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [7,9,6,5,4,3,2,1,8] => ? ∊ {1,7,7,8,8,11,16,17} + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => [9,8,7,6,5,4,3,1,10,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,1,2,8] => [1,7,6,5,4,3,9,8,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,8,4,5,6,1,3,7] => [7,1,5,4,3,8,6,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => [3,6,5,4,8,1,7,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [6,5,1,3,7,4,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 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
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,5,7,3,1,6,2] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [6,5,4,2,7,1,3] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 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
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [5,7,4,1,2,6,3] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 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
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [1,7,2,3,6,5,4] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [6,8,1,4,3,2,7,5] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [6,4,7,3,2,1,5] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [3,7,4,6,2,1,5] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [2,8,5,4,3,7,1,6] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [1,9,6,5,4,3,2,8,7] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => [8,10,7,6,5,4,3,2,1,9] => ? ∊ {0,0,8,8,11,14,15,24,28,30,31,31,39,43,53} + 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 $i\in\{k,\dots,\ell-1\}$.
The following 323 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St000406The number of occurrences of the pattern 3241 in 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. St000241The number of cyclical small excedances. St000488The number of cycles of a permutation of length at most 2. St000516The number of stretching pairs of a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. 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. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001520The number of strict 3-descents. St001549The number of restricted non-inversions between exceedances. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000570The Edelman-Greene number of a permutation. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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 $K[x]/(x^n)$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001890The maximum magnitude of the Möbius function of a poset. 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. St000031The number of cycles in the cycle decomposition of a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. 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. St000914The sum of the values of the Möbius function of 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. St000650The number of 3-rises of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001856The number of edges in the reduced word graph of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000260The radius of a connected graph. St000879The number of long braid edges in the graph of braid moves 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. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001330The hat guessing number of a graph. St000895The number of ones on the main diagonal of an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St000629The defect of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. St001490The number of connected components of a skew partition. 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. St001964The interval resolution global dimension of a poset. St000022The number of fixed points of a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000534The number of 2-rises of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000842The breadth of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000181The number of connected components of the Hasse diagram for the poset. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001846The number of elements which do not have a complement in the lattice. 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. St000058The order of a permutation. St000297The number of leading ones in a binary word. St000455The second largest eigenvalue of a graph if it is integral. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000768The number of peaks in an integer composition. St001435The number of missing boxes in the first row. St000326The position of the first one in a binary word after appending a 1 at the end. St000764The number of strong records in an integer composition. St001487The number of inner corners of a skew partition. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001438The number of missing boxes of a skew partition. 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. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000807The sum of the heights of the valleys of the associated bargraph. 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. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000096The number of spanning trees of a graph. 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. St000221The number of strong fixed points of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000407The number of occurrences of the pattern 2143 in a permutation. St000478Another weight of a partition according to Alladi. 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. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000761The number of ascents in an integer composition. St000787The number of flips required to make a perfect matching noncrossing. St000894The trace of an alternating sign matrix. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. 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. St000943The number of spots the most unlucky car had to go further in a parking function. 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. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. 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. 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. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001381The fertility of a permutation. St001429The number of negative entries in a signed permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001524The degree of symmetry of a binary word. St001552The number of inversions between excedances and fixed points of a permutation. St001577The minimal number of edges to add or remove to make a graph a cograph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001867The number of alignments of type EN of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000056The decomposition (or block) number of a permutation. St000286The number of connected components of the complement of a graph. St000295The length of the border of a binary word. St000657The smallest part of an integer composition. St000694The number of affine bounded permutations that project to a given permutation. St000763The sum of the positions of the strong records of an integer composition. St000788The number of nesting-similar perfect matchings of a perfect matching. St000805The number of peaks of the associated bargraph. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001162The minimum jump 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 $K[x]/(x^n)$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001260The permanent of an alternating sign matrix. St001267The length of the Lyndon factorization of the binary word. 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. St001413Half the length of the longest even length palindromic prefix of a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001481The minimal height of a peak of a Dyck path. St001518The number of graphs with the same ordinary spectrum as the given graph. St001590The crossing number of a perfect matching. St001621The number of atoms of a lattice. St001768The number of reduced words of a signed permutation. 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. St001884The number of borders of a binary word. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000876The number of factors in the Catalan decomposition of a binary word. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001734The lettericity of a graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000627The exponent of a binary word. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000153The number of adjacent cycles of a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001845The number of join irreducibles minus the rank of a lattice. St000296The length of the symmetric border of a binary word. St000317The cycle descent number of a permutation. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001394The genus of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001889The size of the connectivity set of a signed permutation. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St000905The number of different multiplicities of parts of an integer composition. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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 $eA$. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000767The number of runs in an integer composition. St001566The length of the longest arithmetic progression in a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000417The size of the automorphism group of the ordered tree. St001058The breadth of the ordered tree. St000068The number of minimal elements in a poset. St000731The number of double exceedences of a permutation. St001410The minimal entry of a semistandard tableau. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000679The pruning number of an ordered tree. 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. St000091The descent variation of a composition. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000355The number of occurrences of the pattern 21-3. 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. 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. St000732The number of double deficiencies of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 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. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001537The number of cyclic crossings of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. 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. St001781The interlacing number of a set partition. St001857The number of edges in the reduced word graph of a signed permutation. St001903The number of fixed points of a parking function. St000021The number of descents of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000654The first descent of a permutation. St000729The minimal arc length of a set partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001722The number of minimal chains with small intervals between a binary word and the top element. 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. St000084The number of subtrees. 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. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000823The number of unsplittable factors of the set partition. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 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. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000451The length of the longest pattern of the form k 1 2.
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