- FindStat

Your data matches 985 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000002
(load all 766 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000002: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [] => [] => 0
[1,2] => [1] => [1] => 0
[2,1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => 0
[1,3,2] => [1,2] => [1,2] => 0
[2,1,3] => [2,1] => [2,1] => 0
[2,3,1] => [2,1] => [2,1] => 0
[3,1,2] => [1,2] => [1,2] => 0
[3,2,1] => [2,1] => [2,1] => 0
[1,2,3,4] => [1,2,3] => [1,3,2] => 0
[1,2,4,3] => [1,2,3] => [1,3,2] => 0
[1,3,2,4] => [1,3,2] => [1,3,2] => 0
[1,3,4,2] => [1,3,2] => [1,3,2] => 0
[1,4,2,3] => [1,2,3] => [1,3,2] => 0
[1,4,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3,4] => [2,1,3] => [2,1,3] => 0
[2,1,4,3] => [2,1,3] => [2,1,3] => 0
[2,3,1,4] => [2,3,1] => [2,3,1] => 0
[2,3,4,1] => [2,3,1] => [2,3,1] => 0
[2,4,1,3] => [2,1,3] => [2,1,3] => 0
[2,4,3,1] => [2,3,1] => [2,3,1] => 0
[3,1,2,4] => [3,1,2] => [3,1,2] => 0
[3,1,4,2] => [3,1,2] => [3,1,2] => 0
[3,2,1,4] => [3,2,1] => [3,2,1] => 0
[3,2,4,1] => [3,2,1] => [3,2,1] => 0
[3,4,1,2] => [3,1,2] => [3,1,2] => 0
[3,4,2,1] => [3,2,1] => [3,2,1] => 0
[4,1,2,3] => [1,2,3] => [1,3,2] => 0
[4,1,3,2] => [1,3,2] => [1,3,2] => 0
[4,2,1,3] => [2,1,3] => [2,1,3] => 0
[4,2,3,1] => [2,3,1] => [2,3,1] => 0
[4,3,1,2] => [3,1,2] => [3,1,2] => 0
[4,3,2,1] => [3,2,1] => [3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4] => [1,4,3,2] => 0
[1,2,3,5,4] => [1,2,3,4] => [1,4,3,2] => 0
[1,2,4,3,5] => [1,2,4,3] => [1,4,3,2] => 0
[1,2,4,5,3] => [1,2,4,3] => [1,4,3,2] => 0
[1,2,5,3,4] => [1,2,3,4] => [1,4,3,2] => 0
[1,2,5,4,3] => [1,2,4,3] => [1,4,3,2] => 0
[1,3,2,4,5] => [1,3,2,4] => [1,4,3,2] => 0
[1,3,2,5,4] => [1,3,2,4] => [1,4,3,2] => 0
[1,3,4,2,5] => [1,3,4,2] => [1,4,3,2] => 0
[1,3,4,5,2] => [1,3,4,2] => [1,4,3,2] => 0
[1,3,5,2,4] => [1,3,2,4] => [1,4,3,2] => 0
[1,3,5,4,2] => [1,3,4,2] => [1,4,3,2] => 0
[1,4,2,3,5] => [1,4,2,3] => [1,4,3,2] => 0
[1,4,2,5,3] => [1,4,2,3] => [1,4,3,2] => 0
[1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => 0
[1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => 0
[1,4,5,2,3] => [1,4,2,3] => [1,4,3,2] => 0

Description

The number of occurrences of the pattern 123 in a permutation.

Matching statistic: St000052
(load all 104 compositions to match this statistic)

Mapping

Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1,0]
 => 0
[1,2] => [2] => [1,1,0,0]
 => 0
[2,1] => [2] => [1,1,0,0]
 => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
 => 0
[2,1,3] => [3] => [1,1,1,0,0,0]
 => 0
[2,3,1] => [3] => [1,1,1,0,0,0]
 => 0
[3,1,2] => [3] => [1,1,1,0,0,0]
 => 0
[3,2,1] => [2,1] => [1,1,0,0,1,0]
 => 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 0
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 0

Description

The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.

Matching statistic: St000091
(load all 29 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000091: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => [1] => 0
[1,2] => ([],2)
 => [2] => 0
[2,1] => ([(0,1)],2)
 => [1,1] => 0
[1,2,3] => ([],3)
 => [3] => 0
[1,3,2] => ([(1,2)],3)
 => [2,1] => 0
[2,1,3] => ([(1,2)],3)
 => [2,1] => 0
[2,3,1] => ([(0,2),(1,2)],3)
 => [2,1] => 0
[3,1,2] => ([(0,2),(1,2)],3)
 => [2,1] => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => 0
[1,2,3,4] => ([],4)
 => [4] => 0
[1,2,4,3] => ([(2,3)],4)
 => [3,1] => 0
[1,3,2,4] => ([(2,3)],4)
 => [3,1] => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [3,1] => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [3,1] => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 0
[2,1,3,4] => ([(2,3)],4)
 => [3,1] => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2] => 0
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [3,1] => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [3,1] => 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [2,2] => 0
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 0
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [3,1] => 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [2,2] => 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [3,1] => 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => 0
[1,2,3,4,5] => ([],5)
 => [5] => 0
[1,2,3,5,4] => ([(3,4)],5)
 => [4,1] => 0
[1,2,4,3,5] => ([(3,4)],5)
 => [4,1] => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [4,1] => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [4,1] => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 0
[1,3,2,4,5] => ([(3,4)],5)
 => [4,1] => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [3,2] => 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [4,1] => 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1] => 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [3,2] => 0
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [4,1] => 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [3,2] => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => 0

Description

The descent variation of a composition. Defined in [1].

Matching statistic: St000118
(load all 51 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000118: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [.,.]
 => 0
[1,2] => [1,2] => [.,[.,.]]
 => 0
[2,1] => [2,1] => [[.,.],.]
 => 0
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
 => 0
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => 0
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => 0
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
 => 0
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
 => 0
[3,2,1] => [3,2,1] => [[[.,.],.],.]
 => 0
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 0
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 0
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => 0
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => 0
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => 0
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => 0
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => 0
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => 0
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 0
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => 0
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => 0
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => 0
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => 0
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => 0
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
 => 0
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => 0
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
 => 0
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => 0
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0

Description

The number of occurrences of the contiguous pattern {{{[.,[.,[.,.]]]}}} in a binary tree. [[oeis:A001006]] counts binary trees avoiding this pattern.

Matching statistic: St000121
(load all 62 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000121: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [.,.]
 => 0
[1,2] => [1,2] => [.,[.,.]]
 => 0
[2,1] => [2,1] => [[.,.],.]
 => 0
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
 => 0
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => 0
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => 0
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
 => 0
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
 => 0
[3,2,1] => [3,2,1] => [[[.,.],.],.]
 => 0
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 0
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 0
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => 0
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => 0
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => 0
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => 0
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => 0
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => 0
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 0
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => 0
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => 0
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => 0
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => 0
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => 0
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
 => 0
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => 0
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
 => 0
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => 0
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0

Description

The number of occurrences of the contiguous pattern {{{[.,[.,[.,[.,.]]]]}}} in a binary tree. [[oeis:A036765]] counts binary trees avoiding this pattern.

Matching statistic: St000122
(load all 64 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000122: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [.,.]
 => 0
[1,2] => [1,2] => [.,[.,.]]
 => 0
[2,1] => [2,1] => [[.,.],.]
 => 0
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
 => 0
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => 0
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => 0
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
 => 0
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
 => 0
[3,2,1] => [3,2,1] => [[[.,.],.],.]
 => 0
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 0
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 0
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => 0
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => 0
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => 0
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => 0
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => 0
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => 0
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 0
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => 0
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => 0
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => 0
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => 0
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => 0
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
 => 0
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => 0
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
 => 0
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => 0
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0

Description

The number of occurrences of the contiguous pattern {{{[.,[.,[[.,.],.]]]}}} in a binary tree. [[oeis:A086581]] counts binary trees avoiding this pattern.

Matching statistic: St000125
(load all 49 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000125: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [.,.]
 => 0
[1,2] => [1,2] => [.,[.,.]]
 => 0
[2,1] => [1,2] => [.,[.,.]]
 => 0
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
 => 0
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => 0
[2,1,3] => [1,3,2] => [.,[[.,.],.]]
 => 0
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
 => 0
[3,1,2] => [1,2,3] => [.,[.,[.,.]]]
 => 0
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
 => 0
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 0
[1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => 0
[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => 0
[1,3,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
 => 0
[1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
 => 0
[1,4,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
 => 0
[2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
 => 0
[2,1,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
 => 0
[2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
 => 0
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 0
[2,4,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => 0
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => 0
[3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => 0
[3,1,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
 => 0
[3,2,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
 => 0
[3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => 0
[3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 0
[3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 0
[4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 0
[4,1,3,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => 0
[4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => 0
[4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 0
[4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 0
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => 0
[1,2,4,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
 => 0
[1,2,5,3,4] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => 0
[1,2,5,4,3] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => 0
[1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => 0
[1,3,4,2,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
 => 0
[1,3,4,5,2] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
 => 0
[1,3,5,2,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
 => 0
[1,3,5,4,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
 => 0
[1,4,2,3,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => 0
[1,4,2,5,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => 0
[1,4,3,2,5] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => 0
[1,4,3,5,2] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => 0
[1,4,5,2,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => 0

Description

The number of occurrences of the contiguous pattern {{{[.,[[[.,.],.],.]]}}} in a binary tree. [[oeis:A005773]] counts binary trees avoiding this pattern.

Matching statistic: St000126
(load all 90 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000126: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [.,.]
 => 0
[1,2] => [1,2] => [.,[.,.]]
 => 0
[2,1] => [2,1] => [[.,.],.]
 => 0
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
 => 0
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => 0
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => 0
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
 => 0
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
 => 0
[3,2,1] => [3,2,1] => [[[.,.],.],.]
 => 0
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 0
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 0
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => 0
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => 0
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => 0
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => 0
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => 0
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => 0
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 0
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => 0
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => 0
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => 0
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => 0
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => 0
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
 => 0
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => 0
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
 => 0
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => 0
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0

Description

The number of occurrences of the contiguous pattern {{{[.,[.,[.,[.,[.,.]]]]]}}} in a binary tree. [[oeis:A036766]] counts binary trees avoiding this pattern.

Matching statistic: St000127
(load all 94 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000127: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [.,.]
 => 0
[1,2] => [1,2] => [.,[.,.]]
 => 0
[2,1] => [2,1] => [[.,.],.]
 => 0
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
 => 0
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => 0
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => 0
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
 => 0
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
 => 0
[3,2,1] => [3,2,1] => [[[.,.],.],.]
 => 0
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 0
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 0
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => 0
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => 0
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => 0
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => 0
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => 0
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => 0
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 0
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => 0
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => 0
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => 0
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => 0
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => 0
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
 => 0
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => 0
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
 => 0
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => 0
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0

Description

The number of occurrences of the contiguous pattern {{{[.,[.,[.,[[.,.],.]]]]}}} in a binary tree. [[oeis:A159768]] counts binary trees avoiding this pattern.

Matching statistic: St000128
(load all 102 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000128: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [.,.]
 => 0
[1,2] => [1,2] => [.,[.,.]]
 => 0
[2,1] => [2,1] => [[.,.],.]
 => 0
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
 => 0
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => 0
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => 0
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
 => 0
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
 => 0
[3,2,1] => [3,2,1] => [[[.,.],.],.]
 => 0
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 0
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => 0
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => 0
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => 0
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => 0
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => 0
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => 0
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => 0
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 0
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => 0
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => 0
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => 0
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => 0
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => 0
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
 => 0
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => 0
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
 => 0
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => 0
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => 0

Description

The number of occurrences of the contiguous pattern {{{[.,[.,[[.,[.,.]],.]]]}}} in a binary tree. [[oeis:A159769]] counts binary trees avoiding this pattern.

The following 975 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000210Minimum over maximum difference of elements in cycles. St000232The number of crossings of a set partition. St000296The length of the symmetric border of a binary word. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000352The Elizalde-Pak rank of a permutation. St000366The number of double descents of a permutation. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000403The Szeged index minus the Wiener index of a graph. St000405The number of occurrences of the pattern 1324 in a permutation. St000546The number of global descents of a permutation. St000629The defect of a binary word. St000637The length of the longest cycle in a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000731The number of double exceedences of a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000761The number of ascents in an integer composition. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001071The beta invariant of the graph. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001119The length of a shortest maximal path in a graph. St001137Number of simple modules that are 3-regular 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. St001172The number of 1-rises at odd height of a Dyck path. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001305The number of induced cycles on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001395The number of strictly unfriendly partitions of a graph. St001584The area statistic between a Dyck path and its bounce path. St001596The number of two-by-two squares inside a skew partition. St001695The natural comajor index of a standard Young tableau. St001696The natural major 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. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001712The number of natural descents of a standard Young tableau. St001736The total number of cycles in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001797The number of overfull subgraphs of a graph. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000069The number of maximal elements of a poset. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000363The number of minimal vertex covers of a graph. St000501The size of the first part in the decomposition of a permutation. St000505The biggest entry in the block containing the 1. St000542The number of left-to-right-minima of a permutation. St000544The cop number of a graph. St000655The length of the minimal rise of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000701The protection number of a binary tree. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000763The sum of the positions of the strong records of an integer composition. St000764The number of strong records in an integer composition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000781The number of proper colouring schemes of a Ferrers diagram. St000785The number of distinct colouring schemes of a graph. St000805The number of peaks of the associated bargraph. St000913The number of ways to refine the partition into singletons. St000948The chromatic discriminant of a graph. St000971The smallest closer of a set partition. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001196The global dimension of

$A$ minus the global dimension of

$eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001316The domatic number of a graph. St001363The Euler characteristic of a graph according to Knill. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001468The smallest fixpoint of a permutation. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St001597The Frobenius rank of a skew partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000439The position of the first down step of a Dyck path. St000042The number of crossings of a perfect matching. St000057The Shynar inversion number of a standard tableau. St000065The number of entries equal to -1 in an alternating sign matrix. St000090The variation of a composition. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000217The number of occurrences of the pattern 312 in a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000223The number of nestings in the permutation. St000233The number of nestings of a set partition. St000234The number of global ascents of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000297The number of leading ones in a binary word. St000312The number of leaves in a graph. St000347The inversion sum of a binary word. St000356The number of occurrences of the pattern 13-2. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000386The number of factors DDU in a Dyck path. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000448The number of pairs of vertices of a graph with distance 2. St000449The number of pairs of vertices of a graph with distance 4. St000496The rcs statistic of a set partition. St000523The number of 2-protected nodes of a rooted tree. St000534The number of 2-rises of a permutation. St000552The number of cut vertices of a graph. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000660The number of rises of length at least 3 of a Dyck path. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000864The number of circled entries of the shifted recording tableau of a permutation. St000871The number of very big ascents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000921The number of internal inversions of a binary word. St000974The length of the trunk of an ordered tree. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001175The size of a partition minus the hook length of the base cell. St001301The first Betti number of the order complex associated with the poset. St001306The number of induced paths on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001374The Padmakar-Ivan index of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001394The genus of a permutation. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001513The number of nested exceedences of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001577The minimal number of edges to add or remove to make a graph a cograph. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001689The number of celebrities in a graph. St001705The number of occurrences of the pattern 2413 in a permutation. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001764The number of non-convex subsets of vertices in a graph. St001781The interlacing number of a set partition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St001845The number of join irreducibles minus the rank of a lattice. St001847The number of occurrences of the pattern 1432 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000011The number of touch points (or returns) of a Dyck path. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000047The number of standard immaculate tableaux of a given shape. St000068The number of minimal elements in a poset. St000078The number of alternating sign matrices whose left key is the permutation. St000115The single entry in the last row. St000160The multiplicity of the smallest part of a partition. St000183The side length of the Durfee square of an integer partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000382The first part of an integer composition. St000383The last part of an integer composition. St000392The length of the longest run of ones in a binary word. St000517The Kreweras number of an integer partition. St000627The exponent of a binary word. St000657The smallest part of an integer composition. St000679The pruning number of an ordered tree. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000700The protection number of an ordered tree. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000733The row containing the largest entry of a standard tableau. 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. St000765The number of weak records in an integer composition. St000816The number of standard composition tableaux of the composition. St000847The number of standard Young tableaux whose descent set is the binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000920The logarithmic height of a Dyck path. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001267The length of the Lyndon factorization of the binary word. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001437The flex of a binary word. St001518The number of graphs with the same ordinary spectrum as the given graph. St001571The Cartan determinant of the integer partition. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001722The number of minimal chains with small intervals between a binary word and the top element. St001732The number of peaks visible from the left. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001884The number of borders of a binary word. St001933The largest multiplicity of a part in an integer partition. St000397The Strahler number of a rooted tree. St000630The length of the shortest palindromic decomposition of a binary word. St000842The breadth of a permutation. St001330The hat guessing number of a graph. St000317The cycle descent number of a permutation. St000357The number of occurrences of the pattern 12-3. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000379The number of Hamiltonian cycles in a graph. St000478Another weight of a partition according to Alladi. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000699The toughness times the least common multiple of 1,. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000929The constant term of the character polynomial of an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001281The normalized isoperimetric number of a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000054The first entry of the permutation. St000456The monochromatic index of a connected graph. St000487The length of the shortest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000823The number of unsplittable factors of the set partition. St000990The first ascent of a permutation. St000993The multiplicity of the largest part of an integer partition. St001075The minimal size of a block of a set partition. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001592The maximal number of simple paths between any two different vertices of a graph. St000051The size of the left subtree of a binary tree. St000095The number of triangles of a graph. St000133The "bounce" of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000292The number of ascents of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St000310The minimal degree of a vertex of a graph. St000348The non-inversion sum of a binary word. St000358The number of occurrences of the pattern 31-2. St000407The number of occurrences of the pattern 2143 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000502The number of successions of a set partitions. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000562The number of internal points of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. 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. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. 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. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000661The number of rises of length 3 of a Dyck path. St000674The number of hills of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000732The number of double deficiencies of a permutation. St000748The major index of the permutation obtained by flattening the set partition. St000750The number of occurrences of the pattern 4213 in a permutation. St000872The number of very big descents of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to 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. St001139The number of occurrences of hills of size 2 in a Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three 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. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001578The minimal number of edges to add or remove to make a graph a line graph. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001692The number of vertices with higher degree than the average degree in a graph. St001708The number of pairs of vertices of different degree in a graph. St001718The number of non-empty open intervals in a poset. St001742The difference of the maximal and the minimal degree in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001799The number of proper separations of a graph. St001871The number of triconnected components of a graph. St000056The decomposition (or block) number of a permutation. St000096The number of spanning trees of a graph. St000255The number of reduced Kogan faces with the permutation as type. St000273The domination number of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000349The number of different adjacency matrices of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000450The number of edges minus the number of vertices plus 2 of a graph. St000553The number of blocks of a graph. St000659The number of rises of length at least 2 of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. St000729The minimal arc length of a set partition. St000740The last entry of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000762The sum of the positions of the weak records of an integer composition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000908The length of the shortest maximal antichain in a poset. St000916The packing number of a graph. St000991The number of right-to-left minima of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001057The Grundy value of the game of creating an independent set in a graph. St001236The dominant dimension of the corresponding Comp-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. St001256Number of simple reflexive modules that are 2-stable reflexive. 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. St001313The number of Dyck paths above the lattice path given by a binary word. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. St001481The minimal height of a peak of a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001568The smallest positive integer that does not appear twice in the partition. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001642The Prague dimension of a graph. St001734The lettericity of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001765The number of connected components of the friends and strangers graph. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001917The order of toric promotion on the set of labellings of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001471The magnitude of a Dyck path. St000475The number of parts equal to 1 in a partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001561The value of the elementary symmetric function evaluated at 1. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000759The smallest missing part in an integer partition. St001490The number of connected components of a skew partition. St000804The number of occurrences of the vincular pattern |123 in a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000461The rix statistic of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000873The aix statistic of a permutation. St000989The number of final rises of a permutation. St000061The number of nodes on the left branch of a binary tree. St000570The Edelman-Greene number of a permutation. St000640The rank of the largest boolean interval in a poset. St000654The first descent of a permutation. St000914The sum of the values of the Möbius function of a poset. St001204Call 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. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000264The girth of a graph, which is not a tree. St001193The dimension of

$Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001703The villainy of a graph. St000642The size of the smallest orbit of antichains under Panyushev complementation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001487The number of inner corners of a skew partition. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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

$eA$ . St001271The competition number of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001826The maximal number of leaves on a vertex of a graph. St001657The number of twos in an integer partition. St000221The number of strong fixed points of a permutation. St001381The fertility of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000889The number of alternating sign matrices with the same antidiagonal sums. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000879The number of long braid edges in the graph of braid moves of a permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001961The sum of the greatest common divisors of all pairs of parts. St001593This is the number of standard Young tableaux of the given shifted shape. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001783The number of odd automorphisms of a graph. St001479The number of bridges of a graph. St000649The number of 3-excedences of a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001780The order of promotion on the set of standard tableaux of given shape. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000648The number of 2-excedences of a permutation. St000647The number of big descents of a permutation. St001434The number of negative sum pairs of a signed permutation. St000048The multinomial of the parts of a partition. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St000181The number of connected components of the Hasse diagram for the poset. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000247The number of singleton blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000925The number of topologically connected components of a set partition. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000225Difference between largest and smallest parts in a partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001208The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000252The number of nodes of degree 3 of a binary tree. St000355The number of occurrences of the pattern 21-3. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001535The number of cyclic alignments of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000516The number of stretching pairs of a permutation. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000315The number of isolated vertices of a graph. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St001570The minimal number of edges to add to make a graph Hamiltonian. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001411The number of patterns 321 or 3412 in a permutation. St001727The number of invisible inversions of a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000538The number of even inversions of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000836The number of descents of distance 2 of a permutation. St000886The number of permutations with the same antidiagonal sums. St000001The number of reduced words for a permutation. St000883The number of longest increasing subsequences of a permutation. St000214The number of adjacencies of a permutation. St000481The number of upper covers of a partition in dominance order. St000003The number of standard Young tableaux of the partition. St000346The number of coarsenings of a partition. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000618The number of self-evacuating tableaux of given shape. St001432The order dimension of the partition. St000567The sum of the products of all pairs of parts. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000284The Plancherel distribution on integer partitions. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000617The number of global maxima of a Dyck path. St001890The maximum magnitude of the Möbius function of a poset. St001651The Frankl number of a lattice. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001377The major index minus the number of inversions of a permutation. St001536The number of cyclic misalignments of a permutation. St001947The number of ties in a parking function. St001095The number of non-isomorphic posets with precisely one further covering relation. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001720The minimal length of a chain of small intervals in a lattice. St000124The cardinality of the preimage of the Simion-Schmidt map. St000663The number of right floats of a permutation. St001626The number of maximal proper sublattices of a lattice. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000185The weighted size of a partition. St001214The aft of an integer partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001964The interval resolution global dimension of a poset. St000215The number of adjacencies of a permutation, zero appended. St000237The number of small exceedances. St000961The shifted major index of a permutation. St000253The crossing number of a set partition. St001260The permanent of an alternating sign matrix. St000738The first entry in the last row of a standard tableau. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001060The distinguishing index of a graph. St000007The number of saliances of the permutation. St000097The order of the largest clique of the graph. St000675The number of centered multitunnels of a Dyck path. St000374The number of exclusive right-to-left minima of a permutation. St000451The length of the longest pattern of the form k 1 2. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St001735The number of permutations with the same set of runs. St000462The major index minus the number of excedences of a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001052The length of the exterior of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000041The number of nestings of a perfect matching. St000098The chromatic number of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000084The number of subtrees. St000093The cardinality of a maximal independent set of vertices of a graph. St000260The radius of a connected graph. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000081The number of edges of a graph. St000089The absolute variation of a composition. St000171The degree of the graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000268The number of strongly connected orientations of a graph. St000272The treewidth of a graph. St000311The number of vertices of odd degree in a graph. St000313The number of degree 2 vertices of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000350The sum of the vertex degrees of a graph. St000361The second Zagreb index of a graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000422The energy of a graph, if it is integral. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000454The largest eigenvalue of a graph if it is integral. St000465The first Zagreb index of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000571The F-index (or forgotten topological index) of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000915The Ore degree of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001073The number of nowhere zero 3-flows of a graph. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001341The number of edges in the center of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001362The normalized Knill dimension of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001393The induced matching number of a graph. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001512The minimum rank of a graph. St001638The book thickness of a graph. St001644The dimension of a graph. St001649The length of a longest trail in a graph. St001673The degree of asymmetry of an integer composition. St001691The number of kings in a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001812The biclique partition number of a graph. St001869The maximum cut size of a graph. St001962The proper pathwidth of a graph. St000010The length of the partition. St000035The number of left outer peaks of a permutation. St000079The number of alternating sign matrices for a given Dyck path. St000086The number of subgraphs. St000159The number of distinct parts of the integer partition. St000172The Grundy number of a graph. St000259The diameter of a connected graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000277The number of ribbon shaped standard tableaux. St000299The number of nonisomorphic vertex-induced subtrees. St000335The difference of lower and upper interactions. St000343The number of spanning subgraphs of a graph. St000381The largest part of an integer composition. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000669The number of permutations obtained by switching ascents or descents of size 2. St000696The number of cycles in the breakpoint graph of a permutation. St000722The number of different neighbourhoods in a graph. St000767The number of runs in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000808The number of up steps of the associated bargraph. St000820The number of compositions obtained by rotating the composition. St000822The Hadwiger number of the graph. St000843The decomposition number of a perfect matching. St000845The maximal number of elements covered by an element in a poset. St000884The number of isolated descents of a permutation. St000903The number of different parts of an integer composition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000972The composition number of a graph. St000988The orbit size of a permutation under Foata's bijection. St001029The size of the core of a graph. St001048The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001072The evaluation of the Tutte polynomial of the graph at x and y equal to 3. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001110The 3-dynamic chromatic number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001116The game chromatic number of a graph. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001261The Castelnuovo-Mumford regularity of a graph. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001286The annihilation number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001303The number of dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001386The number of prime labellings of a graph. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001484The number of singletons of an integer partition. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001694The number of maximal dissociation sets in a graph. St001716The 1-improper chromatic number of a graph. St001725The harmonious chromatic number of a graph. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001883The mutual visibility number of a graph. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001957The number of Hasse diagrams with a given underlying undirected graph. St001963The tree-depth of a graph. St000258The burning number of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000846The maximal number of elements covering an element of a poset. St000918The 2-limited packing number of a graph. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001429The number of negative entries in a signed permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St001220The width of a permutation. St000485The length of the longest cycle of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St000219The number of occurrences of the pattern 231 in a permutation. St001520The number of strict 3-descents. St001557The number of inversions of the second entry of a permutation. St001948The number of augmented double ascents of a permutation. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000917The open packing number of a graph. St000058The order of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. 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

$eA$ . St000288The number of ones in a binary word. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000753The Grundy value for the game of Kayles on a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001545The second Elser number of a connected graph. St001613The binary logarithm of the size of the center of a lattice. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000754The Grundy value for the game of removing nestings in a perfect matching. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001889The size of the connectivity set of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St001430The number of positive entries in a signed permutation. St000117The number of centered tunnels of a Dyck path. St000142The number of even parts of a partition. St000148The number of odd parts of a partition. St000241The number of cyclical small excedances. St000338The number of pixed points of a permutation. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000549The number of odd partial sums of an integer partition. St000895The number of ones on the main diagonal of an alternating sign matrix. St000954Number of times the corresponding LNakayama algebra has

$Ext^i(D(A),A)=0$ for

$i>0$ . St000995The largest even part of an integer partition. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001092The number of distinct even parts of a partition. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001252Half the sum of the even parts of a partition. St001856The number of edges in the reduced word graph of a permutation. St001866The nesting alignments of a signed permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000958The number of Bruhat factorizations of a permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n−1}]$ by adding

$c_0$ to

$c_{n−1}$ . St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. 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. St001589The nesting number of a perfect matching. St001665The number of pure excedances of a permutation. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001555The order of a signed permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. 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. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001248Sum of the even parts of a partition. St001587Half of the largest even part of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001645The pebbling number of a connected graph. St001621The number of atoms of a lattice. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001249Sum of the odd parts of a partition. St001383The BG-rank of an integer partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001624The breadth of a lattice. St001768The number of reduced words of a signed permutation. St000074The number of special entries. St000894The trace of an alternating sign matrix. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001524The degree of symmetry of a binary word. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St000942The number of critical left to right maxima of the parking functions. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St001927Sparre Andersen's number of positives of a signed permutation. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001857The number of edges in the reduced word graph of a signed permutation. St001410The minimal entry of a semistandard tableau. St000328The maximum number of child nodes in a tree. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000782The indicator function of whether a given perfect matching is an L & P matching.