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Your data matches 19 different statistics following compositions of up to 3 maps.
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Matching statistic: St000052
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000711
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 0
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 0
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,6,4] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,6,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,6,2,4,5] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,2,4,6,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,2,6,5] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,4,6,2,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,2,4,5,6] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,4,2,5,6] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,4,5,2,6] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,3,6] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,5,6,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,5,2,6,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,5,6,2,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,4,6,2,3,5] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,4,2,3,6,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,4,6,3,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,2,3,5,6] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,2,4,3,5,6] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,2,4,5,3,6] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => 0
Description
The number of big exceedences of a permutation.
A big exceedence of a permutation $\pi$ is an index $i$ such that $\pi(i) - i > 1$.
This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.
Matching statistic: St001685
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St001685: 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]
=> [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [2,1] => 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,3,1] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [1,3,2] => [3,1,2] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [3,4,1,2] => 0
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [3,4,2,1] => 0
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,4,1] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,4,3,1] => 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => 0
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [4,2,1,3] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [4,1,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [4,3,1,2] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [4,5,2,3,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,5,3,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [4,5,2,1,3] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,2,1,5,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [4,5,1,3,2] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [4,1,5,3,2] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [4,3,5,1,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,3,1,5,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [4,5,3,2,1] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [4,3,5,2,1] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [3,5,2,4,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,2,5,4,1] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [3,5,2,1,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,2,5,1,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2,1,5,4] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,5,1,4,2] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,1,5,4,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,5,4,1,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [5,3,4,1,2] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [5,3,1,4,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,5,4,2,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [5,3,4,2,1] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [5,3,2,4,1] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [5,3,2,1,4] => 0
Description
The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.
Matching statistic: St000065
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000065: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000065: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [[1]]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0
Description
The number of entries equal to -1 in an alternating sign matrix.
The number of nonzero entries, [[St000890]] is twice this number plus the dimension of the matrix.
Matching statistic: St000358
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [.,.]
=> [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> [1,2] => 0
[1,1,0,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [2,1,3] => 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> [1,2,3] => 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> [3,1,2] => 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [2,3,1] => 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 0
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0
Description
The number of occurrences of the pattern 31-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.
Matching statistic: St000371
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => 0
[1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => 0
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 0
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,3,1,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,5,4,2,3] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,5,2,4,3] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,2,4,1,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 0
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: St000373
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => 0
[1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => 0
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 0
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,5,2,4,3] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 0
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: St000647
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => [3,2,1] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => [3,1,2] => 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => [4,3,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [3,1,4,2,5] => [4,2,1,3,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => [5,4,2,1,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,1,5,2,3] => [4,2,1,5,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => [2,1,5,3,4] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => [5,2,1,3,4] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,2,1,5,4] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => [5,3,2,1,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,3,2,1,5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,4,3,2,1] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [2,4,1,3,5] => [4,3,1,2,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,4,1,2,5] => [3,1,4,2,5] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,5,1,3,4] => [5,4,3,1,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,5,1,2,4] => [3,1,5,4,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,2,3] => [5,3,1,4,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [2,4,1,5,3] => [5,3,1,2,4] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,4,1,5,2] => [3,1,5,2,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => [3,1,2,5,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,2,5,3] => [1,5,3,2,4] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => [3,1,2,4,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,4,3,2,5] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,5,4,3,2] => 0
Description
The number of big descents of a permutation.
For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i)-\pi(i+1) > 1$.
The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation [[Mp00090]], see [Theorem 2.5, 1].
For the number of small descents, see [[St000214]].
Matching statistic: St001167
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001683
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => [2,1] => 0
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => [2,3,1] => 0
[1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => [3,1,2] => 0
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,4,3,1] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,2,4,1] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [3,4,2,1] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [4,1,3,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,4,3,2] => 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [3,4,1,2] => 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => 0
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [4,2,1,3] => 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,2,1,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [3,5,2,4,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [3,2,5,4,1] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [5,3,2,4,1] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [4,3,2,5,1] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [4,3,5,2,1] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => 0
Description
The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000710The number of big deficiencies of a permutation. St000703The number of deficiencies of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St001960The number of descents of a permutation minus one if its first entry is not one. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module.
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