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Your data matches 130 different statistics following compositions of up to 3 maps.
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Matching statistic: St001012
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
St001012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 3 = 0 + 3
[1,1,0,0]
=> 3 = 0 + 3
[1,0,1,0,1,0]
=> 3 = 0 + 3
[1,0,1,1,0,0]
=> 4 = 1 + 3
[1,1,0,0,1,0]
=> 4 = 1 + 3
[1,1,0,1,0,0]
=> 4 = 1 + 3
[1,1,1,0,0,0]
=> 4 = 1 + 3
[1,0,1,0,1,0,1,0]
=> 3 = 0 + 3
[1,0,1,0,1,1,0,0]
=> 4 = 1 + 3
[1,0,1,1,0,0,1,0]
=> 5 = 2 + 3
[1,0,1,1,0,1,0,0]
=> 4 = 1 + 3
[1,0,1,1,1,0,0,0]
=> 5 = 2 + 3
[1,1,0,0,1,0,1,0]
=> 4 = 1 + 3
[1,1,0,0,1,1,0,0]
=> 5 = 2 + 3
[1,1,0,1,0,0,1,0]
=> 4 = 1 + 3
[1,1,0,1,0,1,0,0]
=> 4 = 1 + 3
[1,1,0,1,1,0,0,0]
=> 5 = 2 + 3
[1,1,1,0,0,0,1,0]
=> 5 = 2 + 3
[1,1,1,0,0,1,0,0]
=> 5 = 2 + 3
[1,1,1,0,1,0,0,0]
=> 5 = 2 + 3
[1,1,1,1,0,0,0,0]
=> 5 = 2 + 3
[1,0,1,0,1,0,1,0,1,0]
=> 3 = 0 + 3
[1,0,1,0,1,0,1,1,0,0]
=> 4 = 1 + 3
[1,0,1,0,1,1,0,0,1,0]
=> 5 = 2 + 3
[1,0,1,0,1,1,0,1,0,0]
=> 4 = 1 + 3
[1,0,1,0,1,1,1,0,0,0]
=> 5 = 2 + 3
[1,0,1,1,0,0,1,0,1,0]
=> 5 = 2 + 3
[1,0,1,1,0,0,1,1,0,0]
=> 6 = 3 + 3
[1,0,1,1,0,1,0,0,1,0]
=> 4 = 1 + 3
[1,0,1,1,0,1,0,1,0,0]
=> 4 = 1 + 3
[1,0,1,1,0,1,1,0,0,0]
=> 5 = 2 + 3
[1,0,1,1,1,0,0,0,1,0]
=> 6 = 3 + 3
[1,0,1,1,1,0,0,1,0,0]
=> 6 = 3 + 3
[1,0,1,1,1,0,1,0,0,0]
=> 5 = 2 + 3
[1,0,1,1,1,1,0,0,0,0]
=> 6 = 3 + 3
[1,1,0,0,1,0,1,0,1,0]
=> 4 = 1 + 3
[1,1,0,0,1,0,1,1,0,0]
=> 5 = 2 + 3
[1,1,0,0,1,1,0,0,1,0]
=> 6 = 3 + 3
[1,1,0,0,1,1,0,1,0,0]
=> 5 = 2 + 3
[1,1,0,0,1,1,1,0,0,0]
=> 6 = 3 + 3
[1,1,0,1,0,0,1,0,1,0]
=> 4 = 1 + 3
[1,1,0,1,0,0,1,1,0,0]
=> 5 = 2 + 3
[1,1,0,1,0,1,0,0,1,0]
=> 4 = 1 + 3
[1,1,0,1,0,1,0,1,0,0]
=> 4 = 1 + 3
[1,1,0,1,0,1,1,0,0,0]
=> 5 = 2 + 3
[1,1,0,1,1,0,0,0,1,0]
=> 6 = 3 + 3
[1,1,0,1,1,0,0,1,0,0]
=> 5 = 2 + 3
[1,1,0,1,1,0,1,0,0,0]
=> 5 = 2 + 3
[1,1,0,1,1,1,0,0,0,0]
=> 6 = 3 + 3
[1,1,1,0,0,0,1,0,1,0]
=> 5 = 2 + 3
Description
Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001004
(load all 46 compositions to match this statistic)
(load all 46 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => 2 = 0 + 2
[1,1,0,0]
=> [2,1] => 2 = 0 + 2
[1,0,1,0,1,0]
=> [1,2,3] => 3 = 1 + 2
[1,0,1,1,0,0]
=> [1,3,2] => 3 = 1 + 2
[1,1,0,0,1,0]
=> [2,1,3] => 3 = 1 + 2
[1,1,0,1,0,0]
=> [2,3,1] => 3 = 1 + 2
[1,1,1,0,0,0]
=> [3,2,1] => 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4 = 2 + 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 4 = 2 + 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 4 = 2 + 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4 = 2 + 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3 = 1 + 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 4 = 2 + 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4 = 2 + 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4 = 2 + 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4 = 2 + 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3 = 1 + 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3 = 1 + 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3 = 1 + 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 3 = 1 + 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 5 = 3 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 5 = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 5 = 3 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 5 = 3 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 4 = 2 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 5 = 3 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 5 = 3 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 5 = 3 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 5 = 3 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 4 = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 4 = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 4 = 2 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 5 = 3 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 5 = 3 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 5 = 3 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 5 = 3 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 4 = 2 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 5 = 3 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 5 = 3 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 5 = 3 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5 = 3 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 4 = 2 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 4 = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 4 = 2 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 3 = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 4 = 2 + 2
Description
The number of indices that are either left-to-right maxima or right-to-left minima.
The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].
Matching statistic: St000053
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4 = 2 + 2
Description
The number of valleys of the Dyck path.
Matching statistic: St000203
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000203: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000203: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> 2 = 0 + 2
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [3,2,1] => [[[.,.],.],.]
=> 3 = 1 + 2
[1,0,1,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> 3 = 1 + 2
[1,1,0,0,1,0]
=> [3,1,2] => [[.,[.,.]],.]
=> 2 = 0 + 2
[1,1,0,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> 3 = 1 + 2
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> 3 = 1 + 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> 4 = 2 + 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[[.,.],.],[.,.]]
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 4 = 2 + 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 3 = 1 + 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> 3 = 1 + 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> 4 = 2 + 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 4 = 2 + 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> 2 = 0 + 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> 3 = 1 + 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 4 = 2 + 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 4 = 2 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 5 = 3 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> 5 = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> 4 = 2 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> 5 = 3 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> 5 = 3 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> 4 = 2 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> 4 = 2 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> 5 = 3 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> 5 = 3 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> 3 = 1 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
=> 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> 5 = 3 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> 5 = 3 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> 4 = 2 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> 4 = 2 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> 4 = 2 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [[[.,.],[.,.]],[.,.]]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> 5 = 3 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
=> 5 = 3 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> 3 = 1 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
=> 5 = 3 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> 5 = 3 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> 3 = 1 + 2
Description
The number of external nodes of a binary tree.
That is, the number of nodes that can be reached from the root by only left steps or only right steps, plus $1$ for the root node itself. A counting formula for the number of external node in all binary trees of size $n$ can be found in [1].
Matching statistic: St001318
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001318: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001318: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => ([],2)
=> 2 = 0 + 2
[1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 3 = 1 + 2
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 3 = 1 + 2
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> 3 = 1 + 2
[1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 4 = 2 + 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 4 = 2 + 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 4 = 2 + 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 4 = 2 + 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> 5 = 3 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 5 = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 5 = 3 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 5 = 3 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 5 = 3 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 5 = 3 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 5 = 3 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 5 = 3 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 5 = 3 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 5 = 3 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 5 = 3 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 5 = 3 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 5 = 3 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 5 = 3 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 5 = 3 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 3 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
Description
The number of vertices of the largest induced subforest with the same number of connected components of a graph.
Matching statistic: St001499
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4 = 2 + 2
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.
We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St000676
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 0 + 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 0 + 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 1 + 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 1 + 3
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 1 + 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 1 + 3
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 0 + 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 1 + 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 1 + 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 1 + 3
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 1 + 3
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 0 + 3
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 1 + 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 5 = 2 + 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 5 = 2 + 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 5 = 2 + 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 5 = 2 + 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 4 = 1 + 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 5 = 2 + 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 5 = 2 + 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 5 = 2 + 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 5 = 2 + 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 5 = 2 + 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 4 = 1 + 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 5 = 2 + 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 5 = 2 + 3
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St001068
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 0 + 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 0 + 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 1 + 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 1 + 3
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 1 + 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 1 + 3
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 0 + 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 1 + 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 1 + 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 1 + 3
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 1 + 3
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4 = 1 + 3
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 0 + 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 5 = 2 + 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 5 = 2 + 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 5 = 2 + 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 5 = 2 + 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 5 = 2 + 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 4 = 1 + 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 5 = 2 + 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 5 = 2 + 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 5 = 2 + 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 5 = 2 + 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 5 = 2 + 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 4 = 1 + 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 5 = 2 + 3
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000234
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => [1] => 0
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => [1] => 0
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [1,2] => 1
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [1,2] => 1
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => [2,1] => 0
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [1,2] => 1
[1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => [1,2] => 1
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [1,2,3] => 2
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,3] => 2
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [1,3,2] => 1
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,3] => 2
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,2,3] => 2
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => [2,1,3] => 1
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => [2,1,3] => 1
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [1,3,2] => 1
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,3] => 2
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,2,3] => 2
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => [3,1,2] => 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]]
=> [2,1,4,3] => [2,1,3] => 1
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,2,3] => 2
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [1,2,3] => 2
[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]]
=> [1,2,3,4,5] => [1,2,3,4] => 3
[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]]
=> [1,2,3,5,4] => [1,2,3,4] => 3
[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]]
=> [1,2,4,3,5] => [1,2,4,3] => 2
[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]]
=> [1,2,3,5,4] => [1,2,3,4] => 3
[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]]
=> [1,2,5,3,4] => [1,2,3,4] => 3
[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]]
=> [1,3,2,4,5] => [1,3,2,4] => 2
[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]]
=> [1,3,2,5,4] => [1,3,2,4] => 2
[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]]
=> [1,2,4,3,5] => [1,2,4,3] => 2
[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]]
=> [1,2,3,5,4] => [1,2,3,4] => 3
[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]]
=> [1,2,5,3,4] => [1,2,3,4] => 3
[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]]
=> [1,4,2,3,5] => [1,4,2,3] => 1
[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]]
=> [1,3,2,5,4] => [1,3,2,4] => 2
[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]]
=> [1,2,5,3,4] => [1,2,3,4] => 3
[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]]
=> [1,5,2,3,4] => [1,2,3,4] => 3
[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]]
=> [2,1,3,4,5] => [2,1,3,4] => 2
[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]]
=> [2,1,3,5,4] => [2,1,3,4] => 2
[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]]
=> [2,1,4,3,5] => [2,1,4,3] => 1
[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]]
=> [2,1,3,5,4] => [2,1,3,4] => 2
[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]]
=> [2,1,5,3,4] => [2,1,3,4] => 2
[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]]
=> [1,3,2,4,5] => [1,3,2,4] => 2
[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]]
=> [1,3,2,5,4] => [1,3,2,4] => 2
[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]]
=> [1,2,4,3,5] => [1,2,4,3] => 2
[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]]
=> [1,2,3,5,4] => [1,2,3,4] => 3
[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]]
=> [1,2,5,3,4] => [1,2,3,4] => 3
[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]]
=> [1,4,2,3,5] => [1,4,2,3] => 1
[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]]
=> [1,3,2,5,4] => [1,3,2,4] => 2
[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]]
=> [1,2,5,3,4] => [1,2,3,4] => 3
[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]]
=> [1,5,2,3,4] => [1,2,3,4] => 3
[1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,1,2,4,5] => [3,1,2,4] => 1
Description
The number of global ascents of a permutation.
The global ascents are the integers $i$ such that
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$$
Equivalently, by the pigeonhole principle,
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$$
For $n > 1$ it can also be described as an occurrence of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].
Matching statistic: St000031
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => [1] => 1 = 0 + 1
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => [1] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [1,2] => 2 = 1 + 1
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [1,2] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => [2,1] => 1 = 0 + 1
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [1,2] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => [1,2] => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [1,2,3] => 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,3] => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [1,3,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,2,3] => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => [2,1,3] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => [2,1,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [1,3,2] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,2,3] => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => [3,1,2] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [2,1,3] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,2,3] => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [1,2,3] => 3 = 2 + 1
[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]]
=> [1,2,3,4,5] => [1,2,3,4] => 4 = 3 + 1
[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]]
=> [1,2,3,5,4] => [1,2,3,4] => 4 = 3 + 1
[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]]
=> [1,2,4,3,5] => [1,2,4,3] => 3 = 2 + 1
[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]]
=> [1,2,3,5,4] => [1,2,3,4] => 4 = 3 + 1
[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]]
=> [1,2,5,3,4] => [1,2,3,4] => 4 = 3 + 1
[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]]
=> [1,3,2,4,5] => [1,3,2,4] => 3 = 2 + 1
[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]]
=> [1,3,2,5,4] => [1,3,2,4] => 3 = 2 + 1
[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]]
=> [1,2,4,3,5] => [1,2,4,3] => 3 = 2 + 1
[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]]
=> [1,2,3,5,4] => [1,2,3,4] => 4 = 3 + 1
[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]]
=> [1,2,5,3,4] => [1,2,3,4] => 4 = 3 + 1
[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]]
=> [1,4,2,3,5] => [1,4,2,3] => 2 = 1 + 1
[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]]
=> [1,3,2,5,4] => [1,3,2,4] => 3 = 2 + 1
[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]]
=> [1,2,5,3,4] => [1,2,3,4] => 4 = 3 + 1
[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]]
=> [1,5,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[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]]
=> [2,1,3,4,5] => [2,1,3,4] => 3 = 2 + 1
[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]]
=> [2,1,3,5,4] => [2,1,3,4] => 3 = 2 + 1
[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]]
=> [2,1,4,3,5] => [2,1,4,3] => 2 = 1 + 1
[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]]
=> [2,1,3,5,4] => [2,1,3,4] => 3 = 2 + 1
[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]]
=> [2,1,5,3,4] => [2,1,3,4] => 3 = 2 + 1
[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]]
=> [1,3,2,4,5] => [1,3,2,4] => 3 = 2 + 1
[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]]
=> [1,3,2,5,4] => [1,3,2,4] => 3 = 2 + 1
[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]]
=> [1,2,4,3,5] => [1,2,4,3] => 3 = 2 + 1
[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]]
=> [1,2,3,5,4] => [1,2,3,4] => 4 = 3 + 1
[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]]
=> [1,2,5,3,4] => [1,2,3,4] => 4 = 3 + 1
[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]]
=> [1,4,2,3,5] => [1,4,2,3] => 2 = 1 + 1
[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]]
=> [1,3,2,5,4] => [1,3,2,4] => 3 = 2 + 1
[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]]
=> [1,2,5,3,4] => [1,2,3,4] => 4 = 3 + 1
[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]]
=> [1,5,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,1,2,4,5] => [3,1,2,4] => 2 = 1 + 1
Description
The number of cycles in the cycle decomposition of a permutation.
The following 120 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000056The decomposition (or block) number of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000024The number of double up and double down steps of a Dyck path. St000144The pyramid weight of the Dyck path. St000636The hull number of a graph. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St001654The monophonic hull number of a graph. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001118The acyclic chromatic index of a graph. 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 $eA$. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000744The length of the path to the largest entry in a standard Young tableau. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001875The number of simple modules with projective dimension at most 1. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000907The number of maximal antichains of minimal length in a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001060The distinguishing index of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000702The number of weak deficiencies of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000331The number of upper interactions of a Dyck path. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000062The length of the longest increasing subsequence of the permutation. St000443The number of long tunnels of a Dyck path. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000474Dyson's crank of a partition. St000667The greatest common divisor of the parts of the partition. St000993The multiplicity of the largest part of an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001498The normalised height of a Nakayama algebra with magnitude 1. St001571The Cartan determinant of the integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001432The order dimension of the partition. St001568The smallest positive integer that does not appear twice in the partition. St001645The pebbling number of a connected graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001279The sum of the parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000939The number of characters of the symmetric group whose value on the partition is positive. St000141The maximum drop size of a permutation. St000054The first entry of the permutation. St001863The number of weak excedances of a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000717The number of ordinal summands of a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000381The largest part of an integer composition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000264The girth of a graph, which is not a tree. St000632The jump number of the poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001777The number of weak descents in an integer composition. St001935The number of ascents in a parking function. St000765The number of weak records in an integer composition. St000808The number of up steps of the associated bargraph. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000942The number of critical left to right maxima of the parking functions. St001415The length of the longest palindromic prefix of a binary word. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000653The last descent 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. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra
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