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Your data matches 112 different statistics following compositions of up to 3 maps.
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Matching statistic: St001553
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
St001553: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 0
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> 2
Description
The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.
The statistic returns zero in case that bimodule is the zero module.
Matching statistic: St000996
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0
[1,0,1,0]
=> [1,2] => 0
[1,1,0,0]
=> [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 2
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St000374
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,1,3,2] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [4,2,3,1] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,2,4,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,3,2,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,5,3,4,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,1,2,4,3] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,1,3,2,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,1,3,2,4] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [5,1,3,4,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,1,4,3,2] => 2
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not 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 do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000007
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1 = 0 + 1
[1,0,1,0]
=> [2,1] => [1,2] => 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [2,1] => 2 = 1 + 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,1,3] => [3,2,1] => 3 = 2 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,2,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,4,2,3] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,3,1,2] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,4,1,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,4,3,2] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [4,3,2,1] => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,4,2,1] => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,3,4,2] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4,2,3,1] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,2,4,1] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,1,2,3,4] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,5,2,3,4] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,1,2,3] => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,5,1,2,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,5,3,4] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,2,3] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,5,4,2,3] => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,4,3,1,2] => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [4,5,3,1,2] => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,5,2,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,4,1,2] => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [4,3,5,1,2] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [3,4,5,1,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,5,4] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,1,2,4,3] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,4,3] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [5,4,1,3,2] => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,3,2] => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,5,4,3] => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,1,4,3,2] => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,5,4,3,2] => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [5,4,3,2,1] => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,5,3,2,1] => 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,5,3,2] => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [5,3,4,2,1] => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,3,5,2,1] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [3,4,5,2,1] => 3 = 2 + 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000053
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
Description
The number of valleys of the Dyck path.
Matching statistic: St000306
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
Description
The bounce count of a Dyck path.
For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Matching statistic: St000710
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000710: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000710: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [2,1] => 0
[1,0,1,0]
=> [1,2] => [1,0,1,0]
=> [3,1,2] => 0
[1,1,0,0]
=> [2,1] => [1,1,0,0]
=> [2,3,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [4,3,1,2] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [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]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 2
Description
The number of big deficiencies of a permutation.
A big deficiency of a permutation $\pi$ is an index $i$ such that $i - \pi(i) > 1$.
This statistic is equidistributed with any of the numbers of big exceedences, big descents and big ascents.
Matching statistic: St001142
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001142: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001142: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
Description
The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001169
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
Description
Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001197
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001197: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001197: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
Description
The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
The following 102 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000069The number of maximal elements of a poset. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001720The minimal length of a chain of small intervals in a lattice. St000080The rank of the poset. St000528The height of a poset. St001330The hat guessing number of a graph. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001782The order of rowmotion on the set of order ideals of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001712The number of natural descents of a standard Young tableau. St001905The number of preferred parking spots in a parking function less than the index of the car. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001877Number of indecomposable injective modules with projective dimension 2. 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 $eA$. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001435The number of missing boxes in the first row. St000665The number of rafts of a permutation. St001115The number of even descents of a permutation. St001394The genus of a permutation. St000871The number of very big ascents of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000022The number of fixed points of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000662The staircase size of the code of a permutation. St000451The length of the longest pattern of the form k 1 2. St000317The cycle descent number of a permutation. St000023The number of inner peaks of a permutation. St000091The descent variation of a composition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000353The number of inner valleys of a permutation. St000360The number of occurrences of the pattern 32-1. St000486The number of cycles of length at least 3 of a permutation. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000779The tier of a permutation. St000836The number of descents of distance 2 of a permutation. St000872The number of very big descents of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001114The number of odd descents of a permutation. St001153The number of blocks with even minimum in a set partition. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001665The number of pure excedances of a permutation. St001728The number of invisible descents of a permutation. St001729The number of visible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001871The number of triconnected components of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000062The length of the longest increasing subsequence of the permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000239The number of small weak excedances. St000307The number of rowmotion orbits of a poset. St000308The height of the tree associated to a permutation. St000354The number of recoils of a permutation. St000502The number of successions of a set partitions. St000864The number of circled entries of the shifted recording tableau of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001151The number of blocks with odd minimum. St001152The number of pairs with even minimum in a perfect matching. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001489The maximum of the number of descents and the number of inverse descents. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001737The number of descents of type 2 in a permutation. St000485The length of the longest cycle of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000735The last entry on the main diagonal of a standard tableau. St001644The dimension of a graph. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation.
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