Processing math: 32%

Your data matches 43 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000051
(load all 6 compositions to match this statistic)
Mapping
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000051: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
 => 0
[1,2] => [.,[.,.]]
 => 0
[2,1] => [[.,.],.]
 => 1
[1,2,3] => [.,[.,[.,.]]]
 => 0
[1,3,2] => [.,[[.,.],.]]
 => 0
[2,1,3] => [[.,.],[.,.]]
 => 1
[2,3,1] => [[.,[.,.]],.]
 => 2
[3,1,2] => [[.,.],[.,.]]
 => 1
[3,2,1] => [[[.,.],.],.]
 => 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => 0
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => 0
[1,3,4,2] => [.,[[.,[.,.]],.]]
 => 0
[1,4,2,3] => [.,[[.,.],[.,.]]]
 => 0
[1,4,3,2] => [.,[[[.,.],.],.]]
 => 0
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
 => 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
 => 3
[2,4,1,3] => [[.,[.,.]],[.,.]]
 => 2
[2,4,3,1] => [[.,[[.,.],.]],.]
 => 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
 => 1
[3,1,4,2] => [[.,.],[[.,.],.]]
 => 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => 2
[3,2,4,1] => [[[.,.],[.,.]],.]
 => 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => 2
[3,4,2,1] => [[[.,[.,.]],.],.]
 => 3
[4,1,2,3] => [[.,.],[.,[.,.]]]
 => 1
[4,1,3,2] => [[.,.],[[.,.],.]]
 => 1
[4,2,1,3] => [[[.,.],.],[.,.]]
 => 2
[4,2,3,1] => [[[.,.],[.,.]],.]
 => 3
[4,3,1,2] => [[[.,.],.],[.,.]]
 => 2
[4,3,2,1] => [[[[.,.],.],.],.]
 => 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => 0
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => 0
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
 => 0
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => 0
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => 0
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => 0
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
 => 0
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
 => 0
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
 => 0
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
 => 0
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => 0
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => 0
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => 0
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
 => 0
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => 0
Description
The size of the left subtree of a binary tree.
Matching statistic: St000054
(load all 30 compositions to match this statistic)
Mapping
Mp00066: Permutations —inverse⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 0 + 1
[1,2] => [1,2] => 1 = 0 + 1
[2,1] => [2,1] => 2 = 1 + 1
[1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => 1 = 0 + 1
[2,1,3] => [2,1,3] => 2 = 1 + 1
[2,3,1] => [3,1,2] => 3 = 2 + 1
[3,1,2] => [2,3,1] => 2 = 1 + 1
[3,2,1] => [3,2,1] => 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => 1 = 0 + 1
[1,3,4,2] => [1,4,2,3] => 1 = 0 + 1
[1,4,2,3] => [1,3,4,2] => 1 = 0 + 1
[1,4,3,2] => [1,4,3,2] => 1 = 0 + 1
[2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[2,3,1,4] => [3,1,2,4] => 3 = 2 + 1
[2,3,4,1] => [4,1,2,3] => 4 = 3 + 1
[2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[2,4,3,1] => [4,1,3,2] => 4 = 3 + 1
[3,1,2,4] => [2,3,1,4] => 2 = 1 + 1
[3,1,4,2] => [2,4,1,3] => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
[3,2,4,1] => [4,2,1,3] => 4 = 3 + 1
[3,4,1,2] => [3,4,1,2] => 3 = 2 + 1
[3,4,2,1] => [4,3,1,2] => 4 = 3 + 1
[4,1,2,3] => [2,3,4,1] => 2 = 1 + 1
[4,1,3,2] => [2,4,3,1] => 2 = 1 + 1
[4,2,1,3] => [3,2,4,1] => 3 = 2 + 1
[4,2,3,1] => [4,2,3,1] => 4 = 3 + 1
[4,3,1,2] => [3,4,2,1] => 3 = 2 + 1
[4,3,2,1] => [4,3,2,1] => 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,5,3,4] => 1 = 0 + 1
[1,2,5,3,4] => [1,2,4,5,3] => 1 = 0 + 1
[1,2,5,4,3] => [1,2,5,4,3] => 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => 1 = 0 + 1
[1,3,4,2,5] => [1,4,2,3,5] => 1 = 0 + 1
[1,3,4,5,2] => [1,5,2,3,4] => 1 = 0 + 1
[1,3,5,2,4] => [1,4,2,5,3] => 1 = 0 + 1
[1,3,5,4,2] => [1,5,2,4,3] => 1 = 0 + 1
[1,4,2,3,5] => [1,3,4,2,5] => 1 = 0 + 1
[1,4,2,5,3] => [1,3,5,2,4] => 1 = 0 + 1
[1,4,3,2,5] => [1,4,3,2,5] => 1 = 0 + 1
[1,4,3,5,2] => [1,5,3,2,4] => 1 = 0 + 1
[1,4,5,2,3] => [1,4,5,2,3] => 1 = 0 + 1
Description
The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation π of n, together with its rotations, obtained by conjugating with the long cycle (1,…,n). Drawing the labels 1 to n in this order on a circle, and the arcs (i,π(i)) as straight lines, the rotation of π is obtained by replacing each number i by (imod. Then, \pi(1)-1 is the number of rotations of \pi where the arc (1, \pi(1)) is a deficiency. In particular, if O(\pi) is the orbit of rotations of \pi, then the number of deficiencies of \pi equals \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
Matching statistic: St000740
(load all 83 compositions to match this statistic)
Mapping
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 0 + 1
[1,2] => [2,1] => 1 = 0 + 1
[2,1] => [1,2] => 2 = 1 + 1
[1,2,3] => [2,3,1] => 1 = 0 + 1
[1,3,2] => [3,2,1] => 1 = 0 + 1
[2,1,3] => [1,3,2] => 2 = 1 + 1
[2,3,1] => [1,2,3] => 3 = 2 + 1
[3,1,2] => [3,1,2] => 2 = 1 + 1
[3,2,1] => [2,1,3] => 3 = 2 + 1
[1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[1,2,4,3] => [2,4,3,1] => 1 = 0 + 1
[1,3,2,4] => [3,2,4,1] => 1 = 0 + 1
[1,3,4,2] => [4,2,3,1] => 1 = 0 + 1
[1,4,2,3] => [3,4,2,1] => 1 = 0 + 1
[1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[2,1,3,4] => [1,3,4,2] => 2 = 1 + 1
[2,1,4,3] => [1,4,3,2] => 2 = 1 + 1
[2,3,1,4] => [1,2,4,3] => 3 = 2 + 1
[2,3,4,1] => [1,2,3,4] => 4 = 3 + 1
[2,4,1,3] => [1,4,2,3] => 3 = 2 + 1
[2,4,3,1] => [1,3,2,4] => 4 = 3 + 1
[3,1,2,4] => [3,1,4,2] => 2 = 1 + 1
[3,1,4,2] => [4,1,3,2] => 2 = 1 + 1
[3,2,1,4] => [2,1,4,3] => 3 = 2 + 1
[3,2,4,1] => [2,1,3,4] => 4 = 3 + 1
[3,4,1,2] => [4,1,2,3] => 3 = 2 + 1
[3,4,2,1] => [3,1,2,4] => 4 = 3 + 1
[4,1,2,3] => [3,4,1,2] => 2 = 1 + 1
[4,1,3,2] => [4,3,1,2] => 2 = 1 + 1
[4,2,1,3] => [2,4,1,3] => 3 = 2 + 1
[4,2,3,1] => [2,3,1,4] => 4 = 3 + 1
[4,3,1,2] => [4,2,1,3] => 3 = 2 + 1
[4,3,2,1] => [3,2,1,4] => 4 = 3 + 1
[1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[1,2,3,5,4] => [2,3,5,4,1] => 1 = 0 + 1
[1,2,4,3,5] => [2,4,3,5,1] => 1 = 0 + 1
[1,2,4,5,3] => [2,5,3,4,1] => 1 = 0 + 1
[1,2,5,3,4] => [2,4,5,3,1] => 1 = 0 + 1
[1,2,5,4,3] => [2,5,4,3,1] => 1 = 0 + 1
[1,3,2,4,5] => [3,2,4,5,1] => 1 = 0 + 1
[1,3,2,5,4] => [3,2,5,4,1] => 1 = 0 + 1
[1,3,4,2,5] => [4,2,3,5,1] => 1 = 0 + 1
[1,3,4,5,2] => [5,2,3,4,1] => 1 = 0 + 1
[1,3,5,2,4] => [4,2,5,3,1] => 1 = 0 + 1
[1,3,5,4,2] => [5,2,4,3,1] => 1 = 0 + 1
[1,4,2,3,5] => [3,4,2,5,1] => 1 = 0 + 1
[1,4,2,5,3] => [3,5,2,4,1] => 1 = 0 + 1
[1,4,3,2,5] => [4,3,2,5,1] => 1 = 0 + 1
[1,4,3,5,2] => [5,3,2,4,1] => 1 = 0 + 1
[1,4,5,2,3] => [4,5,2,3,1] => 1 = 0 + 1
Description
The last entry of a permutation. This statistic is undefined for the empty permutation.
Matching statistic: St000133
(load all 2 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => 1
[1,2,3] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [2,3,1] => [2,3,1] => 0
[2,1,3] => [3,1,2] => [3,1,2] => 1
[2,3,1] => [1,3,2] => [1,3,2] => 2
[3,1,2] => [2,1,3] => [2,1,3] => 1
[3,2,1] => [1,2,3] => [1,3,2] => 2
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [3,4,2,1] => [3,4,2,1] => 0
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 0
[1,3,4,2] => [2,4,3,1] => [2,4,3,1] => 0
[1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 0
[1,4,3,2] => [2,3,4,1] => [2,4,3,1] => 0
[2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 1
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 1
[2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 2
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 3
[2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 2
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => 3
[3,1,2,4] => [4,2,1,3] => [4,2,1,3] => 1
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 1
[3,2,1,4] => [4,1,2,3] => [4,1,3,2] => 2
[3,2,4,1] => [1,4,2,3] => [1,4,3,2] => 3
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,2,1] => [1,2,4,3] => [1,4,3,2] => 3
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 1
[4,1,3,2] => [2,3,1,4] => [2,4,1,3] => 1
[4,2,1,3] => [3,1,2,4] => [3,1,4,2] => 2
[4,2,3,1] => [1,3,2,4] => [1,4,3,2] => 3
[4,3,1,2] => [2,1,3,4] => [2,1,4,3] => 2
[4,3,2,1] => [1,2,3,4] => [1,4,3,2] => 3
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => 0
[1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => 0
[1,2,4,5,3] => [3,5,4,2,1] => [3,5,4,2,1] => 0
[1,2,5,3,4] => [4,3,5,2,1] => [4,3,5,2,1] => 0
[1,2,5,4,3] => [3,4,5,2,1] => [3,5,4,2,1] => 0
[1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => 0
[1,3,2,5,4] => [4,5,2,3,1] => [4,5,2,3,1] => 0
[1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => 0
[1,3,4,5,2] => [2,5,4,3,1] => [2,5,4,3,1] => 0
[1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => 0
[1,3,5,4,2] => [2,4,5,3,1] => [2,5,4,3,1] => 0
[1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => 0
[1,4,2,5,3] => [3,5,2,4,1] => [3,5,2,4,1] => 0
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,4,3,1] => 0
[1,4,3,5,2] => [2,5,3,4,1] => [2,5,4,3,1] => 0
[1,4,5,2,3] => [3,2,5,4,1] => [3,2,5,4,1] => 0
Description
The "bounce" of a permutation.
Matching statistic: St000025
(load all 5 compositions to match this statistic)
Mapping
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => 1 = 0 + 1
[1,2] => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[2,1] => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
 => 2 = 1 + 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
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
(load all 3 compositions to match this statistic)
Mapping
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ 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
[1,2] => [.,[.,.]]
 => [1,0,1,0]
 => 1 = 0 + 1
[2,1] => [[.,.],.]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[3,1,2] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
Description
The position of the first return of a Dyck path.
Matching statistic: St001291
(load all 9 compositions to match this statistic)
Mapping
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => 1 = 0 + 1
[1,2] => [2,1] => [1,1,0,0]
 => 1 = 0 + 1
[2,1] => [1,2] => [1,0,1,0]
 => 2 = 1 + 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => 1 = 0 + 1
[2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 2 = 1 + 1
[2,3,1] => [2,1,3] => [1,1,0,0,1,0]
 => 3 = 2 + 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[2,4,3,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[3,2,4,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. Let A be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of D(A) \otimes D(A), where D(A) is the natural dual of A.
Matching statistic: St001497
(load all 4 compositions to match this statistic)
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001497: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 = 0 + 1
[1,2] => [1,2] => [2,1] => 1 = 0 + 1
[2,1] => [2,1] => [1,2] => 2 = 1 + 1
[1,2,3] => [1,3,2] => [3,1,2] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [3,1,2] => 1 = 0 + 1
[2,1,3] => [2,1,3] => [2,3,1] => 2 = 1 + 1
[2,3,1] => [2,3,1] => [2,1,3] => 3 = 2 + 1
[3,1,2] => [3,1,2] => [1,3,2] => 2 = 1 + 1
[3,2,1] => [3,2,1] => [1,2,3] => 3 = 2 + 1
[1,2,3,4] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,2,4,3] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,3,2,4] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2 = 1 + 1
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => 4 = 3 + 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => 4 = 3 + 1
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 2 = 1 + 1
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 3 = 2 + 1
[3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 4 = 3 + 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 3 = 2 + 1
[3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 4 = 3 + 1
[4,1,2,3] => [4,1,3,2] => [1,4,2,3] => 2 = 1 + 1
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => 2 = 1 + 1
[4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 3 = 2 + 1
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 4 = 3 + 1
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 3 = 2 + 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 4 = 3 + 1
[1,2,3,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 0 + 1
[1,2,3,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 0 + 1
[1,2,4,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 0 + 1
[1,2,4,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 0 + 1
[1,2,5,3,4] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 0 + 1
[1,2,5,4,3] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 0 + 1
[1,3,2,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 0 + 1
[1,3,2,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 0 + 1
[1,3,4,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 0 + 1
[1,3,4,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 0 + 1
[1,3,5,2,4] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 0 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 0 + 1
[1,4,2,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 0 + 1
[1,4,2,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 0 + 1
[1,4,3,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 0 + 1
[1,4,3,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 0 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 0 + 1
Description
The position of the largest weak excedence of a permutation.
Matching statistic: St000439
(load all 5 compositions to match this statistic)
Mapping
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => 2 = 0 + 2
[1,2] => [1,2] => [1,0,1,0]
 => 2 = 0 + 2
[2,1] => [2,1] => [1,1,0,0]
 => 3 = 1 + 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2 = 0 + 2
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 0 + 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 3 = 1 + 2
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
 => 4 = 2 + 2
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
 => 3 = 1 + 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 4 = 2 + 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 2 = 0 + 2
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 2 = 0 + 2
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 3 = 1 + 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 3 = 1 + 2
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 4 = 2 + 2
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 5 = 3 + 2
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 4 = 2 + 2
[2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 5 = 3 + 2
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 3 = 1 + 2
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 3 = 1 + 2
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 4 = 2 + 2
[3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 5 = 3 + 2
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 4 = 2 + 2
[3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 5 = 3 + 2
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 3 = 1 + 2
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 4 = 2 + 2
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 5 = 3 + 2
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 4 = 2 + 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 5 = 3 + 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 0 + 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 0 + 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 0 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 0 + 2
[1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => 2 = 0 + 2
[1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 0 + 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 0 + 2
Description
The position of the first down step of a Dyck path.
Matching statistic: St000141
(load all 2 compositions to match this statistic)
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [1,2] => 0
[2,1] => [2,1] => [1,2] => [2,1] => 1
[1,2,3] => [1,3,2] => [3,2,1] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,2,1] => [1,2,3] => 0
[2,1,3] => [2,1,3] => [1,3,2] => [2,3,1] => 1
[2,3,1] => [2,3,1] => [1,2,3] => [3,2,1] => 2
[3,1,2] => [3,1,2] => [3,1,2] => [2,1,3] => 1
[3,2,1] => [3,2,1] => [2,1,3] => [3,1,2] => 2
[1,2,3,4] => [1,4,3,2] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,4,3] => [1,4,3,2] => [4,3,2,1] => [1,2,3,4] => 0
[1,3,2,4] => [1,4,3,2] => [4,3,2,1] => [1,2,3,4] => 0
[1,3,4,2] => [1,4,3,2] => [4,3,2,1] => [1,2,3,4] => 0
[1,4,2,3] => [1,4,3,2] => [4,3,2,1] => [1,2,3,4] => 0
[1,4,3,2] => [1,4,3,2] => [4,3,2,1] => [1,2,3,4] => 0
[2,1,3,4] => [2,1,4,3] => [1,4,3,2] => [2,3,4,1] => 1
[2,1,4,3] => [2,1,4,3] => [1,4,3,2] => [2,3,4,1] => 1
[2,3,1,4] => [2,4,1,3] => [1,4,2,3] => [3,2,4,1] => 2
[2,3,4,1] => [2,4,3,1] => [1,3,2,4] => [4,2,3,1] => 3
[2,4,1,3] => [2,4,1,3] => [1,4,2,3] => [3,2,4,1] => 2
[2,4,3,1] => [2,4,3,1] => [1,3,2,4] => [4,2,3,1] => 3
[3,1,2,4] => [3,1,4,2] => [4,1,3,2] => [2,3,1,4] => 1
[3,1,4,2] => [3,1,4,2] => [4,1,3,2] => [2,3,1,4] => 1
[3,2,1,4] => [3,2,1,4] => [2,1,4,3] => [3,4,1,2] => 2
[3,2,4,1] => [3,2,4,1] => [2,1,3,4] => [4,3,1,2] => 3
[3,4,1,2] => [3,4,1,2] => [4,1,2,3] => [3,2,1,4] => 2
[3,4,2,1] => [3,4,2,1] => [3,1,2,4] => [4,2,1,3] => 3
[4,1,2,3] => [4,1,3,2] => [4,3,1,2] => [2,1,3,4] => 1
[4,1,3,2] => [4,1,3,2] => [4,3,1,2] => [2,1,3,4] => 1
[4,2,1,3] => [4,2,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[4,2,3,1] => [4,2,3,1] => [2,3,1,4] => [4,1,3,2] => 3
[4,3,1,2] => [4,3,1,2] => [4,2,1,3] => [3,1,2,4] => 2
[4,3,2,1] => [4,3,2,1] => [3,2,1,4] => [4,1,2,3] => 3
[1,2,3,4,5] => [1,5,4,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,5,4,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,5,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,5,4,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,5,4,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,3,5,4,2] => [1,5,4,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,4,2,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,4,2,5,3] => [1,5,4,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,4,3,2,5] => [1,5,4,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 0
Description
The maximum drop size of a permutation. The maximum drop size of a permutation \pi of [n]=\{1,2,\ldots, n\} is defined to be the maximum value of i-\pi(i).
The following 33 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000147The largest part of an integer partition. St000316The number of non-left-to-right-maxima of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000069The number of maximal elements of a poset. St000297The number of leading ones in a binary word. St000382The first part of an integer composition. 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. St000734The last entry in the first row of a standard tableau. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module S_0 in the special CNakayama algebra corresponding to the Dyck path. St000738The first entry in the last row of a standard tableau. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000653The last descent of a permutation. St000989The number of final rises of a permutation. St001480The number of simple summands of the module J^2/J^3. St000061The number of nodes on the left branch of a binary tree. St000840The number of closers smaller than the largest opener in a perfect matching. St001434The number of negative sum pairs of a signed permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. 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. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St000454The largest eigenvalue of a graph if it is integral.