- FindStat

Your data matches 40 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000012
(load all 5 compositions to match this statistic)

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1,0]
 => 0
[.,[.,.]]
 => [1,1,0,0]
 => 1
[[.,.],.]
 => [1,0,1,0]
 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => 3
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => 2
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => 6
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => 5
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => 4
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => 4
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => 3
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 3
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 2
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 2
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => 3
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => 2
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => 10
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 9
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 8
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => 8
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => 7
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 7
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => 6
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => 6
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => 5
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => 7
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => 6
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => 5
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => 5
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 5
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 4
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 4
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 4
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 4
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1

Description

The area of a Dyck path. This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic. 1. Dyck paths are bijection with '''area sequences'''

$(a_1,\ldots,a_n)$ such that

$a_1 = 0, a_{k+1} \leq a_k + 1$ . 2. The generating function

$\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence

$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$ 3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of

$q,t$ -Catalan numbers.

Matching statistic: St000008

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1] => [1] => [1] => 0
[.,[.,.]]
 => [2,1] => [2,1] => [1,1] => 1
[[.,.],.]
 => [1,2] => [1,2] => [2] => 0
[.,[.,[.,.]]]
 => [3,2,1] => [3,2,1] => [1,1,1] => 3
[.,[[.,.],.]]
 => [2,3,1] => [2,3,1] => [2,1] => 2
[[.,.],[.,.]]
 => [1,3,2] => [3,1,2] => [1,2] => 1
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [1,2] => 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [3] => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,3,2,1] => [1,1,1,1] => 6
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,4,2,1] => [2,1,1] => 5
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [4,2,3,1] => [1,2,1] => 4
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [3,2,4,1] => [1,2,1] => 4
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [2,3,4,1] => [3,1] => 3
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [4,3,1,2] => [1,1,2] => 3
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [3,4,1,2] => [2,2] => 2
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,4,1,3] => [2,2] => 2
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [4,1,2,3] => [1,3] => 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1,4] => [1,1,2] => 3
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [2,3,1,4] => [2,2] => 2
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [3,1,2,4] => [1,3] => 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => [1,3] => 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [4] => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1] => 10
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,5,3,2,1] => [2,1,1,1] => 9
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [5,3,4,2,1] => [1,2,1,1] => 8
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [4,3,5,2,1] => [1,2,1,1] => 8
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,4,5,2,1] => [3,1,1] => 7
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [5,4,2,3,1] => [1,1,2,1] => 7
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [4,5,2,3,1] => [2,2,1] => 6
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [3,5,2,4,1] => [2,2,1] => 6
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [5,2,3,4,1] => [1,3,1] => 5
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [4,3,2,5,1] => [1,1,2,1] => 7
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [3,4,2,5,1] => [2,2,1] => 6
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [4,2,3,5,1] => [1,3,1] => 5
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [3,2,4,5,1] => [1,3,1] => 5
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [2,3,4,5,1] => [4,1] => 4
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,2] => 6
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [4,5,3,1,2] => [2,1,2] => 5
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [5,3,4,1,2] => [1,2,2] => 4
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [4,3,5,1,2] => [1,2,2] => 4
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [3,4,5,1,2] => [3,2] => 3
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [5,2,4,1,3] => [1,2,2] => 4
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,4,5,1,3] => [3,2] => 3
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => 3
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [4,5,1,2,3] => [2,3] => 2
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,2,5,1,4] => [1,2,2] => 4
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [2,3,5,1,4] => [3,2] => 3
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [3,5,1,2,4] => [2,3] => 2
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,5,1,3,4] => [2,3] => 2
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [5,1,2,3,4] => [1,4] => 1
[[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,6,5,8,7,4,3,2] => [6,8,5,7,4,3,1,2] => ? => ? = 17
[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
 => [1,5,4,8,7,6,3,2] => [8,5,7,4,6,3,1,2] => ? => ? = 15
[[.,.],[.,[[.,[.,.]],[[.,.],.]]]]
 => [1,5,4,7,8,6,3,2] => [5,7,8,4,6,3,1,2] => ? => ? = 14
[[.,.],[.,[[.,[.,[.,.]]],[.,.]]]]
 => [1,6,5,4,8,7,3,2] => [6,5,8,4,7,3,1,2] => ? => ? = 15
[[.,.],[.,[[.,[[.,.],.]],[.,.]]]]
 => [1,5,6,4,8,7,3,2] => ? => ? => ? = 14
[[.,.],[.,[[[.,[.,.]],.],[.,.]]]]
 => [1,5,4,6,8,7,3,2] => [5,8,4,6,7,3,1,2] => ? => ? = 13
[[.,.],[.,[[[[.,.],.],.],[.,.]]]]
 => [1,4,5,6,8,7,3,2] => ? => ? => ? = 12
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
 => [1,5,7,6,4,8,3,2] => ? => ? => ? = 15
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
 => [1,5,6,7,4,8,3,2] => [5,6,7,4,8,3,1,2] => ? => ? = 14
[[.,.],[.,[[[.,[.,.]],[.,.]],.]]]
 => [1,5,4,7,6,8,3,2] => [5,7,4,6,8,3,1,2] => ? => ? = 13
[[.,.],[.,[[[[.,.],[.,.]],.],.]]]
 => [1,4,6,5,7,8,3,2] => ? => ? => ? = 12
[[.,.],[[.,.],[.,[[.,.],[.,.]]]]]
 => [1,3,6,8,7,5,4,2] => [8,6,7,5,3,4,1,2] => ? => ? = 14
[[.,.],[[[.,.],[.,.]],[.,[.,.]]]]
 => [1,3,5,4,8,7,6,2] => [8,5,7,3,4,6,1,2] => ? => ? = 10
[[.,.],[[[.,[.,.]],.],[.,[.,.]]]]
 => [1,4,3,5,8,7,6,2] => [8,4,7,3,5,6,1,2] => ? => ? = 10
[[.,.],[[[.,.],[.,[.,.]]],[.,.]]]
 => [1,3,6,5,4,8,7,2] => [6,5,8,3,4,7,1,2] => ? => ? = 10
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
 => [1,3,5,4,6,8,7,2] => [5,8,3,4,6,7,1,2] => ? => ? = 8
[[.,.],[[[.,.],[.,[.,[.,.]]]],.]]
 => [1,3,7,6,5,4,8,2] => ? => ? => ? = 12
[[.,.],[[[.,.],[.,[[.,.],.]]],.]]
 => [1,3,6,7,5,4,8,2] => ? => ? => ? = 11
[[.,.],[[[.,[.,.]],[.,[.,.]]],.]]
 => [1,4,3,7,6,5,8,2] => ? => ? => ? = 10
[[.,.],[[[.,[[.,.],.]],[.,.]],.]]
 => [1,4,5,3,7,6,8,2] => ? => ? => ? = 9
[[.,.],[[[.,[.,[[.,.],.]]],.],.]]
 => [1,5,6,4,3,7,8,2] => ? => ? => ? = 11
[[.,.],[[[.,[[.,.],[.,.]]],.],.]]
 => [1,4,6,5,3,7,8,2] => [6,4,5,3,7,8,1,2] => ? => ? = 10
[[.,.],[[[[.,.],[.,[.,.]]],.],.]]
 => [1,3,6,5,4,7,8,2] => ? => ? => ? = 9
[[.,[.,.]],[.,[.,[.,[[.,.],.]]]]]
 => [2,1,7,8,6,5,4,3] => [7,8,6,5,2,4,1,3] => ? => ? = 15
[[.,[.,.]],[.,[.,[[.,.],[.,.]]]]]
 => [2,1,6,8,7,5,4,3] => [8,6,7,5,2,4,1,3] => ? => ? = 14
[[.,[.,.]],[[.,[[[.,.],.],.]],.]]
 => [2,1,5,6,7,4,8,3] => [5,6,7,2,4,8,1,3] => ? => ? = 9
[[.,[.,.]],[[[.,.],[[.,.],.]],.]]
 => [2,1,4,6,7,5,8,3] => [6,7,2,4,5,8,1,3] => ? => ? = 8
[[[.,.],.],[.,[.,[[.,.],[.,.]]]]]
 => [1,2,6,8,7,5,4,3] => [8,6,7,5,4,1,2,3] => ? => ? = 13
[[[.,.],.],[.,[[.,.],[.,[.,.]]]]]
 => [1,2,5,8,7,6,4,3] => [8,7,5,6,4,1,2,3] => ? => ? = 12
[[[.,.],.],[[.,.],[[.,[.,.]],.]]]
 => [1,2,4,7,6,8,5,3] => [7,6,8,4,5,1,2,3] => ? => ? = 9
[[[.,.],.],[[.,[.,.]],[.,[.,.]]]]
 => [1,2,5,4,8,7,6,3] => [8,5,7,4,6,1,2,3] => ? => ? = 9
[[[.,.],.],[[.,[[.,.],.]],[.,.]]]
 => [1,2,5,6,4,8,7,3] => [5,6,8,4,7,1,2,3] => ? => ? = 8
[[[.,.],.],[[[.,[.,.]],.],[.,.]]]
 => [1,2,5,4,6,8,7,3] => [5,8,4,6,7,1,2,3] => ? => ? = 7
[[[.,.],.],[[[[.,.],.],[.,.]],.]]
 => [1,2,4,5,7,6,8,3] => ? => ? => ? = 6
[[[.,[.,.]],.],[[.,[[.,.],.]],.]]
 => [2,1,3,6,7,5,8,4] => [6,7,2,5,8,1,3,4] => ? => ? = 7
[[[.,.],[[.,.],.]],[[.,.],[.,.]]]
 => [1,3,4,2,6,8,7,5] => [8,3,4,6,7,1,2,5] => ? => ? = 6
[[[[.,.],[.,.]],.],[.,[[.,.],.]]]
 => [1,3,2,4,7,8,6,5] => [7,8,3,6,1,2,4,5] => ? => ? = 6
[[[[.,.],[.,.]],.],[[.,.],[.,.]]]
 => [1,3,2,4,6,8,7,5] => [8,3,6,7,1,2,4,5] => ? => ? = 5
[[[.,.],[[.,.],[.,.]]],[.,[.,.]]]
 => [1,3,5,4,2,8,7,6] => [5,8,3,4,7,1,2,6] => ? => ? = 7
[[[[.,.],[[.,.],.]],.],[.,[.,.]]]
 => [1,3,4,2,5,8,7,6] => [8,3,4,7,1,2,5,6] => ? => ? = 5
[[[.,.],[.,[[.,.],[.,.]]]],[.,.]]
 => [1,4,6,5,3,2,8,7] => ? => ? => ? = 9
[[[.,.],[[.,[.,.]],[.,.]]],[.,.]]
 => [1,4,3,6,5,2,8,7] => ? => ? => ? = 7
[[[.,.],[[.,[.,[.,.]]],.]],[.,.]]
 => [1,5,4,3,6,2,8,7] => ? => ? => ? = 8
[[[.,.],[[.,[[.,.],.]],.]],[.,.]]
 => [1,4,5,3,6,2,8,7] => ? => ? => ? = 7
[[[.,.],[[[.,.],[.,.]],.]],[.,.]]
 => [1,3,5,4,6,2,8,7] => ? => ? => ? = 6
[[[.,.],[[[.,[.,.]],.],.]],[.,.]]
 => [1,4,3,5,6,2,8,7] => ? => ? => ? = 6
[[[[.,.],[.,[[.,.],.]]],.],[.,.]]
 => [1,4,5,3,2,6,8,7] => [4,5,3,8,1,2,6,7] => ? => ? = 6
[[[.,.],[.,[[.,.],[[.,.],.]]]],.]
 => [1,4,6,7,5,3,2,8] => [6,7,4,5,3,1,2,8] => ? => ? = 11
[[[.,.],[.,[[.,[.,[.,.]]],.]]],.]
 => [1,6,5,4,7,3,2,8] => ? => ? => ? = 12
[[[.,.],[.,[[.,[[.,.],.]],.]]],.]
 => [1,5,6,4,7,3,2,8] => [5,6,4,7,3,1,2,8] => ? => ? = 11

Description

The major index of the composition. The descents of a composition

$[c_1,c_2,\dots,c_k]$ are the partial sums

$c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$ , excluding the sum of all parts. The major index of a composition is the sum of its descents. For details about the major index see [[Permutations/Descents-Major]].

Matching statistic: St001161

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1,0]
 => [1,0]
 => [1,0]
 => 0
[.,[.,.]]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
[[.,.],.]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 6
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 5
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 4
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 4
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 10
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 9
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 8
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 8
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 7
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 7
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 6
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 6
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 5
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 7
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 6
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 5
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 5
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 6
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 5
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 18
[[.,.],[.,[.,[[[[.,.],.],.],.]]]]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 15
[[.,.],[.,[[.,.],[[[.,.],.],.]]]]
 => [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[[.,.],[.,[[.,[.,.]],[[.,.],.]]]]
 => [1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[[.,.],[.,[[.,[[.,.],.]],[.,.]]]]
 => [1,0,1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
 => [1,0,1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[[.,.],[.,[[[[[.,.],.],.],.],.]]]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[[.,.],[[.,.],[.,[[[.,.],.],.]]]]
 => [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 13
[[.,.],[[.,.],[[[[.,.],.],.],.]]]
 => [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[[.,.],[[.,[.,.]],[[[.,.],.],.]]]
 => [1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[[.,.],[[[.,.],.],[[[.,.],.],.]]]
 => [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[.,[[.,.],.]],[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[[.,.],[[[.,.],[.,.]],[[.,.],.]]]
 => [1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[[.,[.,.]],.],[[.,.],.]]]
 => [1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[.,[[[.,.],.],.]],[.,.]]]
 => [1,0,1,1,1,0,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[[.,.],[[[.,.],[[.,.],.]],[.,.]]]
 => [1,0,1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[[.,[.,.]],[.,.]],[.,.]]]
 => [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[[.,[[.,.],.]],.],[.,.]]]
 => [1,0,1,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[.,[.,[[[.,.],.],.]]],.]]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 13
[[.,.],[[.,[[[[.,.],.],.],.]],.]]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[[.,.],[[[.,.],[[[.,.],.],.]],.]]
 => [1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[[.,[.,.]],[[.,.],.]],.]]
 => [1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[[.,[[.,.],.]],[.,.]],.]]
 => [1,0,1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[[.,[[[.,.],.],.]],.],.]]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[[[[[.,.],.],.],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[[.,[.,.]],[.,[.,[[[.,.],.],.]]]]
 => [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 13
[[.,[.,.]],[.,[[[[.,.],.],.],.]]]
 => [1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[[.,[.,.]],[[.,.],[[[.,.],.],.]]]
 => [1,1,0,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,[.,.]],[[.,[.,.]],[[.,.],.]]]
 => [1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,[.,.]],[[.,[[.,.],.]],[.,.]]]
 => [1,1,0,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,[.,.]],[[.,[[[.,.],.],.]],.]]
 => [1,1,0,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[[[.,.],.],[.,[.,[[[.,.],.],.]]]]
 => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 12
[[[.,.],.],[.,[[[[.,.],.],.],.]]]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[[[.,.],.],[[.,.],[[[.,.],.],.]]]
 => [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[[[.,.],.],[[.,[.,.]],[[.,.],.]]]
 => [1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[[[.,.],.],[[.,[[.,.],.]],[.,.]]]
 => [1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[[[.,.],.],[[.,[[[.,.],.],.]],.]]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[[[.,.],.],[[[[[.,.],.],.],.],.]]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[[[.,.],[.,.]],[.,[[[.,.],.],.]]]
 => [1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[[[.,.],[.,.]],[[[[.,.],.],.],.]]
 => [1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[[[.,[.,.]],.],[.,[[[.,.],.],.]]]
 => [1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[[[.,[.,.]],.],[[[[.,.],.],.],.]]
 => [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[[[[.,.],.],.],[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7
[[[[.,.],.],.],[[[[.,.],.],.],.]]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[[[.,.],[.,[.,.]]],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[[[.,.],[[.,.],.]],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[[[.,[.,.]],[.,.]],[[[.,.],.],.]]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[[[[.,.],.],[.,.]],[[[.,.],.],.]]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[[[.,[[.,.],.]],.],[[[.,.],.],.]]
 => [1,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5

Description

The major index north count of a Dyck path. The descent set

$\operatorname{des}(D)$ of a Dyck path

$D = D_1 \cdots D_{2n}$ with

$D_i \in \{N,E\}$ is given by all indices

$i$ such that

$D_i = E$ and

$D_{i+1} = N$ . This is, the positions of the valleys of

$D$ . The '''major index''' of a Dyck path is then the sum of the positions of the valleys,

$\sum_{i \in \operatorname{des}(D)} i$ , see [[St000027]]. The '''major index north count''' is given by

$\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = N\}$ .

Matching statistic: St000947

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1,0]
 => [1,0]
 => [1,0]
 => ? = 0
[.,[.,.]]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
[[.,.],.]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 6
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 5
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 4
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 4
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 10
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 9
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 8
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 8
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 7
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 7
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 6
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 6
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 5
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 7
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 6
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 5
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 5
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 6
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 5
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 6
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 18
[[.,.],[.,[.,[[[[.,.],.],.],.]]]]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 15
[[.,.],[.,[[.,.],[[[.,.],.],.]]]]
 => [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[[.,.],[.,[[.,[.,.]],[[.,.],.]]]]
 => [1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[[.,.],[.,[[.,[[.,.],.]],[.,.]]]]
 => [1,0,1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
 => [1,0,1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[[.,.],[.,[[[[[.,.],.],.],.],.]]]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[[.,.],[[.,.],[.,[[[.,.],.],.]]]]
 => [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 13
[[.,.],[[.,.],[[[[.,.],.],.],.]]]
 => [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[[.,.],[[.,[.,.]],[[[.,.],.],.]]]
 => [1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[[.,.],[[[.,.],.],[[[.,.],.],.]]]
 => [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[.,[[.,.],.]],[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[[.,.],[[[.,.],[.,.]],[[.,.],.]]]
 => [1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[[.,[.,.]],.],[[.,.],.]]]
 => [1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[.,[[[.,.],.],.]],[.,.]]]
 => [1,0,1,1,1,0,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[[.,.],[[[.,.],[[.,.],.]],[.,.]]]
 => [1,0,1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[[.,[.,.]],[.,.]],[.,.]]]
 => [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[[.,[[.,.],.]],.],[.,.]]]
 => [1,0,1,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[.,[.,[[[.,.],.],.]]],.]]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 13
[[.,.],[[.,[[[[.,.],.],.],.]],.]]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[[.,.],[[[.,.],[[[.,.],.],.]],.]]
 => [1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[[.,[.,.]],[[.,.],.]],.]]
 => [1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[[.,[[.,.],.]],[.,.]],.]]
 => [1,0,1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[[.,[[[.,.],.],.]],.],.]]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,.],[[[[[[.,.],.],.],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[[.,[.,.]],[.,[.,[[[.,.],.],.]]]]
 => [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 13
[[.,[.,.]],[.,[[[[.,.],.],.],.]]]
 => [1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[[.,[.,.]],[[.,.],[[[.,.],.],.]]]
 => [1,1,0,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,[.,.]],[[.,[.,.]],[[.,.],.]]]
 => [1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,[.,.]],[[.,[[.,.],.]],[.,.]]]
 => [1,1,0,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,[.,.]],[[.,[[[.,.],.],.]],.]]
 => [1,1,0,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[[[.,.],.],[.,[.,[[[.,.],.],.]]]]
 => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 12
[[[.,.],.],[.,[[[[.,.],.],.],.]]]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[[[.,.],.],[[.,.],[[[.,.],.],.]]]
 => [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[[[.,.],.],[[.,[.,.]],[[.,.],.]]]
 => [1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[[[.,.],.],[[.,[[.,.],.]],[.,.]]]
 => [1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[[[.,.],.],[[.,[[[.,.],.],.]],.]]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[[[.,.],.],[[[[[.,.],.],.],.],.]]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[[[.,.],[.,.]],[.,[[[.,.],.],.]]]
 => [1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[[[.,.],[.,.]],[[[[.,.],.],.],.]]
 => [1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[[[.,[.,.]],.],[.,[[[.,.],.],.]]]
 => [1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[[[.,[.,.]],.],[[[[.,.],.],.],.]]
 => [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[[[[.,.],.],.],[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7
[[[[.,.],.],.],[[[[.,.],.],.],.]]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[[[.,.],[.,[.,.]]],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[[[.,.],[[.,.],.]],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[[[.,[.,.]],[.,.]],[[[.,.],.],.]]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[[[[.,.],.],[.,.]],[[[.,.],.],.]]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4

Description

The major index east count of a Dyck path. The descent set

$\operatorname{des}(D)$ of a Dyck path

$D = D_1 \cdots D_{2n}$ with

$D_i \in \{N,E\}$ is given by all indices

$i$ such that

$D_i = E$ and

$D_{i+1} = N$ . This is, the positions of the valleys of

$D$ . The '''major index''' of a Dyck path is then the sum of the positions of the valleys,

$\sum_{i \in \operatorname{des}(D)} i$ , see [[St000027]]. The '''major index east count''' is given by

$\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$ .

Matching statistic: St000391

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1] => [1] =>  => ? = 0
[.,[.,.]]
 => [2,1] => [2,1] => 1 => 1
[[.,.],.]
 => [1,2] => [1,2] => 0 => 0
[.,[.,[.,.]]]
 => [3,2,1] => [3,2,1] => 11 => 3
[.,[[.,.],.]]
 => [2,3,1] => [2,3,1] => 01 => 2
[[.,.],[.,.]]
 => [1,3,2] => [3,1,2] => 10 => 1
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => 10 => 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 00 => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,3,2,1] => 111 => 6
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,4,2,1] => 011 => 5
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [4,2,3,1] => 101 => 4
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [3,2,4,1] => 101 => 4
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [2,3,4,1] => 001 => 3
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [4,3,1,2] => 110 => 3
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [3,4,1,2] => 010 => 2
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,4,1,3] => 010 => 2
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [4,1,2,3] => 100 => 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1,4] => 110 => 3
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [2,3,1,4] => 010 => 2
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [3,1,2,4] => 100 => 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => 100 => 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 000 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,4,3,2,1] => 1111 => 10
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,5,3,2,1] => 0111 => 9
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [5,3,4,2,1] => 1011 => 8
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [4,3,5,2,1] => 1011 => 8
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,4,5,2,1] => 0011 => 7
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [5,4,2,3,1] => 1101 => 7
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [4,5,2,3,1] => 0101 => 6
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [3,5,2,4,1] => 0101 => 6
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [5,2,3,4,1] => 1001 => 5
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [4,3,2,5,1] => 1101 => 7
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [3,4,2,5,1] => 0101 => 6
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [4,2,3,5,1] => 1001 => 5
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [3,2,4,5,1] => 1001 => 5
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [2,3,4,5,1] => 0001 => 4
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [5,4,3,1,2] => 1110 => 6
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [4,5,3,1,2] => 0110 => 5
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [5,3,4,1,2] => 1010 => 4
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [4,3,5,1,2] => 1010 => 4
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [3,4,5,1,2] => 0010 => 3
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [5,2,4,1,3] => 1010 => 4
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,4,5,1,3] => 0010 => 3
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [5,4,1,2,3] => 1100 => 3
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [4,5,1,2,3] => 0100 => 2
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,2,5,1,4] => 1010 => 4
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [2,3,5,1,4] => 0010 => 3
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [3,5,1,2,4] => 0100 => 2
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,5,1,3,4] => 0100 => 2
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [5,1,2,3,4] => 1000 => 1
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,3,2,1,5] => 1110 => 6
[[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,6,5,8,7,4,3,2] => [6,8,5,7,4,3,1,2] => ? => ? = 17
[[.,.],[.,[.,[[[.,.],.],[.,.]]]]]
 => [1,5,6,8,7,4,3,2] => [8,5,6,7,4,3,1,2] => ? => ? = 16
[[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
 => [1,4,7,6,8,5,3,2] => [7,6,8,4,5,3,1,2] => ? => ? = 15
[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
 => [1,5,4,8,7,6,3,2] => [8,5,7,4,6,3,1,2] => ? => ? = 15
[[.,.],[.,[[.,[.,.]],[[.,.],.]]]]
 => [1,5,4,7,8,6,3,2] => [5,7,8,4,6,3,1,2] => ? => ? = 14
[[.,.],[.,[[[.,.],.],[.,[.,.]]]]]
 => [1,4,5,8,7,6,3,2] => [8,7,4,5,6,3,1,2] => ? => ? = 14
[[.,.],[.,[[[.,.],.],[[.,.],.]]]]
 => [1,4,5,7,8,6,3,2] => [7,8,4,5,6,3,1,2] => ? => ? = 13
[[.,.],[.,[[.,[.,[.,.]]],[.,.]]]]
 => [1,6,5,4,8,7,3,2] => [6,5,8,4,7,3,1,2] => ? => ? = 15
[[.,.],[.,[[.,[[.,.],.]],[.,.]]]]
 => [1,5,6,4,8,7,3,2] => ? => ? => ? = 14
[[.,.],[.,[[[.,.],[.,.]],[.,.]]]]
 => [1,4,6,5,8,7,3,2] => [6,8,4,5,7,3,1,2] => ? => ? = 13
[[.,.],[.,[[[.,[.,.]],.],[.,.]]]]
 => [1,5,4,6,8,7,3,2] => [5,8,4,6,7,3,1,2] => ? => ? = 13
[[.,.],[.,[[[[.,.],.],.],[.,.]]]]
 => [1,4,5,6,8,7,3,2] => ? => ? => ? = 12
[[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
 => [1,6,7,5,4,8,3,2] => [6,7,5,4,8,3,1,2] => ? => ? = 16
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
 => [1,5,7,6,4,8,3,2] => ? => ? => ? = 15
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
 => [1,5,6,7,4,8,3,2] => [5,6,7,4,8,3,1,2] => ? => ? = 14
[[.,.],[.,[[[.,[.,.]],[.,.]],.]]]
 => [1,5,4,7,6,8,3,2] => [5,7,4,6,8,3,1,2] => ? => ? = 13
[[.,.],[.,[[[[.,.],.],[.,.]],.]]]
 => [1,4,5,7,6,8,3,2] => [7,4,5,6,8,3,1,2] => ? => ? = 12
[[.,.],[.,[[[[.,.],[.,.]],.],.]]]
 => [1,4,6,5,7,8,3,2] => ? => ? => ? = 12
[[.,.],[[.,.],[.,[[.,.],[.,.]]]]]
 => [1,3,6,8,7,5,4,2] => [8,6,7,5,3,4,1,2] => ? => ? = 14
[[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
 => [1,4,3,6,8,7,5,2] => [8,4,6,7,3,5,1,2] => ? => ? = 11
[[.,.],[[[.,.],.],[.,[[.,.],.]]]]
 => [1,3,4,7,8,6,5,2] => [7,8,6,3,4,5,1,2] => ? => ? = 11
[[.,.],[[[.,.],.],[[.,[.,.]],.]]]
 => [1,3,4,7,6,8,5,2] => [7,6,8,3,4,5,1,2] => ? => ? = 10
[[.,.],[[[.,.],.],[[[.,.],.],.]]]
 => [1,3,4,6,7,8,5,2] => [6,7,8,3,4,5,1,2] => ? => ? = 9
[[.,.],[[.,[[.,.],.]],[[.,.],.]]]
 => [1,4,5,3,7,8,6,2] => [4,5,7,8,3,6,1,2] => ? => ? = 10
[[.,.],[[[.,.],[.,.]],[.,[.,.]]]]
 => [1,3,5,4,8,7,6,2] => [8,5,7,3,4,6,1,2] => ? => ? = 10
[[.,.],[[[.,[.,.]],.],[.,[.,.]]]]
 => [1,4,3,5,8,7,6,2] => [8,4,7,3,5,6,1,2] => ? => ? = 10
[[.,.],[[.,[[.,[.,.]],.]],[.,.]]]
 => [1,5,4,6,3,8,7,2] => [5,4,6,8,3,7,1,2] => ? => ? = 11
[[.,.],[[.,[[[.,.],.],.]],[.,.]]]
 => [1,4,5,6,3,8,7,2] => [4,5,6,8,3,7,1,2] => ? => ? = 10
[[.,.],[[[.,.],[.,[.,.]]],[.,.]]]
 => [1,3,6,5,4,8,7,2] => [6,5,8,3,4,7,1,2] => ? => ? = 10
[[.,.],[[[.,.],[[.,.],.]],[.,.]]]
 => [1,3,5,6,4,8,7,2] => [5,6,8,3,4,7,1,2] => ? => ? = 9
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
 => [1,3,5,4,6,8,7,2] => [5,8,3,4,6,7,1,2] => ? => ? = 8
[[.,.],[[.,[.,[.,[[.,.],.]]]],.]]
 => [1,6,7,5,4,3,8,2] => [6,7,5,4,3,8,1,2] => ? => ? = 15
[[.,.],[[.,[[.,.],[.,[.,.]]]],.]]
 => [1,4,7,6,5,3,8,2] => [7,6,4,5,3,8,1,2] => ? => ? = 13
[[.,.],[[.,[[[.,.],.],[.,.]]],.]]
 => [1,4,5,7,6,3,8,2] => [7,4,5,6,3,8,1,2] => ? => ? = 11
[[.,.],[[.,[[.,[.,[.,.]]],.]],.]]
 => [1,6,5,4,7,3,8,2] => [6,5,4,7,3,8,1,2] => ? => ? = 13
[[.,.],[[.,[[[.,.],[.,.]],.]],.]]
 => [1,4,6,5,7,3,8,2] => [6,4,5,7,3,8,1,2] => ? => ? = 11
[[.,.],[[[.,.],[.,[.,[.,.]]]],.]]
 => [1,3,7,6,5,4,8,2] => ? => ? => ? = 12
[[.,.],[[[.,.],[.,[[.,.],.]]],.]]
 => [1,3,6,7,5,4,8,2] => ? => ? => ? = 11
[[.,.],[[[.,.],[[.,[.,.]],.]],.]]
 => [1,3,6,5,7,4,8,2] => [6,5,7,3,4,8,1,2] => ? => ? = 10
[[.,.],[[[.,[.,.]],[.,[.,.]]],.]]
 => [1,4,3,7,6,5,8,2] => ? => ? => ? = 10
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
 => [1,3,4,6,7,5,8,2] => [6,7,3,4,5,8,1,2] => ? => ? = 8
[[.,.],[[[.,[[.,.],.]],[.,.]],.]]
 => [1,4,5,3,7,6,8,2] => ? => ? => ? = 9
[[.,.],[[[[.,[.,.]],.],[.,.]],.]]
 => [1,4,3,5,7,6,8,2] => [4,7,3,5,6,8,1,2] => ? => ? = 8
[[.,.],[[[.,[.,[[.,.],.]]],.],.]]
 => [1,5,6,4,3,7,8,2] => ? => ? => ? = 11
[[.,.],[[[.,[[.,.],[.,.]]],.],.]]
 => [1,4,6,5,3,7,8,2] => [6,4,5,3,7,8,1,2] => ? => ? = 10
[[.,.],[[[[.,.],[.,[.,.]]],.],.]]
 => [1,3,6,5,4,7,8,2] => ? => ? => ? = 9
[[.,[.,.]],[.,[.,[.,[[.,.],.]]]]]
 => [2,1,7,8,6,5,4,3] => [7,8,6,5,2,4,1,3] => ? => ? = 15
[[.,[.,.]],[.,[.,[[.,.],[.,.]]]]]
 => [2,1,6,8,7,5,4,3] => [8,6,7,5,2,4,1,3] => ? => ? = 14
[[.,[.,.]],[.,[[[.,.],.],[.,.]]]]
 => [2,1,5,6,8,7,4,3] => [8,5,6,7,2,4,1,3] => ? => ? = 11

Description

The sum of the positions of the ones in a binary word.

Matching statistic: St000330
(load all 2 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1] => [1] => [[1]]
 => 0
[.,[.,.]]
 => [2,1] => [2,1] => [[1],[2]]
 => 1
[[.,.],.]
 => [1,2] => [1,2] => [[1,2]]
 => 0
[.,[.,[.,.]]]
 => [3,2,1] => [3,2,1] => [[1],[2],[3]]
 => 3
[.,[[.,.],.]]
 => [2,3,1] => [2,3,1] => [[1,2],[3]]
 => 2
[[.,.],[.,.]]
 => [1,3,2] => [3,1,2] => [[1,3],[2]]
 => 1
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [[1,3],[2]]
 => 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [[1,2,3]]
 => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
 => 6
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
 => 5
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [4,2,3,1] => [[1,3],[2],[4]]
 => 4
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [3,2,4,1] => [[1,3],[2],[4]]
 => 4
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [2,3,4,1] => [[1,2,3],[4]]
 => 3
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [4,3,1,2] => [[1,4],[2],[3]]
 => 3
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [3,4,1,2] => [[1,2],[3,4]]
 => 2
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,4,1,3] => [[1,2],[3,4]]
 => 2
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [4,1,2,3] => [[1,3,4],[2]]
 => 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
 => 3
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [2,3,1,4] => [[1,2,4],[3]]
 => 2
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [3,1,2,4] => [[1,3,4],[2]]
 => 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
 => 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
 => 10
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,5,3,2,1] => [[1,2],[3],[4],[5]]
 => 9
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [5,3,4,2,1] => [[1,3],[2],[4],[5]]
 => 8
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [4,3,5,2,1] => [[1,3],[2],[4],[5]]
 => 8
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,4,5,2,1] => [[1,2,3],[4],[5]]
 => 7
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [5,4,2,3,1] => [[1,4],[2],[3],[5]]
 => 7
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [4,5,2,3,1] => [[1,2],[3,4],[5]]
 => 6
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [3,5,2,4,1] => [[1,2],[3,4],[5]]
 => 6
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [5,2,3,4,1] => [[1,3,4],[2],[5]]
 => 5
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [4,3,2,5,1] => [[1,4],[2],[3],[5]]
 => 7
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [3,4,2,5,1] => [[1,2,4],[3],[5]]
 => 6
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [4,2,3,5,1] => [[1,3,4],[2],[5]]
 => 5
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [3,2,4,5,1] => [[1,3,4],[2],[5]]
 => 5
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [2,3,4,5,1] => [[1,2,3,4],[5]]
 => 4
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [5,4,3,1,2] => [[1,5],[2],[3],[4]]
 => 6
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [4,5,3,1,2] => [[1,2],[3,5],[4]]
 => 5
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [5,3,4,1,2] => [[1,3],[2,5],[4]]
 => 4
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [4,3,5,1,2] => [[1,3],[2,5],[4]]
 => 4
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [3,4,5,1,2] => [[1,2,3],[4,5]]
 => 3
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [5,2,4,1,3] => [[1,3],[2,5],[4]]
 => 4
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,4,5,1,3] => [[1,2,3],[4,5]]
 => 3
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [5,4,1,2,3] => [[1,4,5],[2],[3]]
 => 3
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [4,5,1,2,3] => [[1,2,5],[3,4]]
 => 2
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,2,5,1,4] => [[1,3],[2,5],[4]]
 => 4
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [2,3,5,1,4] => [[1,2,3],[4,5]]
 => 3
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [3,5,1,2,4] => [[1,2,5],[3,4]]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,5,1,3,4] => [[1,2,5],[3,4]]
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [5,1,2,3,4] => [[1,3,4,5],[2]]
 => 1
[[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,6,5,8,7,4,3,2] => [6,8,5,7,4,3,1,2] => ?
 => ? = 17
[[.,.],[.,[.,[[[.,.],.],[.,.]]]]]
 => [1,5,6,8,7,4,3,2] => [8,5,6,7,4,3,1,2] => ?
 => ? = 16
[[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
 => [1,4,7,6,8,5,3,2] => [7,6,8,4,5,3,1,2] => ?
 => ? = 15
[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
 => [1,5,4,8,7,6,3,2] => [8,5,7,4,6,3,1,2] => ?
 => ? = 15
[[.,.],[.,[[.,[.,.]],[[.,.],.]]]]
 => [1,5,4,7,8,6,3,2] => [5,7,8,4,6,3,1,2] => ?
 => ? = 14
[[.,.],[.,[[[.,.],.],[.,[.,.]]]]]
 => [1,4,5,8,7,6,3,2] => [8,7,4,5,6,3,1,2] => ?
 => ? = 14
[[.,.],[.,[[[.,.],.],[[.,.],.]]]]
 => [1,4,5,7,8,6,3,2] => [7,8,4,5,6,3,1,2] => ?
 => ? = 13
[[.,.],[.,[[.,[.,[.,.]]],[.,.]]]]
 => [1,6,5,4,8,7,3,2] => [6,5,8,4,7,3,1,2] => ?
 => ? = 15
[[.,.],[.,[[.,[[.,.],.]],[.,.]]]]
 => [1,5,6,4,8,7,3,2] => ? => ?
 => ? = 14
[[.,.],[.,[[[.,.],[.,.]],[.,.]]]]
 => [1,4,6,5,8,7,3,2] => [6,8,4,5,7,3,1,2] => ?
 => ? = 13
[[.,.],[.,[[[.,[.,.]],.],[.,.]]]]
 => [1,5,4,6,8,7,3,2] => [5,8,4,6,7,3,1,2] => ?
 => ? = 13
[[.,.],[.,[[[[.,.],.],.],[.,.]]]]
 => [1,4,5,6,8,7,3,2] => ? => ?
 => ? = 12
[[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
 => [1,6,7,5,4,8,3,2] => [6,7,5,4,8,3,1,2] => ?
 => ? = 16
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
 => [1,5,7,6,4,8,3,2] => ? => ?
 => ? = 15
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
 => [1,5,6,7,4,8,3,2] => [5,6,7,4,8,3,1,2] => ?
 => ? = 14
[[.,.],[.,[[[.,[.,.]],[.,.]],.]]]
 => [1,5,4,7,6,8,3,2] => [5,7,4,6,8,3,1,2] => ?
 => ? = 13
[[.,.],[.,[[[[.,.],.],[.,.]],.]]]
 => [1,4,5,7,6,8,3,2] => [7,4,5,6,8,3,1,2] => ?
 => ? = 12
[[.,.],[.,[[[[.,.],[.,.]],.],.]]]
 => [1,4,6,5,7,8,3,2] => ? => ?
 => ? = 12
[[.,.],[[.,.],[.,[[.,.],[.,.]]]]]
 => [1,3,6,8,7,5,4,2] => [8,6,7,5,3,4,1,2] => ?
 => ? = 14
[[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
 => [1,4,3,6,8,7,5,2] => [8,4,6,7,3,5,1,2] => ?
 => ? = 11
[[.,.],[[[.,.],.],[.,[[.,.],.]]]]
 => [1,3,4,7,8,6,5,2] => [7,8,6,3,4,5,1,2] => ?
 => ? = 11
[[.,.],[[[.,.],.],[[.,[.,.]],.]]]
 => [1,3,4,7,6,8,5,2] => [7,6,8,3,4,5,1,2] => ?
 => ? = 10
[[.,.],[[[.,.],.],[[[.,.],.],.]]]
 => [1,3,4,6,7,8,5,2] => [6,7,8,3,4,5,1,2] => ?
 => ? = 9
[[.,.],[[.,[[.,.],.]],[[.,.],.]]]
 => [1,4,5,3,7,8,6,2] => [4,5,7,8,3,6,1,2] => ?
 => ? = 10
[[.,.],[[[.,.],[.,.]],[.,[.,.]]]]
 => [1,3,5,4,8,7,6,2] => [8,5,7,3,4,6,1,2] => ?
 => ? = 10
[[.,.],[[[.,.],[.,.]],[[.,.],.]]]
 => [1,3,5,4,7,8,6,2] => [5,7,8,3,4,6,1,2] => ?
 => ? = 9
[[.,.],[[[.,[.,.]],.],[.,[.,.]]]]
 => [1,4,3,5,8,7,6,2] => [8,4,7,3,5,6,1,2] => ?
 => ? = 10
[[.,.],[[.,[[.,[.,.]],.]],[.,.]]]
 => [1,5,4,6,3,8,7,2] => [5,4,6,8,3,7,1,2] => ?
 => ? = 11
[[.,.],[[.,[[[.,.],.],.]],[.,.]]]
 => [1,4,5,6,3,8,7,2] => [4,5,6,8,3,7,1,2] => ?
 => ? = 10
[[.,.],[[[.,.],[.,[.,.]]],[.,.]]]
 => [1,3,6,5,4,8,7,2] => [6,5,8,3,4,7,1,2] => ?
 => ? = 10
[[.,.],[[[.,.],[[.,.],.]],[.,.]]]
 => [1,3,5,6,4,8,7,2] => [5,6,8,3,4,7,1,2] => ?
 => ? = 9
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
 => [1,3,5,4,6,8,7,2] => [5,8,3,4,6,7,1,2] => ?
 => ? = 8
[[.,.],[[.,[.,[.,[[.,.],.]]]],.]]
 => [1,6,7,5,4,3,8,2] => [6,7,5,4,3,8,1,2] => ?
 => ? = 15
[[.,.],[[.,[.,[[.,[.,.]],.]]],.]]
 => [1,6,5,7,4,3,8,2] => [6,5,7,4,3,8,1,2] => ?
 => ? = 14
[[.,.],[[.,[[.,.],[.,[.,.]]]],.]]
 => [1,4,7,6,5,3,8,2] => [7,6,4,5,3,8,1,2] => ?
 => ? = 13
[[.,.],[[.,[[[.,.],.],[.,.]]],.]]
 => [1,4,5,7,6,3,8,2] => [7,4,5,6,3,8,1,2] => ?
 => ? = 11
[[.,.],[[.,[[[.,.],[.,.]],.]],.]]
 => [1,4,6,5,7,3,8,2] => [6,4,5,7,3,8,1,2] => ?
 => ? = 11
[[.,.],[[.,[[[.,[.,.]],.],.]],.]]
 => [1,5,4,6,7,3,8,2] => [5,4,6,7,3,8,1,2] => ?
 => ? = 11
[[.,.],[[[.,.],[.,[.,[.,.]]]],.]]
 => [1,3,7,6,5,4,8,2] => ? => ?
 => ? = 12
[[.,.],[[[.,.],[.,[[.,.],.]]],.]]
 => [1,3,6,7,5,4,8,2] => ? => ?
 => ? = 11
[[.,.],[[[.,.],[[.,.],[.,.]]],.]]
 => [1,3,5,7,6,4,8,2] => [7,5,6,3,4,8,1,2] => ?
 => ? = 10
[[.,.],[[[.,.],[[.,[.,.]],.]],.]]
 => [1,3,6,5,7,4,8,2] => [6,5,7,3,4,8,1,2] => ?
 => ? = 10
[[.,.],[[[.,.],[[[.,.],.],.]],.]]
 => [1,3,5,6,7,4,8,2] => [5,6,7,3,4,8,1,2] => ?
 => ? = 9
[[.,.],[[[.,[.,.]],[.,[.,.]]],.]]
 => [1,4,3,7,6,5,8,2] => ? => ?
 => ? = 10
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
 => [1,3,4,6,7,5,8,2] => [6,7,3,4,5,8,1,2] => ?
 => ? = 8
[[.,.],[[[.,[[.,.],.]],[.,.]],.]]
 => [1,4,5,3,7,6,8,2] => ? => ?
 => ? = 9
[[.,.],[[[[.,.],[.,.]],[.,.]],.]]
 => [1,3,5,4,7,6,8,2] => [5,7,3,4,6,8,1,2] => ?
 => ? = 8
[[.,.],[[[[.,[.,.]],.],[.,.]],.]]
 => [1,4,3,5,7,6,8,2] => [4,7,3,5,6,8,1,2] => ?
 => ? = 8
[[.,.],[[[.,[.,[[.,.],.]]],.],.]]
 => [1,5,6,4,3,7,8,2] => ? => ?
 => ? = 11
[[.,.],[[[.,[[.,.],[.,.]]],.],.]]
 => [1,4,6,5,3,7,8,2] => [6,4,5,3,7,8,1,2] => ?
 => ? = 10

Description

The (standard) major index of a standard tableau. A descent of a standard tableau

$T$ is an index

$i$ such that

$i+1$ appears in a row strictly below the row of

$i$ . The (standard) major index is the the sum of the descents.

Matching statistic: St000378

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1,0]
 => [1,0]
 => []
 => 0
[.,[.,.]]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1]
 => 1
[[.,.],.]
 => [1,0,1,0]
 => [1,1,0,0]
 => []
 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [2,1]
 => 3
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [2]
 => 2
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1]
 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 6
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 5
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 4
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 4
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 3
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 3
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 2
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 2
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 3
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 2
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 10
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 9
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 8
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => 8
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 7
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 7
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 6
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => 6
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 5
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => 7
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => 6
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => 5
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => 5
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 4
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 6
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 5
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 4
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => 4
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 3
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => 4
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => 3
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 3
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 2
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => 4
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 3
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,1,1]
 => ? = 17
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,1]
 => ? = 18
[.,[.,[.,[[.,[[.,.],.]],.]]]]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,1]
 => ? = 17
[.,[.,[.,[[[.,.],[.,.]],.]]]]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,1]
 => ? = 16
[.,[.,[.,[[[.,[.,.]],.],.]]]]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,2]
 => ? = 16
[.,[.,[[.,.],[[.,[.,.]],.]]]]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,1]
 => ? = 15
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,2,1]
 => ? = 15
[.,[.,[[.,[.,.]],[[.,.],.]]]]
 => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,2]
 => ? = 14
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,1,1]
 => ? = 15
[.,[.,[[.,[[.,.],.]],[.,.]]]]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,1,1]
 => ? = 14
[.,[.,[[[.,.],[.,.]],[.,.]]]]
 => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2,1,1]
 => ? = 13
[.,[.,[[[.,[.,.]],.],[.,.]]]]
 => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,1,1]
 => ? = 13
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,1]
 => ? = 17
[.,[.,[[.,[.,[[.,.],.]]],.]]]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,1]
 => ? = 16
[.,[.,[[.,[[.,.],[.,.]]],.]]]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2,1]
 => ? = 15
[.,[.,[[.,[[.,[.,.]],.]],.]]]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,1]
 => ? = 15
[.,[.,[[.,[[[.,.],.],.]],.]]]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,1]
 => ? = 14
[.,[.,[[[.,.],[.,[.,.]]],.]]]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2]
 => ? = 14
[.,[.,[[[.,.],[[.,.],.]],.]]]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2]
 => ? = 13
[.,[.,[[[.,[.,.]],[.,.]],.]]]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,1]
 => ? = 13
[.,[.,[[[[.,.],.],[.,.]],.]]]
 => [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2]
 => ? = 12
[.,[.,[[[.,[.,[.,.]]],.],.]]]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,2]
 => ? = 14
[.,[.,[[[.,[[.,.],.]],.],.]]]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2,2]
 => ? = 13
[.,[.,[[[[.,[.,.]],.],.],.]]]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,2]
 => ? = 12
[.,[[.,.],[.,[[.,[.,.]],.]]]]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2]
 => ? = 14
[.,[[.,.],[[.,[.,.]],[.,.]]]]
 => [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [6,5,3,1,1,1]
 => ? = 12
[.,[[.,.],[[.,[.,[.,.]]],.]]]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [6,5,3,1,1]
 => ? = 13
[.,[[.,.],[[.,[[.,.],.]],.]]]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
 => [6,4,4,1,1]
 => ? = 12
[.,[[.,.],[[[.,.],[.,.]],.]]]
 => [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
 => [5,5,4,1,1]
 => ? = 11
[.,[[.,.],[[[.,[.,.]],.],.]]]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [6,4,2,2,2]
 => ? = 11
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,2,1]
 => ? = 13
[.,[[.,[.,.]],[.,[[.,.],.]]]]
 => [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,2]
 => ? = 12
[.,[[.,[.,.]],[[.,.],[.,.]]]]
 => [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,1,1]
 => ? = 11
[.,[[.,[.,.]],[[.,[.,.]],.]]]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
 => [6,5,3,1]
 => ? = 11
[.,[[.,[.,.]],[[[.,.],.],.]]]
 => [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0,1,0]
 => [6,4,4,1]
 => ? = 10
[.,[[[.,.],.],[[.,[.,.]],.]]]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
 => [6,5,2,2]
 => ? = 10
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,2,1]
 => ? = 12
[.,[[.,[.,[.,.]]],[[.,.],.]]]
 => [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,3,2]
 => ? = 11
[.,[[.,[[.,.],.]],[.,[.,.]]]]
 => [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2,2,1]
 => ? = 11
[.,[[[.,.],[.,.]],[.,[.,.]]]]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [6,4,2,2,2,1]
 => ? = 10
[.,[[[.,.],[.,.]],[[.,.],.]]]
 => [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,3,3,2]
 => ? = 9
[.,[[[.,[.,.]],.],[.,[.,.]]]]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,2,1]
 => ? = 10
[.,[[[.,[.,.]],.],[[.,.],.]]]
 => [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,1,0,0]
 => [5,5,4,1]
 => ? = 9
[.,[[.,[.,[.,[.,.]]]],[.,.]]]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,1,1]
 => ? = 13
[.,[[.,[.,[[.,.],.]]],[.,.]]]
 => [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2,1,1]
 => ? = 12
[.,[[.,[[.,.],[.,.]]],[.,.]]]
 => [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
 => [6,4,3,1,1,1]
 => ? = 11
[.,[[.,[[.,[.,.]],.]],[.,.]]]
 => [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,1,1]
 => ? = 11
[.,[[.,[[[.,.],.],.]],[.,.]]]
 => [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [5,5,3,1,1,1]
 => ? = 10
[.,[[[.,.],[.,[.,.]]],[.,.]]]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [6,4,2,2,1,1]
 => ? = 10
[.,[[[.,.],[[.,.],.]],[.,.]]]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [5,5,2,2,1,1]
 => ? = 9

Description

The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells

$c$ in the diagram of an integer partition

$\lambda$ for which

$\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$ . See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].

Matching statistic: St000041
(load all 2 compositions to match this statistic)

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000041: Perfect matchings ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1,0]
 => [(1,2)]
 => 0
[.,[.,.]]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => 1
[[.,.],.]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 3
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => 2
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => 6
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => 5
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => 4
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => 4
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => 3
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => 3
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => 2
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => 2
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => 3
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => 2
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => 10
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => 9
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => 8
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => 8
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => 7
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => 7
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => 6
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => 6
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => 5
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => 7
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => 6
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => 5
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => 5
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => 4
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => 6
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => 5
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => 4
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => 4
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => 3
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => 4
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9)]
 => 3
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => 3
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => 2
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => 4
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9)]
 => 3
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => 1
[[.,.],[.,[[[.,.],.],[.,.]]]]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [(1,2),(3,14),(4,13),(5,6),(7,8),(9,12),(10,11)]
 => ? = 10
[[.,.],[.,[[[.,.],[.,.]],.]]]
 => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [(1,2),(3,14),(4,13),(5,6),(7,10),(8,9),(11,12)]
 => ? = 10
[[.,.],[.,[[[.,[.,.]],.],.]]]
 => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [(1,2),(3,14),(4,13),(5,8),(6,7),(9,10),(11,12)]
 => ? = 10
[[.,.],[.,[[[[.,.],.],.],.]]]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [(1,2),(3,14),(4,13),(5,6),(7,8),(9,10),(11,12)]
 => ? = 9
[[.,.],[[.,.],[.,[[.,.],.]]]]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [(1,2),(3,14),(4,5),(6,13),(7,12),(8,9),(10,11)]
 => ? = 10
[[.,.],[[.,.],[[.,.],[.,.]]]]
 => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [(1,2),(3,14),(4,5),(6,13),(7,8),(9,12),(10,11)]
 => ? = 9
[[.,.],[[.,.],[[.,[.,.]],.]]]
 => [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [(1,2),(3,14),(4,5),(6,13),(7,10),(8,9),(11,12)]
 => ? = 9
[[.,.],[[.,.],[[[.,.],.],.]]]
 => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [(1,2),(3,14),(4,5),(6,13),(7,8),(9,10),(11,12)]
 => ? = 8
[[.,.],[[.,[.,.]],[.,[.,.]]]]
 => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [(1,2),(3,14),(4,7),(5,6),(8,13),(9,12),(10,11)]
 => ? = 9
[[.,.],[[.,[.,.]],[[.,.],.]]]
 => [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [(1,2),(3,14),(4,7),(5,6),(8,13),(9,10),(11,12)]
 => ? = 8
[[.,.],[[[.,.],.],[.,[.,.]]]]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [(1,2),(3,14),(4,5),(6,7),(8,13),(9,12),(10,11)]
 => ? = 8
[[.,.],[[[.,.],.],[[.,.],.]]]
 => [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [(1,2),(3,14),(4,5),(6,7),(8,13),(9,10),(11,12)]
 => ? = 7
[[.,.],[[.,[.,[.,.]]],[.,.]]]
 => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [(1,2),(3,14),(4,9),(5,8),(6,7),(10,13),(11,12)]
 => ? = 9
[[.,.],[[.,[[.,.],.]],[.,.]]]
 => [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [(1,2),(3,14),(4,9),(5,6),(7,8),(10,13),(11,12)]
 => ? = 8
[[.,.],[[[.,.],[.,.]],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [(1,2),(3,14),(4,5),(6,9),(7,8),(10,13),(11,12)]
 => ? = 7
[[.,.],[[[.,[.,.]],.],[.,.]]]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [(1,2),(3,14),(4,7),(5,6),(8,9),(10,13),(11,12)]
 => ? = 7
[[.,.],[[[[.,.],.],.],[.,.]]]
 => [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [(1,2),(3,14),(4,5),(6,7),(8,9),(10,13),(11,12)]
 => ? = 6
[[.,.],[[.,[[.,[.,.]],.]],.]]
 => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [(1,2),(3,14),(4,11),(5,8),(6,7),(9,10),(12,13)]
 => ? = 9
[[.,.],[[[.,.],[.,[.,.]]],.]]
 => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [(1,2),(3,14),(4,5),(6,11),(7,10),(8,9),(12,13)]
 => ? = 8
[[.,.],[[[.,.],[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [(1,2),(3,14),(4,5),(6,11),(7,8),(9,10),(12,13)]
 => ? = 7
[[.,.],[[[.,[.,.]],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [(1,2),(3,14),(4,7),(5,6),(8,11),(9,10),(12,13)]
 => ? = 7
[[.,.],[[[[.,.],.],[.,.]],.]]
 => [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [(1,2),(3,14),(4,5),(6,7),(8,11),(9,10),(12,13)]
 => ? = 6
[[.,.],[[[[.,.],[.,.]],.],.]]
 => [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,5),(6,9),(7,8),(10,11),(12,13)]
 => ? = 6
[[.,[.,.]],[.,[.,[[.,.],.]]]]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [(1,4),(2,3),(5,14),(6,13),(7,12),(8,9),(10,11)]
 => ? = 10
[[.,[.,.]],[.,[[.,.],[.,.]]]]
 => [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [(1,4),(2,3),(5,14),(6,13),(7,8),(9,12),(10,11)]
 => ? = 9
[[.,[.,.]],[.,[[.,[.,.]],.]]]
 => [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [(1,4),(2,3),(5,14),(6,13),(7,10),(8,9),(11,12)]
 => ? = 9
[[.,[.,.]],[.,[[[.,.],.],.]]]
 => [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [(1,4),(2,3),(5,14),(6,13),(7,8),(9,10),(11,12)]
 => ? = 8
[[.,[.,.]],[[.,.],[.,[.,.]]]]
 => [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [(1,4),(2,3),(5,14),(6,7),(8,13),(9,12),(10,11)]
 => ? = 8
[[.,[.,.]],[[.,.],[[.,.],.]]]
 => [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => [(1,4),(2,3),(5,14),(6,7),(8,13),(9,10),(11,12)]
 => ? = 7
[[.,[.,.]],[[.,[.,.]],[.,.]]]
 => [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [(1,4),(2,3),(5,14),(6,9),(7,8),(10,13),(11,12)]
 => ? = 7
[[.,[.,.]],[[[.,.],.],[.,.]]]
 => [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [(1,4),(2,3),(5,14),(6,7),(8,9),(10,13),(11,12)]
 => ? = 6
[[.,[.,.]],[[.,[.,[.,.]]],.]]
 => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => [(1,4),(2,3),(5,14),(6,11),(7,10),(8,9),(12,13)]
 => ? = 8
[[.,[.,.]],[[.,[[.,.],.]],.]]
 => [1,1,0,0,1,1,1,0,1,0,0,1,0,0]
 => [(1,4),(2,3),(5,14),(6,11),(7,8),(9,10),(12,13)]
 => ? = 7
[[.,[.,.]],[[[.,.],[.,.]],.]]
 => [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => [(1,4),(2,3),(5,14),(6,7),(8,11),(9,10),(12,13)]
 => ? = 6
[[.,[.,.]],[[[.,[.,.]],.],.]]
 => [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,4),(2,3),(5,14),(6,9),(7,8),(10,11),(12,13)]
 => ? = 6
[[.,[.,.]],[[[[.,.],.],.],.]]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,4),(2,3),(5,14),(6,7),(8,9),(10,11),(12,13)]
 => ? = 5
[[[.,.],.],[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [(1,2),(3,4),(5,14),(6,13),(7,10),(8,9),(11,12)]
 => ? = 8
[[[.,.],.],[[.,.],[.,[.,.]]]]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [(1,2),(3,4),(5,14),(6,7),(8,13),(9,12),(10,11)]
 => ? = 7
[[[.,.],.],[[.,.],[[.,.],.]]]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [(1,2),(3,4),(5,14),(6,7),(8,13),(9,10),(11,12)]
 => ? = 6
[[[.,.],.],[[.,[.,.]],[.,.]]]
 => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [(1,2),(3,4),(5,14),(6,9),(7,8),(10,13),(11,12)]
 => ? = 6
[[[.,.],.],[[[.,.],.],[.,.]]]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [(1,2),(3,4),(5,14),(6,7),(8,9),(10,13),(11,12)]
 => ? = 5
[[[.,.],.],[[[.,.],[.,.]],.]]
 => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,14),(6,7),(8,11),(9,10),(12,13)]
 => ? = 5
[[.,[.,[.,.]]],[.,[[.,.],.]]]
 => [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
 => [(1,6),(2,5),(3,4),(7,14),(8,13),(9,10),(11,12)]
 => ? = 8
[[.,[.,[.,.]]],[[.,.],[.,.]]]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [(1,6),(2,5),(3,4),(7,14),(8,9),(10,13),(11,12)]
 => ? = 7
[[.,[.,[.,.]]],[[.,[.,.]],.]]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [(1,6),(2,5),(3,4),(7,14),(8,11),(9,10),(12,13)]
 => ? = 7
[[.,[.,[.,.]]],[[[.,.],.],.]]
 => [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
 => [(1,6),(2,5),(3,4),(7,14),(8,9),(10,11),(12,13)]
 => ? = 6
[[.,[[.,.],.]],[.,[.,[.,.]]]]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,13),(9,12),(10,11)]
 => ? = 8
[[.,[[.,.],.]],[.,[[.,.],.]]]
 => [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,13),(9,10),(11,12)]
 => ? = 7
[[.,[[.,.],.]],[[.,.],[.,.]]]
 => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,9),(10,13),(11,12)]
 => ? = 6
[[.,[[.,.],.]],[[.,[.,.]],.]]
 => [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,11),(9,10),(12,13)]
 => ? = 6

Description

The number of nestings of a perfect matching. This is the number of pairs of edges

$((a,b), (c,d))$ such that

$a\le c\le d\le b$ . i.e., the edge

$(c,d)$ is nested inside

$(a,b)$ .

Matching statistic: St001397
(load all 4 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001397: Posets ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1] => ([],1)
 => 0
[.,[.,.]]
 => [2,1] => ([],2)
 => 1
[[.,.],.]
 => [1,2] => ([(0,1)],2)
 => 0
[.,[.,[.,.]]]
 => [3,2,1] => ([],3)
 => 3
[.,[[.,.],.]]
 => [2,3,1] => ([(1,2)],3)
 => 2
[[.,.],[.,.]]
 => [1,3,2] => ([(0,1),(0,2)],3)
 => 1
[[.,[.,.]],.]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => 1
[[[.,.],.],.]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([],4)
 => 6
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(2,3)],4)
 => 5
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => ([(1,2),(1,3)],4)
 => 4
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 4
[.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 3
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 3
[[.,.],[[.,.],.]]
 => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 2
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[[[.,.],.],[.,.]]
 => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
[[[.,.],[.,.]],.]
 => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => ([],5)
 => 10
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => ([(3,4)],5)
 => 9
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => 8
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => 8
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => 7
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => 7
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => 6
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 6
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => 5
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => 7
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => 6
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => 5
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => 5
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => 4
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 6
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => 5
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 4
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 4
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 3
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 4
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => 3
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 3
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 2
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => 3
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
 => [5,6,4,7,3,2,1] => ([(3,6),(4,5),(5,6)],7)
 => ? = 17
[.,[.,[[.,.],[[[.,.],.],.]]]]
 => [3,5,6,7,4,2,1] => ([(2,4),(2,6),(5,3),(6,5)],7)
 => ? = 14
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [4,3,7,6,5,2,1] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ? = 15
[.,[.,[[[.,.],.],[[.,.],.]]]]
 => [3,4,6,7,5,2,1] => ([(2,6),(5,4),(6,3),(6,5)],7)
 => ? = 13
[.,[.,[[.,[[.,.],.]],[.,.]]]]
 => [4,5,3,7,6,2,1] => ([(2,5),(2,6),(3,4),(4,5),(4,6)],7)
 => ? = 14
[.,[.,[[.,[[.,.],[.,.]]],.]]]
 => [4,6,5,3,7,2,1] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 15
[.,[.,[[.,[[[.,.],.],.]],.]]]
 => [4,5,6,3,7,2,1] => ([(2,6),(3,4),(4,5),(5,6)],7)
 => ? = 14
[.,[.,[[[[.,.],.],[.,.]],.]]]
 => [3,4,6,5,7,2,1] => ([(2,3),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 12
[.,[.,[[[.,[[.,.],.]],.],.]]]
 => [4,5,3,6,7,2,1] => ([(2,6),(3,4),(4,6),(6,5)],7)
 => ? = 13
[.,[.,[[[[[.,.],.],.],.],.]]]
 => [3,4,5,6,7,2,1] => ([(2,6),(4,5),(5,3),(6,4)],7)
 => ? = 11
[.,[[.,.],[[.,.],[[.,.],.]]]]
 => [2,4,6,7,5,3,1] => ([(1,4),(1,6),(5,3),(6,2),(6,5)],7)
 => ? = 12
[.,[[.,.],[[[.,.],.],[.,.]]]]
 => [2,4,5,7,6,3,1] => ([(1,4),(1,5),(5,6),(6,2),(6,3)],7)
 => ? = 11
[.,[[.,.],[[[.,.],[.,.]],.]]]
 => [2,4,6,5,7,3,1] => ([(1,2),(1,5),(3,6),(4,6),(5,3),(5,4)],7)
 => ? = 11
[.,[[.,.],[[[[.,.],.],.],.]]]
 => [2,4,5,6,7,3,1] => ([(1,3),(1,6),(4,5),(5,2),(6,4)],7)
 => ? = 10
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
 => [3,2,7,6,5,4,1] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 13
[.,[[.,[.,.]],[.,[[.,.],.]]]]
 => [3,2,6,7,5,4,1] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(6,3)],7)
 => ? = 12
[.,[[.,[.,.]],[[.,[.,.]],.]]]
 => [3,2,6,5,7,4,1] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(5,3),(6,3)],7)
 => ? = 11
[.,[[[.,.],.],[.,[[.,.],.]]]]
 => [2,3,6,7,5,4,1] => ([(1,6),(5,4),(6,2),(6,3),(6,5)],7)
 => ? = 11
[.,[[[.,.],.],[[.,.],[.,.]]]]
 => [2,3,5,7,6,4,1] => ([(1,6),(5,3),(5,4),(6,2),(6,5)],7)
 => ? = 10
[.,[[[.,.],.],[[.,[.,.]],.]]]
 => [2,3,6,5,7,4,1] => ([(1,5),(3,6),(4,6),(5,2),(5,3),(5,4)],7)
 => ? = 10
[.,[[[.,.],.],[[[.,.],.],.]]]
 => [2,3,5,6,7,4,1] => ([(1,6),(4,5),(5,3),(6,2),(6,4)],7)
 => ? = 9
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
 => [4,3,2,7,6,5,1] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ? = 12
[.,[[.,[[.,.],.]],[.,[.,.]]]]
 => [3,4,2,7,6,5,1] => ([(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
 => ? = 11
[.,[[.,[[.,.],.]],[[.,.],.]]]
 => [3,4,2,6,7,5,1] => ([(1,5),(1,6),(2,4),(4,5),(4,6),(6,3)],7)
 => ? = 10
[.,[[[[.,.],.],.],[[.,.],.]]]
 => [2,3,4,6,7,5,1] => ([(1,5),(4,3),(5,6),(6,2),(6,4)],7)
 => ? = 8
[.,[[.,[[.,.],[.,.]]],[.,.]]]
 => [3,5,4,2,7,6,1] => ([(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 11
[.,[[.,[[[.,.],.],.]],[.,.]]]
 => [3,4,5,2,7,6,1] => ([(1,5),(1,6),(2,3),(3,4),(4,5),(4,6)],7)
 => ? = 10
[.,[[[.,.],[[.,.],.]],[.,.]]]
 => [2,4,5,3,7,6,1] => ([(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,2)],7)
 => ? = 9
[.,[[[[.,.],.],[.,.]],[.,.]]]
 => [2,3,5,4,7,6,1] => ([(1,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3)],7)
 => ? = 8
[.,[[[.,[[.,.],.]],.],[.,.]]]
 => [3,4,2,5,7,6,1] => ([(1,6),(2,3),(3,6),(6,4),(6,5)],7)
 => ? = 9
[.,[[[[[.,.],.],.],.],[.,.]]]
 => [2,3,4,5,7,6,1] => ([(1,5),(4,6),(5,4),(6,2),(6,3)],7)
 => ? = 7
[.,[[.,[[.,.],[[.,.],.]]],.]]
 => [3,5,6,4,2,7,1] => ([(1,6),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ? = 12
[.,[[.,[[[.,.],.],[.,.]]],.]]
 => [3,4,6,5,2,7,1] => ([(1,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 11
[.,[[.,[[.,[[.,.],.]],.]],.]]
 => [4,5,3,6,2,7,1] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
 => ? = 12
[.,[[.,[[[.,.],[.,.]],.]],.]]
 => [3,5,4,6,2,7,1] => ([(1,5),(2,3),(2,4),(3,6),(4,6),(6,5)],7)
 => ? = 11
[.,[[[.,.],[[.,.],[.,.]]],.]]
 => [2,4,6,5,3,7,1] => ([(1,4),(1,5),(2,6),(3,6),(4,6),(5,2),(5,3)],7)
 => ? = 10
[.,[[[.,.],[[[.,.],.],.]],.]]
 => [2,4,5,6,3,7,1] => ([(1,3),(1,5),(2,6),(3,6),(4,2),(5,4)],7)
 => ? = 9
[.,[[[.,[.,.]],[.,[.,.]]],.]]
 => [3,2,6,5,4,7,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
 => ? = 10
[.,[[[[.,.],.],[.,[.,.]]],.]]
 => [2,3,6,5,4,7,1] => ([(1,5),(2,6),(3,6),(4,6),(5,2),(5,3),(5,4)],7)
 => ? = 9
[.,[[[[.,.],.],[[.,.],.]],.]]
 => [2,3,5,6,4,7,1] => ([(1,5),(2,6),(3,6),(4,3),(5,2),(5,4)],7)
 => ? = 8
[.,[[[.,[[.,.],.]],[.,.]],.]]
 => [3,4,2,6,5,7,1] => ([(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
 => ? = 9
[.,[[[.,[[.,.],[.,.]]],.],.]]
 => [3,5,4,2,6,7,1] => ([(1,6),(2,3),(2,4),(3,6),(4,6),(6,5)],7)
 => ? = 10
[.,[[[.,[[[.,.],.],.]],.],.]]
 => [3,4,5,2,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
 => ? = 9
[.,[[[[[.,.],.],[.,.]],.],.]]
 => [2,3,5,4,6,7,1] => ([(1,5),(2,6),(3,6),(5,2),(5,3),(6,4)],7)
 => ? = 7
[.,[[[[.,[[.,.],.]],.],.],.]]
 => [3,4,2,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => ? = 8
[.,[[[[[.,.],[.,.]],.],.],.]]
 => [2,4,3,5,6,7,1] => ([(1,3),(1,4),(3,6),(4,6),(5,2),(6,5)],7)
 => ? = 7
[.,[[[[[[.,.],.],.],.],.],.]]
 => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ? = 6
[[.,.],[.,[[.,.],[[.,.],.]]]]
 => [1,4,6,7,5,3,2] => ([(0,3),(0,4),(0,6),(5,2),(6,1),(6,5)],7)
 => ? = 11
[[.,.],[.,[[[[.,.],.],.],.]]]
 => [1,4,5,6,7,3,2] => ([(0,2),(0,3),(0,6),(4,5),(5,1),(6,4)],7)
 => ? = 9
[[.,.],[[.,.],[.,[[.,.],.]]]]
 => [1,3,6,7,5,4,2] => ([(0,4),(0,6),(5,3),(6,1),(6,2),(6,5)],7)
 => ? = 10

Description

Number of pairs of incomparable elements in a finite poset. For a finite poset

$(P,\leq)$ , this is the number of unordered pairs

$\{x,y\} \in \binom{P}{2}$ with

$x \not\leq y$ and

$y \not\leq x$ .

Matching statistic: St000081
(load all 2 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 100%

Values

[.,.]
 => [1] => ([],1)
 => 0
[.,[.,.]]
 => [2,1] => ([(0,1)],2)
 => 1
[[.,.],.]
 => [1,2] => ([],2)
 => 0
[.,[.,[.,.]]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[.,[[.,.],.]]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
[[.,.],[.,.]]
 => [1,3,2] => ([(1,2)],3)
 => 1
[[.,[.,.]],.]
 => [2,1,3] => ([(1,2)],3)
 => 1
[[[.,.],.],.]
 => [1,2,3] => ([],3)
 => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 5
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[[.,.],[[.,.],.]]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 2
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2
[[[.,.],.],[.,.]]
 => [1,2,4,3] => ([(2,3)],4)
 => 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3
[[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2
[[[.,.],[.,.]],.]
 => [1,3,2,4] => ([(2,3)],4)
 => 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(2,3)],4)
 => 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => ([],4)
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 10
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 9
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 8
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 8
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 7
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 7
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 6
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 7
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 4
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 3
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 2
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 4
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 3
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 2
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 2
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => ([(3,4)],5)
 => 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => [5,4,7,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 17
[.,[.,[[.,[.,.]],[[.,.],.]]]]
 => [4,3,6,7,5,2,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 14
[.,[.,[[.,[[.,.],.]],[.,.]]]]
 => [4,5,3,7,6,2,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 14
[.,[.,[[[.,.],[.,.]],[.,.]]]]
 => [3,5,4,7,6,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 13
[.,[.,[[[.,[.,.]],.],[.,.]]]]
 => [4,3,5,7,6,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 13
[.,[.,[[[.,[.,.]],[.,.]],.]]]
 => [4,3,6,5,7,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 13
[.,[[.,.],[[.,[.,.]],[.,.]]]]
 => [2,5,4,7,6,3,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 12
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
 => [3,2,7,6,5,4,1] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 13
[.,[[.,[.,.]],[.,[[.,.],.]]]]
 => [3,2,6,7,5,4,1] => ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 12
[.,[[.,[.,.]],[[.,.],[.,.]]]]
 => [3,2,5,7,6,4,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 11
[.,[[.,[.,.]],[[.,[.,.]],.]]]
 => [3,2,6,5,7,4,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 11
[.,[[.,[.,.]],[[[.,.],.],.]]]
 => [3,2,5,6,7,4,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 10
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
 => [4,3,2,7,6,5,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 12
[.,[[.,[.,[.,.]]],[[.,.],.]]]
 => [4,3,2,6,7,5,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 11
[.,[[.,[[.,.],.]],[.,[.,.]]]]
 => [3,4,2,7,6,5,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 11
[.,[[.,[[.,.],.]],[[.,.],.]]]
 => [3,4,2,6,7,5,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 10
[.,[[[.,.],[.,.]],[.,[.,.]]]]
 => [2,4,3,7,6,5,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 10
[.,[[[.,.],[.,.]],[[.,.],.]]]
 => [2,4,3,6,7,5,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 9
[.,[[[.,[.,.]],.],[.,[.,.]]]]
 => [3,2,4,7,6,5,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 10
[.,[[[.,[.,.]],.],[[.,.],.]]]
 => [3,2,4,6,7,5,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 9
[.,[[.,[.,[.,[.,.]]]],[.,.]]]
 => [5,4,3,2,7,6,1] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 13
[.,[[.,[.,[[.,.],.]]],[.,.]]]
 => [4,5,3,2,7,6,1] => ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 12
[.,[[.,[[.,.],[.,.]]],[.,.]]]
 => [3,5,4,2,7,6,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 11
[.,[[.,[[.,[.,.]],.]],[.,.]]]
 => [4,3,5,2,7,6,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 11
[.,[[.,[[[.,.],.],.]],[.,.]]]
 => [3,4,5,2,7,6,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 10
[.,[[[.,.],[.,[.,.]]],[.,.]]]
 => [2,5,4,3,7,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 10
[.,[[[.,.],[[.,.],.]],[.,.]]]
 => [2,4,5,3,7,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 9
[.,[[[[.,.],.],[.,.]],[.,.]]]
 => [2,3,5,4,7,6,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 8
[.,[[[.,[.,[.,.]]],.],[.,.]]]
 => [4,3,2,5,7,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 10
[.,[[[.,[[.,.],.]],.],[.,.]]]
 => [3,4,2,5,7,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 9
[.,[[[[.,.],[.,.]],.],[.,.]]]
 => [2,4,3,5,7,6,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 8
[.,[[[[.,[.,.]],.],.],[.,.]]]
 => [3,2,4,5,7,6,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 8
[.,[[.,[[.,[.,.]],[.,.]]],.]]
 => [4,3,6,5,2,7,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 12
[.,[[[.,[.,.]],[.,[.,.]]],.]]
 => [3,2,6,5,4,7,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 10
[.,[[[.,[.,.]],[[.,.],.]],.]]
 => [3,2,5,6,4,7,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 9
[.,[[[.,[.,[.,.]]],[.,.]],.]]
 => [4,3,2,6,5,7,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 10
[.,[[[.,[[.,.],.]],[.,.]],.]]
 => [3,4,2,6,5,7,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 9
[.,[[[[.,.],[.,.]],[.,.]],.]]
 => [2,4,3,6,5,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 8
[.,[[[[.,[.,.]],.],[.,.]],.]]
 => [3,2,4,6,5,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 8
[.,[[[[.,[.,.]],[.,.]],.],.]]
 => [3,2,5,4,6,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 8
[[.,.],[.,[[.,[.,.]],[.,.]]]]
 => [1,5,4,7,6,3,2] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 11
[[.,.],[[.,[.,.]],[.,[.,.]]]]
 => [1,4,3,7,6,5,2] => ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 9
[[.,.],[[.,[.,.]],[[.,.],.]]]
 => [1,4,3,6,7,5,2] => ([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 8
[[.,.],[[.,[.,[.,.]]],[.,.]]]
 => [1,5,4,3,7,6,2] => ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 9
[[.,.],[[.,[[.,.],.]],[.,.]]]
 => [1,4,5,3,7,6,2] => ([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 8
[[.,.],[[[.,.],[.,.]],[.,.]]]
 => [1,3,5,4,7,6,2] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 7
[[.,.],[[[.,[.,.]],.],[.,.]]]
 => [1,4,3,5,7,6,2] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 7
[[.,.],[[[.,[.,.]],[.,.]],.]]
 => [1,4,3,6,5,7,2] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 7
[[.,[.,.]],[.,[.,[.,[.,.]]]]]
 => [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 11
[[.,[.,.]],[.,[.,[[.,.],.]]]]
 => [2,1,6,7,5,4,3] => ([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 10

Description

The number of edges of a graph.

The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000161The sum of the sizes of the right subtrees of a binary tree. St000246The number of non-inversions of a permutation. St000018The number of inversions of a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St000795The mad of a permutation. St000833The comajor index of a permutation. St000067The inversion number of the alternating sign matrix. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000057The Shynar inversion number of a standard tableau. St000332The positive inversions of an alternating sign matrix. St001428The number of B-inversions of a signed permutation. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St000004The major index of a permutation. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000042The number of crossings of a perfect matching. St000233The number of nestings of a set partition. St000305The inverse major index of a permutation. St000496The rcs statistic of a set partition. St001311The cyclomatic number of a graph. St001718The number of non-empty open intervals in a poset. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000450The number of edges minus the number of vertices plus 2 of a graph. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001862The number of crossings of a signed permutation. St000359The number of occurrences of the pattern 23-1. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St001433The flag major index of a signed permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.