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Your data matches 66 different statistics following compositions of up to 3 maps.
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Matching statistic: St000371
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => [1] => 0
[1,0,1,0]
=> [3,1,2] => [2,3,1] => [2,1] => 0
[1,1,0,0]
=> [2,3,1] => [3,1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [2,3,1] => 0
[1,0,1,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => [2,1,3] => 0
[1,1,0,0,1,0]
=> [2,4,1,3] => [3,1,4,2] => [3,1,2] => 0
[1,1,0,1,0,0]
=> [4,3,1,2] => [3,4,2,1] => [3,2,1] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,1] => 0
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,3,5,1,4] => [2,3,1,4] => 0
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [2,4,1,5,3] => [2,4,1,3] => 0
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [2,4,5,3,1] => [2,4,3,1] => 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [2,5,1,3,4] => [2,1,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => [3,1,4,2] => 0
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => [3,1,2,4] => 0
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,4,2,5,1] => [3,4,2,1] => 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,4,5,2,1] => [3,4,2,1] => 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [3,5,2,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => [4,1,2,3] => 0
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,1,5,3,2] => [4,1,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [4,5,2,3,1] => [4,2,3,1] => 2
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [2,3,4,6,1,5] => [2,3,4,1,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [2,3,5,1,6,4] => [2,3,5,1,4] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [2,3,5,6,4,1] => [2,3,5,4,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [2,3,6,1,4,5] => [2,3,1,4,5] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [2,4,1,5,6,3] => [2,4,1,5,3] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [2,4,1,6,3,5] => [2,4,1,3,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [2,4,5,3,6,1] => [2,4,5,3,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [2,4,5,6,3,1] => [2,4,5,3,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [2,4,6,3,1,5] => [2,4,3,1,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [2,5,1,3,6,4] => [2,5,1,3,4] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [2,5,1,6,4,3] => [2,5,1,4,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [2,5,6,3,4,1] => [2,5,3,4,1] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [2,6,1,3,4,5] => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [3,1,4,5,6,2] => [3,1,4,5,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [3,1,4,6,2,5] => [3,1,4,2,5] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [3,1,5,2,6,4] => [3,1,5,2,4] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [3,1,5,6,4,2] => [3,1,5,4,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [3,1,6,2,4,5] => [3,1,2,4,5] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [3,4,2,5,6,1] => [3,4,2,5,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [3,4,2,6,1,5] => [3,4,2,1,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [3,4,5,2,6,1] => [3,4,5,2,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,4,5,6,1,2] => [3,4,5,1,2] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [3,4,6,2,1,5] => [3,4,2,1,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [3,5,2,1,6,4] => [3,5,2,1,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [3,5,2,6,4,1] => [3,5,2,4,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,5,6,2,4,1] => [3,5,2,4,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [3,6,2,1,4,5] => [3,2,1,4,5] => 1
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
Matching statistic: St000375
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => [2,1] => 0
[1,0,1,0]
=> [3,1,2] => [2,3,1] => [3,2,1] => 0
[1,1,0,0]
=> [2,3,1] => [3,1,2] => [1,3,2] => 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [3,2,4,1] => 0
[1,0,1,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 0
[1,1,0,0,1,0]
=> [2,4,1,3] => [3,1,4,2] => [1,3,4,2] => 0
[1,1,0,1,0,0]
=> [4,3,1,2] => [3,4,2,1] => [4,3,2,1] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,4,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => [3,2,4,5,1] => 0
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,3,5,1,4] => [3,2,5,1,4] => 0
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [2,4,1,5,3] => [4,2,1,5,3] => 0
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [2,4,5,3,1] => [4,2,5,3,1] => 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [2,5,1,3,4] => [5,2,1,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => [1,3,4,5,2] => 0
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => [1,3,5,2,4] => 0
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,4,2,5,1] => [4,3,2,5,1] => 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,4,5,2,1] => [4,3,5,2,1] => 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [3,5,2,1,4] => [5,3,2,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => [1,4,2,5,3] => 0
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,1,5,3,2] => [1,4,5,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [4,5,2,3,1] => [5,4,2,3,1] => 2
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,5,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => [3,2,4,5,6,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [2,3,4,6,1,5] => [3,2,4,6,1,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [2,3,5,1,6,4] => [3,2,5,1,6,4] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [2,3,5,6,4,1] => [3,2,5,6,4,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [2,3,6,1,4,5] => [3,2,6,1,4,5] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [2,4,1,5,6,3] => [4,2,1,5,6,3] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [2,4,1,6,3,5] => [4,2,1,6,3,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [2,4,5,3,6,1] => [4,2,5,3,6,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [2,4,5,6,3,1] => [4,2,5,6,3,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [2,4,6,3,1,5] => [4,2,6,3,1,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [2,5,1,3,6,4] => [5,2,1,3,6,4] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [2,5,1,6,4,3] => [5,2,1,6,4,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [2,5,6,3,4,1] => [5,2,6,3,4,1] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [2,6,1,3,4,5] => [6,2,1,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [3,1,4,5,6,2] => [1,3,4,5,6,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [3,1,4,6,2,5] => [1,3,4,6,2,5] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [3,1,5,2,6,4] => [1,3,5,2,6,4] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [3,1,5,6,4,2] => [1,3,5,6,4,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [3,1,6,2,4,5] => [1,3,6,2,4,5] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [3,4,2,5,6,1] => [4,3,2,5,6,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [3,4,2,6,1,5] => [4,3,2,6,1,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [3,4,5,2,6,1] => [4,3,5,2,6,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,4,5,6,1,2] => [4,3,5,6,1,2] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [3,4,6,2,1,5] => [4,3,6,2,1,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [3,5,2,1,6,4] => [5,3,2,1,6,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [3,5,2,6,4,1] => [5,3,2,6,4,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,5,6,2,4,1] => [5,3,6,2,4,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [3,6,2,1,4,5] => [6,3,2,1,4,5] => 1
Description
The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St001089
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
St001089: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 0
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> 1
[]
=> ? = 0
Description
Number 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.
Matching statistic: St000372
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000372: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000372: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,2] => [1] => 0
[1,0,1,0]
=> [3,1,2] => [1,3,2] => [1,2] => 0
[1,1,0,0]
=> [2,3,1] => [2,1,3] => [2,1] => 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [1,3,2] => 0
[1,0,1,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => [2,1,3] => 0
[1,1,0,0,1,0]
=> [2,4,1,3] => [3,1,4,2] => [3,1,2] => 0
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,2,4,3] => [1,2,3] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,4,3,2] => 0
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,5,4,1,3] => [2,4,1,3] => 0
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,5,1,4,2] => [3,1,4,2] => 0
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,5,2,1,4] => [3,2,1,4] => 0
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [4,1,5,3,2] => [4,1,3,2] => 0
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [4,2,1,3] => 0
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,4,2] => [1,3,4,2] => 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => [1,2,4,3] => 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [2,3,5,1,4] => [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,3,1,5,2] => [4,3,1,2] => 0
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,1,2,5,3] => [4,1,2,3] => 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,3,2,5,4] => [1,3,2,4] => 2
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,5,4,3,2] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [2,6,5,4,1,3] => [2,5,4,1,3] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [3,6,5,1,4,2] => [3,5,1,4,2] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,5,2,4,3] => [1,5,2,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [3,6,5,2,1,4] => [3,5,2,1,4] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,6,1,5,3,2] => [4,1,5,3,2] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,6,2,5,1,3] => [4,2,5,1,3] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,3,5,4,2] => [1,3,5,4,2] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,2,5,4,3] => [1,2,5,4,3] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [2,6,3,5,1,4] => [2,3,5,1,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [4,6,3,1,5,2] => [4,3,1,5,2] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [4,6,1,2,5,3] => [4,1,2,5,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [4,6,3,2,1,5] => [4,3,2,1,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [5,1,6,4,3,2] => [5,1,4,3,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [5,2,6,4,1,3] => [5,2,4,1,3] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [5,3,6,1,4,2] => [5,3,1,4,2] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [5,1,6,2,4,3] => [5,1,2,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [5,3,6,2,1,4] => [5,3,2,1,4] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,4,6,5,3,2] => [1,4,5,3,2] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [2,4,6,5,1,3] => [2,4,5,1,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,3,6,5,4,2] => [1,3,5,4,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => [2,1,5,4,3] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [2,3,6,5,1,4] => [2,3,5,1,4] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [3,4,6,1,5,2] => [3,4,1,5,2] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,4,6,2,5,3] => [1,4,2,5,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,3,6,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [3,4,6,2,1,5] => [3,4,2,1,5] => 1
[]
=> [1] => [1] => [] => ? = 0
Description
The number of mid points of increasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) < \pi(j) < \pi(k)$.
The generating function is given by [1].
Matching statistic: St000562
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000562: Set partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000562: Set partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => {{1,2}}
=> 0
[1,0,1,0]
=> [3,1,2] => [3,1,2] => {{1,3},{2}}
=> 0
[1,1,0,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => {{1,4},{2},{3}}
=> 0
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,1,3,2] => {{1,4},{2},{3}}
=> 0
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
[1,1,0,1,0,0]
=> [4,3,1,2] => [3,4,2,1] => {{1,3},{2,4}}
=> 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 0
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 0
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [4,1,5,3,2] => {{1,4},{2},{3,5}}
=> 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 0
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 0
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 0
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,5,2,1,4] => {{1,3},{2,5},{4}}
=> 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,5,2,3,1] => {{1,4},{2,5},{3}}
=> 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [3,5,2,4,1] => {{1,3},{2,5},{4}}
=> 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => {{1,5},{2},{3},{4}}
=> 0
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,2,5,3,1] => {{1,4},{2},{3,5}}
=> 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [3,5,4,2,1] => {{1,3,4},{2,5}}
=> 2
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => {{1,6},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,1,2,3,5,4] => {{1,6},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6,1,2,4,3,5] => {{1,6},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [5,1,2,6,4,3] => {{1,5},{2},{3},{4,6}}
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6,1,2,4,5,3] => {{1,6},{2},{3},{4},{5}}
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6,1,3,2,4,5] => {{1,6},{2},{3},{4},{5}}
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6,1,3,2,5,4] => {{1,6},{2},{3},{4},{5}}
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [4,1,6,3,2,5] => {{1,4},{2},{3,6},{5}}
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [5,1,6,3,4,2] => {{1,5},{2},{3,6},{4}}
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [4,1,6,3,5,2] => {{1,4},{2},{3,6},{5}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6,1,3,4,2,5] => {{1,6},{2},{3},{4},{5}}
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [5,1,3,6,4,2] => {{1,5},{2},{3},{4,6}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [4,1,6,5,3,2] => {{1,4,5},{2},{3,6}}
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,1,3,4,5,2] => {{1,6},{2},{3},{4},{5}}
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6,2,1,3,4,5] => {{1,6},{2},{3},{4},{5}}
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,2,1,3,5,4] => {{1,6},{2},{3},{4},{5}}
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6,2,1,4,3,5] => {{1,6},{2},{3},{4},{5}}
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [5,2,1,6,4,3] => {{1,5},{2},{3},{4,6}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6,2,1,4,5,3] => {{1,6},{2},{3},{4},{5}}
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [3,6,2,1,4,5] => {{1,3},{2,6},{4},{5}}
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [3,6,2,1,5,4] => {{1,3},{2,6},{4},{5}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,6,2,3,1,5] => {{1,4},{2,6},{3},{5}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [6,5,2,3,4,1] => {{1,6},{2,5},{3},{4}}
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,6,2,3,5,1] => {{1,4},{2,6},{3},{5}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [3,6,2,4,1,5] => {{1,3},{2,6},{4},{5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [3,5,2,6,4,1] => {{1,3},{2,5},{4,6}}
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [4,6,2,5,3,1] => {{1,4,5},{2,6},{3}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [3,6,2,4,5,1] => {{1,3},{2,6},{4},{5}}
=> 1
[]
=> [1] => [1] => {{1}}
=> ? = 0
Description
The number of internal points of a set partition.
An element $e$ is internal, if there are $f < e < g$ such that the blocks of $f$ and $g$ have larger minimal element than the block of $e$. See Section 5.5 of [1]
Matching statistic: St001687
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,0,1,0]
=> [3,1,2] => [3,1,2] => [3,1,2] => 0
[1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [3,4,1,2] => [4,1,3,2] => 0
[1,0,1,1,0,0]
=> [3,1,4,2] => [4,1,3,2] => [3,4,1,2] => 0
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,4,2,3] => [1,4,2,3] => 0
[1,1,0,1,0,0]
=> [4,3,1,2] => [4,2,1,3] => [2,4,1,3] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [3,4,5,1,2] => [5,1,4,3,2] => 0
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [3,5,1,4,2] => [4,5,1,3,2] => 0
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,1,5,3,2] => [5,3,4,1,2] => 0
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [4,5,3,1,2] => [3,5,1,4,2] => 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => [3,4,5,1,2] => 0
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,4,5,2,3] => [1,5,2,4,3] => 0
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,5,2,4,3] => [1,4,5,2,3] => 0
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [4,2,5,1,3] => [2,5,1,4,3] => 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,5,2,1,3] => [5,2,4,1,3] => 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,2,1,4,3] => [2,4,5,1,3] => 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,5,3,2,4] => [1,3,5,2,4] => 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5,2,3,1,4] => [2,3,5,1,4] => 2
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [3,4,5,6,1,2] => [6,1,5,4,3,2] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [3,4,6,1,5,2] => [5,6,1,4,3,2] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [3,5,1,6,4,2] => [6,4,5,1,3,2] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [3,5,6,4,1,2] => [4,6,1,5,3,2] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [3,6,1,4,5,2] => [4,5,6,1,3,2] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,1,5,6,3,2] => [6,3,5,4,1,2] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,1,6,3,5,2] => [5,6,3,4,1,2] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [4,5,3,6,1,2] => [3,6,1,5,4,2] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,5,6,3,1,2] => [6,3,5,1,4,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [4,6,3,1,5,2] => [3,5,6,1,4,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [5,1,3,6,4,2] => [3,6,4,5,1,2] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [5,1,6,4,3,2] => [4,6,3,5,1,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [5,6,3,4,1,2] => [3,4,6,1,5,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,1,3,4,5,2] => [3,4,5,6,1,2] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,4,5,6,2,3] => [1,6,2,5,4,3] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,4,6,2,5,3] => [1,5,6,2,4,3] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,5,2,6,4,3] => [1,6,4,5,2,3] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,5,6,4,2,3] => [1,4,6,2,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,6,2,4,5,3] => [1,4,5,6,2,3] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [4,2,5,6,1,3] => [2,6,1,5,4,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [4,2,6,1,5,3] => [2,5,6,1,4,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,5,2,6,1,3] => [6,2,5,4,1,3] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [4,5,6,1,2,3] => [6,1,5,2,4,3] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,6,2,1,5,3] => [5,6,2,4,1,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [5,2,1,6,4,3] => [2,6,4,5,1,3] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [5,2,6,4,1,3] => [2,4,6,1,5,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [5,6,2,4,1,3] => [4,6,2,5,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6,2,1,4,5,3] => [2,4,5,6,1,3] => 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [1,4,5,6,7,2,3] => [1,7,2,6,5,4,3] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [1,4,5,7,2,6,3] => [1,6,7,2,5,4,3] => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [1,4,6,2,7,5,3] => [1,7,5,6,2,4,3] => ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [1,4,6,7,5,2,3] => [1,5,7,2,6,4,3] => ? = 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [1,4,7,2,5,6,3] => [1,5,6,7,2,4,3] => ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [1,5,2,6,7,4,3] => [1,7,4,6,5,2,3] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [1,5,2,7,4,6,3] => [1,6,7,4,5,2,3] => ? = 0
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [1,5,6,4,7,2,3] => [1,4,7,2,6,5,3] => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [1,5,6,7,4,2,3] => [1,7,4,6,2,5,3] => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [1,5,7,4,2,6,3] => [1,4,6,7,2,5,3] => ? = 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [1,6,2,4,7,5,3] => [1,4,7,5,6,2,3] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => [1,6,2,7,5,4,3] => [1,5,7,4,6,2,3] => ? = 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [1,5,6,3,7,2,4] => [1,7,3,6,5,2,4] => ? = 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [1,5,6,7,2,3,4] => [1,7,2,6,3,5,4] => ? = 0
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [1,5,7,3,2,6,4] => [1,6,7,3,5,2,4] => ? = 1
Description
The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.
Matching statistic: St000373
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [2,1] => 0
[1,0,1,0]
=> [1,0,1,0]
=> [3,1,2] => 0
[1,1,0,0]
=> [1,1,0,0]
=> [2,3,1] => 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 0
[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] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => ? = 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 0
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => ? = 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ? = 2
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? = 0
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 0
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => ? = 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 0
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => ? = 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => ? = 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => ? = 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => ? = 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => ? = 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => ? = 2
Description
The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j \geq j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St001876
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 74%●distinct values known / distinct values provided: 50%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 74%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1,0]
=> ([],1)
=> ([(0,1)],2)
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? = 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
=> ? = 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> ? = 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> ? = 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ? = 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? = 3
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
=> ? = 0
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ([(0,6),(1,10),(2,10),(4,9),(5,9),(6,7),(7,4),(7,5),(8,1),(8,2),(9,8),(10,3)],11)
=> ? = 2
[]
=> []
=> ?
=> ?
=> ? = 0
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St000454
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 75%
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0
[1,0,1,0]
=> [2,1] => [1,2] => ([],2)
=> 0
[1,1,0,0]
=> [1,2] => [1,2] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => ([],3)
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => ([],3)
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => ([],3)
=> 0
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => ([],4)
=> 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => ([],4)
=> 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => ([],4)
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => ([],4)
=> 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,2,4] => ([(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => ([],4)
=> 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => ([],5)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => ([],5)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => ([],5)
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => ([],5)
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,3,4,5] => ([],5)
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,3,4,5] => ([],5)
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,4,5] => ([],5)
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 2
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [1,2,3,4,5,6] => ([],6)
=> 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [1,2,3,4,5,6] => ([],6)
=> 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => [1,2,3,4,5,6] => ([],6)
=> 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,1,2] => [1,2,3,4,6,5] => ([(4,5)],6)
=> 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,1,2] => [1,2,3,4,5,6] => ([],6)
=> 0
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,1,2] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,1,2] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,1,2] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? = 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1,2,5] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,2,6] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ? = 0
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,5,3,1,2,6] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? = 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,4,1,2,6] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2
[]
=> [] => ? => ?
=> ? = 0
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001624
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 53%●distinct values known / distinct values provided: 50%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 53%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1,0]
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
=> ? = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> ? = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ? = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? = 3 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(2,8),(3,8),(4,7),(5,1),(6,4),(7,2),(7,3),(8,5)],9)
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(2,8),(3,8),(4,7),(5,1),(6,4),(7,2),(7,3),(8,5)],9)
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,7),(1,9),(2,10),(4,11),(5,8),(6,1),(6,10),(7,5),(8,2),(8,6),(9,11),(10,4),(10,9),(11,3)],12)
=> ? = 2 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9)
=> ? = 1 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
=> ? = 1 + 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
=> ? = 1 + 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
=> ? = 0 + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
=> ? = 1 + 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ([(0,6),(1,10),(2,10),(4,9),(5,9),(6,7),(7,4),(7,5),(8,1),(8,2),(9,8),(10,3)],11)
=> ? = 2 + 1
[]
=> []
=> ?
=> ?
=> ? = 0 + 1
Description
The breadth of a lattice.
The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
The following 56 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001877Number of indecomposable injective modules with projective dimension 2. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000731The number of double exceedences of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000451The length of the longest pattern of the form k 1 2. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000223The number of nestings in the permutation. St000877The depth of the binary word interpreted as a path. St001513The number of nested exceedences of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000058The order of a permutation. St001394The genus of a permutation. St000352The Elizalde-Pak rank of a permutation. St000534The number of 2-rises of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. 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. St001115The number of even descents of a permutation. St000669The number of permutations obtained by switching ascents or descents of size 2. St000359The number of occurrences of the pattern 23-1. St000366The number of double descents of a permutation. St001866The nesting alignments of a signed permutation. St000007The number of saliances of the permutation. St000068The number of minimal elements in a poset. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000842The breadth of a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St001868The number of alignments of type NE of a signed permutation. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000648The number of 2-excedences of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001867The number of alignments of type EN of a signed permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000805The number of peaks of the associated bargraph. St000942The number of critical left to right maxima of the parking functions. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001904The length of the initial strictly increasing segment of a parking function.
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