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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St000335
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000335: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000335: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> 1
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> 2
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
Description
The difference of lower and upper interactions.
An ''upper interaction'' in a Dyck path is the occurrence of a factor $0^k 1^k$ with $k \geq 1$ (see [[St000331]]), and a ''lower interaction'' is the occurrence of a factor $1^k 0^k$ with $k \geq 1$. In both cases, $1$ denotes an up-step $0$ denotes a a down-step.
Matching statistic: St001226
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001226: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001226: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> 2 = 1 + 1
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
Description
The 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.
That is the number of i such that $Ext_A^1(J,e_i J)=0$.
Matching statistic: St000011
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> [1,0]
=> 1
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000025
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> [1,0]
=> 1
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 2
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000026
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> [1,0]
=> 1
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 2
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
Description
The position of the first return of a Dyck path.
Matching statistic: St000056
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> [1] => 1
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> [1,2] => 2
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 3
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> [3,1,2] => 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 3
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,3,2] => 2
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> [2,3,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
Description
The decomposition (or block) number of a permutation.
For $\pi \in \mathcal{S}_n$, this is given by
$$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$
This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum.
This is one plus [[St000234]].
Matching statistic: St000883
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> [1] => 1
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> [2,1] => 2
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> [1,2] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 3
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => 3
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 2
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
Description
The number of longest increasing subsequences of a permutation.
Matching statistic: St001135
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> [1,0]
=> 1
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
Description
The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001201
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001201: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001201: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> [1,0]
=> 1
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
Description
The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path.
Matching statistic: St001461
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001461: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001461: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> [1] => 1
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> [1,2] => 2
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> [2,1] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 3
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> [3,1,2] => 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 3
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,3,2] => 2
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> [2,3,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
Description
The number of topologically connected components of the chord diagram of a permutation.
The chord diagram of a permutation $\pi\in\mathfrak S_n$ is obtained by placing labels $1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from $i$ to $\pi(i)$ for every label $i$.
This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component.
The permutation $\pi\in\mathfrak S_n$ stabilizes an interval $I=\{a,a+1,\dots,b\}$ if $\pi(I)=I$. It is stabilized-interval-free, if the only interval $\pi$ stablizes is $\{1,\dots,n\}$. Thus, this statistic is $1$ if $\pi$ is stabilized-interval-free.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000234The number of global ascents of a permutation. St000439The position of the first down step of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000990The first ascent of a permutation. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000199The column of the unique '1' in the last row of the alternating sign matrix.
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