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Your data matches 504 different statistics following compositions of up to 3 maps.
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Matching statistic: St000688
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000688: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000688: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> 0 = -1 + 1
[2]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
Description
The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path.
The global dimension is given by [[St000684]] and the dominant dimension is given by [[St000685]]. To every Dyck path there is an LNakayama algebra associated as described in [[St000684]].
Dyck paths for which the global dimension and the dominant dimension of the the LNakayama algebra coincide and both dimensions at least $2$ correspond to the LNakayama algebras that are higher Auslander algebras in the sense of [1].
Matching statistic: St000970
(load all 44 compositions to match this statistic)
(load all 44 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000970: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000970: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> 0 = -1 + 1
[2]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
Description
Number of peaks minus the dominant dimension of the corresponding LNakayama algebra.
Matching statistic: St001026
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> 0 = -1 + 1
[2]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
Description
The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001012
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 3 = -1 + 4
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 4 = 0 + 4
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 0 + 4
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 4 = 0 + 4
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4 = 0 + 4
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 0 + 4
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 4 = 0 + 4
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 0 + 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4 = 0 + 4
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4 = 0 + 4
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 4 = 0 + 4
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5 = 1 + 4
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 0 + 4
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 0 + 4
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 4 = 0 + 4
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 5 = 1 + 4
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 4 = 0 + 4
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 4 = 0 + 4
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 4 = 0 + 4
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 4 = 0 + 4
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 4 = 0 + 4
[4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 4 = 0 + 4
[3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 4 = 0 + 4
Description
Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000455
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [2,1] => ([(0,1)],2)
=> -1
[2]
=> [1,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0
[1,1]
=> [1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 0
[3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 0
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 0
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000204
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> 0 = -1 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[.,.],.],[.,.]],.]
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [[[[[.,.],.],.],[.,.]],.]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [[[[[[.,.],.],.],.],[.,.]],.]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[.,.],[.,[.,.]]],.]
=> 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 1 = 0 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> 1 = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 2 = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[[[.,.],.],.],[[.,.],.]]
=> 1 = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> 1 = 0 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,[[.,.],.]]]
=> 1 = 0 + 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[[[[.,.],.],.],.],[[.,.],.]]
=> 1 = 0 + 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> 1 = 0 + 1
[4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,[[.,.],.]]]
=> 1 = 0 + 1
[3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,[.,[[.,.],.]]]]
=> 1 = 0 + 1
Description
The number of internal nodes of a binary tree.
That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.
Matching statistic: St000225
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2]
=> 0 = -1 + 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [2,1,1,1,1]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,1,1,1,1]
=> 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [2,1,1,1,1,1]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,1]
=> 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 1 = 0 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,1,1,1,1,1]
=> 1 = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> 2 = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [2,1,1,1,1]
=> 1 = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [2,1,1,1,1]
=> 1 = 0 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [2,1,1,1,1]
=> 1 = 0 + 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => [2,1,1,1,1,1]
=> 1 = 0 + 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => [2,1,1,1,1,1]
=> 1 = 0 + 1
[4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,6,7,1,2,3,4] => [2,1,1,1,1,1]
=> 1 = 0 + 1
[3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => [2,1,1,1,1,1]
=> 1 = 0 + 1
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000291
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00272: Binary words —Gray next⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00272: Binary words —Gray next⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 11 => 01 => 0 = -1 + 1
[2]
=> 100 => 110 => 010 => 1 = 0 + 1
[1,1]
=> 110 => 010 => 110 => 1 = 0 + 1
[3]
=> 1000 => 1100 => 0100 => 1 = 0 + 1
[1,1,1]
=> 1110 => 1010 => 0010 => 1 = 0 + 1
[4]
=> 10000 => 11000 => 01000 => 1 = 0 + 1
[2,2]
=> 1100 => 0100 => 1100 => 1 = 0 + 1
[1,1,1,1]
=> 11110 => 01110 => 11110 => 1 = 0 + 1
[5]
=> 100000 => 110000 => 010000 => 1 = 0 + 1
[1,1,1,1,1]
=> 111110 => 101110 => 001110 => 1 = 0 + 1
[6]
=> 1000000 => 1100000 => 0100000 => 1 = 0 + 1
[4,2]
=> 100100 => 000100 => 100100 => 2 = 1 + 1
[3,3]
=> 11000 => 01000 => 11000 => 1 = 0 + 1
[2,2,2]
=> 11100 => 10100 => 00100 => 1 = 0 + 1
[1,1,1,1,1,1]
=> 1111110 => 0111110 => 1111110 => 1 = 0 + 1
[2,2,2,1]
=> 111010 => 011010 => 111010 => 2 = 1 + 1
[4,4]
=> 110000 => 010000 => 110000 => 1 = 0 + 1
[2,2,2,2]
=> 111100 => 011100 => 111100 => 1 = 0 + 1
[3,3,3]
=> 111000 => 101000 => 001000 => 1 = 0 + 1
[5,5]
=> 1100000 => 0100000 => 1100000 => 1 = 0 + 1
[2,2,2,2,2]
=> 1111100 => 1011100 => 0011100 => 1 = 0 + 1
[4,4,4]
=> 1110000 => 1010000 => 0010000 => 1 = 0 + 1
[3,3,3,3]
=> 1111000 => 0111000 => 1111000 => 1 = 0 + 1
Description
The number of descents of a binary word.
Matching statistic: St000292
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00135: Binary words —rotate front-to-back⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 01 => 10 => 0 = -1 + 1
[2]
=> 100 => 001 => 010 => 1 = 0 + 1
[1,1]
=> 110 => 101 => 011 => 1 = 0 + 1
[3]
=> 1000 => 0001 => 0010 => 1 = 0 + 1
[1,1,1]
=> 1110 => 1101 => 1011 => 1 = 0 + 1
[4]
=> 10000 => 00001 => 00010 => 1 = 0 + 1
[2,2]
=> 1100 => 1001 => 0011 => 1 = 0 + 1
[1,1,1,1]
=> 11110 => 11101 => 11011 => 1 = 0 + 1
[5]
=> 100000 => 000001 => 000010 => 1 = 0 + 1
[1,1,1,1,1]
=> 111110 => 111101 => 111011 => 1 = 0 + 1
[6]
=> 1000000 => 0000001 => 0000010 => 1 = 0 + 1
[4,2]
=> 100100 => 001001 => 010010 => 2 = 1 + 1
[3,3]
=> 11000 => 10001 => 00011 => 1 = 0 + 1
[2,2,2]
=> 11100 => 11001 => 10011 => 1 = 0 + 1
[1,1,1,1,1,1]
=> 1111110 => 1111101 => 1111011 => 1 = 0 + 1
[2,2,2,1]
=> 111010 => 110101 => 101011 => 2 = 1 + 1
[4,4]
=> 110000 => 100001 => 000011 => 1 = 0 + 1
[2,2,2,2]
=> 111100 => 111001 => 110011 => 1 = 0 + 1
[3,3,3]
=> 111000 => 110001 => 100011 => 1 = 0 + 1
[5,5]
=> 1100000 => 1000001 => 0000011 => 1 = 0 + 1
[2,2,2,2,2]
=> 1111100 => 1111001 => 1110011 => 1 = 0 + 1
[4,4,4]
=> 1110000 => 1100001 => 1000011 => 1 = 0 + 1
[3,3,3,3]
=> 1111000 => 1110001 => 1100011 => 1 = 0 + 1
Description
The number of ascents of a binary word.
Matching statistic: St000374
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [1,2] => 0 = -1 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => [1,3,2] => 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => [1,2,3,4,6,5] => 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [1,3,4,5,6,2] => 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [1,4,2,5,3] => 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 1 = 0 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => 1 = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,4,5,2,3] => 2 = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => [1,2,3,5,4,6] => 1 = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => [1,3,2,4,5,6] => 1 = 0 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => [1,2,4,3,5,6] => 1 = 0 + 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => 1 = 0 + 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => 1 = 0 + 1
[4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => 1 = 0 + 1
[3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => 1 = 0 + 1
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
The following 494 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000442The maximal area to the right of an up step of a Dyck path. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000624The normalized sum of the minimal distances to a greater element. St000628The balance of a binary word. St000659The number of rises of length at least 2 of a Dyck path. St000662The staircase size of the code of a permutation. St000665The number of rafts of a permutation. St000703The number of deficiencies of a permutation. St000766The number of inversions of an integer composition. St000829The Ulam distance of a permutation to the identity permutation. St000834The number of right outer peaks of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000919The number of maximal left branches of a binary tree. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001214The aft of an integer partition. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St000013The height of a Dyck path. St000236The number of cyclical small weak excedances. St000259The diameter of a connected graph. St000388The number of orbits of vertices of a graph under automorphisms. St000444The length of the maximal rise of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000820The number of compositions obtained by rotating the composition. St000891The number of distinct diagonal sums of a permutation matrix. St001128The exponens consonantiae of a partition. St001471The magnitude of a Dyck path. St001486The number of corners of the ribbon associated with an integer composition. St001512The minimum rank of a graph. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001093The detour number of a graph. St000486The number of cycles of length at least 3 of a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000661The number of rises of length 3 of a Dyck path. St000872The number of very big descents of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000619The number of cyclic descents of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001246The maximal difference between two consecutive entries of a permutation. St000216The absolute length of a permutation. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St000376The bounce deficit of a Dyck path. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000516The number of stretching pairs of a permutation. St000538The number of even inversions of a permutation. St000539The number of odd inversions of a permutation. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000649The number of 3-excedences of a permutation. St000691The number of changes of a binary word. St000709The number of occurrences of 14-2-3 or 14-3-2. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000779The tier of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000809The reduced reflection length of the permutation. St000836The number of descents of distance 2 of a permutation. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001114The number of odd descents of a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001130The number of two successive successions in a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001394The genus of a permutation. St001731The factorization defect of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000402Half the size of the symmetry class of a permutation. St000485The length of the longest cycle of a permutation. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000568The hook number of a binary tree. St000630The length of the shortest palindromic decomposition of a binary word. St000647The number of big descents of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000847The number of standard Young tableaux whose descent set is the binary word. St000886The number of permutations with the same antidiagonal sums. St000983The length of the longest alternating subword. St000988The orbit size of a permutation under Foata's bijection. St001081The number of minimal length factorizations of a permutation into star transpositions. St001220The width of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000028The number of stack-sorts needed to sort a permutation. St001469The holeyness of a permutation. St000141The maximum drop size of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001737The number of descents of type 2 in a permutation. St001884The number of borders of a binary word. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000742The number of big ascents of a permutation after prepending zero. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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 $eA$. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000023The number of inner peaks of a permutation. St000079The number of alternating sign matrices for a given Dyck path. St000242The number of indices that are not cyclical small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000873The aix statistic of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001388The number of non-attacking neighbors of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St001874Lusztig's a-function for the symmetric group. St000099The number of valleys of a permutation, including the boundary. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001642The Prague dimension of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St000915The Ore degree of a graph. St000219The number of occurrences of the pattern 231 in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001520The number of strict 3-descents. St001524The degree of symmetry of a binary word. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St001344The neighbouring number of a permutation. St001656The monophonic position number of a graph. St000355The number of occurrences of the pattern 21-3. St000462The major index minus the number of excedences of a permutation. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000570The Edelman-Greene number of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000914The sum of the values of the Möbius function of a poset. St001877Number of indecomposable injective modules with projective dimension 2. St000045The number of linear extensions of a binary tree. St000741The Colin de Verdière graph invariant. St001571The Cartan determinant of the integer partition. St001625The Möbius invariant of a lattice. St001811The Castelnuovo-Mumford regularity of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001890The maximum magnitude of the Möbius function of a poset. 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 $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000928The sum of the coefficients of the character polynomial of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000117The number of centered tunnels of a Dyck path. St000348The non-inversion sum of a binary word. St000477The weight of a partition according to Alladi. St000682The Grundy value of Welter's game on a binary word. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001910The height of the middle non-run of a Dyck path. St000053The number of valleys of the Dyck path. St000306The bounce count of a Dyck path. St000390The number of runs of ones in a binary word. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000675The number of centered multitunnels of a Dyck path. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001568The smallest positive integer that does not appear twice in the partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001732The number of peaks visible from the left. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001933The largest multiplicity of a part in an integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000015The number of peaks of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001488The number of corners of a skew partition. St001530The depth of a Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000369The dinv deficit of a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000893The number of distinct diagonal sums of an alternating sign matrix. St000905The number of different multiplicities of parts of an integer composition. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001557The number of inversions of the second entry of a permutation. St001569The maximal modular displacement of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000679The pruning number of an ordered tree. St000264The girth of a graph, which is not a tree. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000934The 2-degree of an integer partition. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001410The minimal entry of a semistandard tableau. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000254The nesting number of a set partition. St000260The radius of a connected graph. St000768The number of peaks in an integer composition. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000092The number of outer peaks of a permutation. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St001096The size of the overlap set of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001424The number of distinct squares in a binary word. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001624The breadth of a lattice. St000638The number of up-down runs of a permutation. St001618The cardinality of the Frattini sublattice of a lattice. St000940The number of characters of the symmetric group whose value on the partition is zero. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000522The number of 1-protected nodes of a rooted tree. St000521The number of distinct subtrees of an ordered tree. St000782The indicator function of whether a given perfect matching is an L & P matching. St001095The number of non-isomorphic posets with precisely one further covering relation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000075The orbit size of a standard tableau under promotion. St000080The rank of the poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001902The number of potential covers of a poset. St000528The height of a poset. St000906The length of the shortest maximal chain in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001782The order of rowmotion on the set of order ideals of a poset. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000043The number of crossings plus two-nestings of a perfect matching. St000089The absolute variation of a composition. St000101The cocharge of a semistandard tableau. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000233The number of nestings of a set partition. St000317The cycle descent number of a permutation. St000360The number of occurrences of the pattern 32-1. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000454The largest eigenvalue of a graph if it is integral. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000632The jump number of the poset. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001423The number of distinct cubes in a binary word. St001470The cyclic holeyness of a permutation. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001565The number of arithmetic progressions of length 2 in a permutation. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001822The number of alignments of a signed permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001843The Z-index of a set partition. St001856The number of edges in the reduced word graph of a permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001948The number of augmented double ascents of a permutation. St000133The "bounce" of a permutation. St000168The number of internal nodes of an ordered tree. St000222The number of alignments in the permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000640The rank of the largest boolean interval in a poset. St000739The first entry in the last row of a semistandard tableau. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000942The number of critical left to right maxima of the parking functions. St000989The number of final rises of a permutation. St001401The number of distinct entries in a semistandard tableau. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001516The number of cyclic bonds of a permutation. St001535The number of cyclic alignments of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001768The number of reduced words of a signed permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001841The number of inversions of a set partition. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001911A descent variant minus the number of inversions. St000004The major index of a permutation. St000021The number of descents of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000105The number of blocks in the set partition. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000166The depth minus 1 of an ordered tree. St000211The rank of the set partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000251The number of nonsingleton blocks of a set partition. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000461The rix statistic of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000493The los statistic of a set partition. St000499The rcb statistic of a set partition. St000504The cardinality of the first block of a set partition. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000653The last descent of a permutation. St000654The first descent of a permutation. St000702The number of weak deficiencies of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000730The maximal arc length of a set partition. St000747A variant of the major index of a set partition. St000748The major index of the permutation obtained by flattening the set partition. St000794The mak of a permutation. St000798The makl of a permutation. St000823The number of unsplittable factors of the set partition. St000833The comajor index of a permutation. St000925The number of topologically connected components of a set partition. St000956The maximal displacement of a permutation. St000991The number of right-to-left minima of a permutation. St001060The distinguishing index of a graph. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001285The number of primes in the column sums of the two line notation of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001405The number of bonds in a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001497The position of the largest weak excedence of a permutation. St001517The length of a longest pair of twins in a permutation. St001566The length of the longest arithmetic progression in a permutation. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001665The number of pure excedances of a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001729The number of visible descents of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001769The reflection length of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001928The number of non-overlapping descents in a permutation. St000135The number of lucky cars of the parking function. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000519The largest length of a factor maximising the subword complexity. St000744The length of the path to the largest entry in a standard Young tableau. St000863The length of the first row of the shifted shape of a permutation. St000922The minimal number such that all substrings of this length are unique. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001375The pancake length of a permutation. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001821The sorting index of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000642The size of the smallest orbit of antichains under Panyushev complementation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001267The length of the Lyndon factorization of the binary word.
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