- FindStat

Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000099
(load all 6 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of valleys of a permutation, including the boundary. The number of valleys excluding the boundary is [[St000353]].

Matching statistic: St000023
(load all 6 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of inner peaks of a permutation. The number of peaks including the boundary is [[St000092]].

Matching statistic: St000092
(load all 4 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of outer peaks of a permutation. An outer peak in a permutation

$w = [w_1,..., w_n]$ is either a position

$i$ such that

$w_{i-1} < w_i > w_{i+1}$ or

$1$ if

$w_1 > w_2$ or

$n$ if

$w_{n} > w_{n-1}$ . In other words, it is a peak in the word

$[0,w_1,..., w_n,0]$ .

Matching statistic: St000201
(load all 2 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of leaf nodes in a binary tree. Equivalently, the number of cherries [1] in the complete binary tree. The number of binary trees of size

$n$ , at least

$1$ , with exactly one leaf node for is

$2^{n-1}$ , see [2]. The number of binary tree of size

$n$ , at least

$3$ , with exactly two leaf nodes is

$n(n+1)2^{n-2}$ , see [3].

Matching statistic: St000390

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1] => [1] => 1 => 1
[2,1] => [1] => [1] => 1 => 1
[1,2,3] => [1,2] => [2] => 10 => 1
[1,3,2] => [1,2] => [2] => 10 => 1
[2,1,3] => [2,1] => [1,1] => 11 => 1
[2,3,1] => [2,1] => [1,1] => 11 => 1
[3,1,2] => [1,2] => [2] => 10 => 1
[3,2,1] => [2,1] => [1,1] => 11 => 1
[1,2,3,4] => [1,2,3] => [3] => 100 => 1
[1,2,4,3] => [1,2,3] => [3] => 100 => 1
[1,3,2,4] => [1,3,2] => [2,1] => 101 => 2
[1,3,4,2] => [1,3,2] => [2,1] => 101 => 2
[1,4,2,3] => [1,2,3] => [3] => 100 => 1
[1,4,3,2] => [1,3,2] => [2,1] => 101 => 2
[2,1,3,4] => [2,1,3] => [1,2] => 110 => 1
[2,1,4,3] => [2,1,3] => [1,2] => 110 => 1
[2,3,1,4] => [2,3,1] => [2,1] => 101 => 2
[2,3,4,1] => [2,3,1] => [2,1] => 101 => 2
[2,4,1,3] => [2,1,3] => [1,2] => 110 => 1
[2,4,3,1] => [2,3,1] => [2,1] => 101 => 2
[3,1,2,4] => [3,1,2] => [1,2] => 110 => 1
[3,1,4,2] => [3,1,2] => [1,2] => 110 => 1
[3,2,1,4] => [3,2,1] => [1,1,1] => 111 => 1
[3,2,4,1] => [3,2,1] => [1,1,1] => 111 => 1
[3,4,1,2] => [3,1,2] => [1,2] => 110 => 1
[3,4,2,1] => [3,2,1] => [1,1,1] => 111 => 1
[4,1,2,3] => [1,2,3] => [3] => 100 => 1
[4,1,3,2] => [1,3,2] => [2,1] => 101 => 2
[4,2,1,3] => [2,1,3] => [1,2] => 110 => 1
[4,2,3,1] => [2,3,1] => [2,1] => 101 => 2
[4,3,1,2] => [3,1,2] => [1,2] => 110 => 1
[4,3,2,1] => [3,2,1] => [1,1,1] => 111 => 1
[1,2,3,4,5] => [1,2,3,4] => [4] => 1000 => 1
[1,2,3,5,4] => [1,2,3,4] => [4] => 1000 => 1
[1,2,4,3,5] => [1,2,4,3] => [3,1] => 1001 => 2
[1,2,4,5,3] => [1,2,4,3] => [3,1] => 1001 => 2
[1,2,5,3,4] => [1,2,3,4] => [4] => 1000 => 1
[1,2,5,4,3] => [1,2,4,3] => [3,1] => 1001 => 2
[1,3,2,4,5] => [1,3,2,4] => [2,2] => 1010 => 2
[1,3,2,5,4] => [1,3,2,4] => [2,2] => 1010 => 2
[1,3,4,2,5] => [1,3,4,2] => [3,1] => 1001 => 2
[1,3,4,5,2] => [1,3,4,2] => [3,1] => 1001 => 2
[1,3,5,2,4] => [1,3,2,4] => [2,2] => 1010 => 2
[1,3,5,4,2] => [1,3,4,2] => [3,1] => 1001 => 2
[1,4,2,3,5] => [1,4,2,3] => [2,2] => 1010 => 2
[1,4,2,5,3] => [1,4,2,3] => [2,2] => 1010 => 2
[1,4,3,2,5] => [1,4,3,2] => [2,1,1] => 1011 => 2
[1,4,3,5,2] => [1,4,3,2] => [2,1,1] => 1011 => 2
[1,4,5,2,3] => [1,4,2,3] => [2,2] => 1010 => 2
[1,4,5,3,2] => [1,4,3,2] => [2,1,1] => 1011 => 2

Description

The number of runs of ones in a binary word.

Matching statistic: St000196
(load all 2 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree. Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by

$2^{n-2}$ .

Matching statistic: St000292
(load all 3 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1] => [1] => 1 => 0 = 1 - 1
[2,1] => [1] => [1] => 1 => 0 = 1 - 1
[1,2,3] => [1,2] => [2] => 10 => 0 = 1 - 1
[1,3,2] => [1,2] => [2] => 10 => 0 = 1 - 1
[2,1,3] => [2,1] => [1,1] => 11 => 0 = 1 - 1
[2,3,1] => [2,1] => [1,1] => 11 => 0 = 1 - 1
[3,1,2] => [1,2] => [2] => 10 => 0 = 1 - 1
[3,2,1] => [2,1] => [1,1] => 11 => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [3] => 100 => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [3] => 100 => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [2,1] => 101 => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [2,1] => 101 => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [3] => 100 => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [2,1] => 101 => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [1,2] => 110 => 0 = 1 - 1
[2,1,4,3] => [2,1,3] => [1,2] => 110 => 0 = 1 - 1
[2,3,1,4] => [2,3,1] => [2,1] => 101 => 1 = 2 - 1
[2,3,4,1] => [2,3,1] => [2,1] => 101 => 1 = 2 - 1
[2,4,1,3] => [2,1,3] => [1,2] => 110 => 0 = 1 - 1
[2,4,3,1] => [2,3,1] => [2,1] => 101 => 1 = 2 - 1
[3,1,2,4] => [3,1,2] => [1,2] => 110 => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [1,2] => 110 => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [1,1,1] => 111 => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [1,1,1] => 111 => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [1,2] => 110 => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [1,1,1] => 111 => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [3] => 100 => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [2,1] => 101 => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [1,2] => 110 => 0 = 1 - 1
[4,2,3,1] => [2,3,1] => [2,1] => 101 => 1 = 2 - 1
[4,3,1,2] => [3,1,2] => [1,2] => 110 => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [1,1,1] => 111 => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [4] => 1000 => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [4] => 1000 => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [3,1] => 1001 => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [3,1] => 1001 => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [4] => 1000 => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [3,1] => 1001 => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [2,2] => 1010 => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [2,2] => 1010 => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [3,1] => 1001 => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [3,1] => 1001 => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [2,2] => 1010 => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [3,1] => 1001 => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [2,2] => 1010 => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [2,2] => 1010 => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [2,1,1] => 1011 => 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1] => 1011 => 1 = 2 - 1
[1,4,5,2,3] => [1,4,2,3] => [2,2] => 1010 => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [2,1,1] => 1011 => 1 = 2 - 1

Description

The number of ascents of a binary word.

Matching statistic: St000353
(load all 4 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of inner valleys of a permutation. The number of valleys including the boundary is [[St000099]].

Matching statistic: St000291
(load all 2 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1] =>  =>  => ? = 1 - 1
[2,1] => [1] =>  =>  => ? = 1 - 1
[1,2,3] => [1,2] => 0 => 0 => 0 = 1 - 1
[1,3,2] => [1,2] => 0 => 0 => 0 = 1 - 1
[2,1,3] => [2,1] => 1 => 1 => 0 = 1 - 1
[2,3,1] => [2,1] => 1 => 1 => 0 = 1 - 1
[3,1,2] => [1,2] => 0 => 0 => 0 = 1 - 1
[3,2,1] => [2,1] => 1 => 1 => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => 01 => 10 => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => 01 => 10 => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => 01 => 10 => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => 10 => 01 => 0 = 1 - 1
[2,1,4,3] => [2,1,3] => 10 => 01 => 0 = 1 - 1
[2,3,1,4] => [2,3,1] => 01 => 10 => 1 = 2 - 1
[2,3,4,1] => [2,3,1] => 01 => 10 => 1 = 2 - 1
[2,4,1,3] => [2,1,3] => 10 => 01 => 0 = 1 - 1
[2,4,3,1] => [2,3,1] => 01 => 10 => 1 = 2 - 1
[3,1,2,4] => [3,1,2] => 10 => 01 => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => 10 => 01 => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => 11 => 11 => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => 11 => 11 => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => 10 => 01 => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => 11 => 11 => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => 01 => 10 => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => 10 => 01 => 0 = 1 - 1
[4,2,3,1] => [2,3,1] => 01 => 10 => 1 = 2 - 1
[4,3,1,2] => [3,1,2] => 10 => 01 => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => 11 => 11 => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => 001 => 100 => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => 001 => 100 => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => 001 => 100 => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => 010 => 010 => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => 010 => 010 => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => 001 => 100 => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => 001 => 100 => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => 010 => 010 => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => 001 => 100 => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => 010 => 010 => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => 010 => 010 => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => 011 => 110 => 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,2] => 011 => 110 => 1 = 2 - 1
[1,4,5,2,3] => [1,4,2,3] => 010 => 010 => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => 011 => 110 => 1 = 2 - 1
[1,5,2,3,4] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,5,2,4,3] => [1,2,4,3] => 001 => 100 => 1 = 2 - 1

Description

The number of descents of a binary word.

Matching statistic: St001487

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1] => [1] => [[1],[]]
 => 1
[2,1] => [1] => [1] => [[1],[]]
 => 1
[1,2,3] => [1,2] => [2] => [[2],[]]
 => 1
[1,3,2] => [1,2] => [2] => [[2],[]]
 => 1
[2,1,3] => [2,1] => [1,1] => [[1,1],[]]
 => 1
[2,3,1] => [2,1] => [1,1] => [[1,1],[]]
 => 1
[3,1,2] => [1,2] => [2] => [[2],[]]
 => 1
[3,2,1] => [2,1] => [1,1] => [[1,1],[]]
 => 1
[1,2,3,4] => [1,2,3] => [3] => [[3],[]]
 => 1
[1,2,4,3] => [1,2,3] => [3] => [[3],[]]
 => 1
[1,3,2,4] => [1,3,2] => [2,1] => [[2,2],[1]]
 => 2
[1,3,4,2] => [1,3,2] => [2,1] => [[2,2],[1]]
 => 2
[1,4,2,3] => [1,2,3] => [3] => [[3],[]]
 => 1
[1,4,3,2] => [1,3,2] => [2,1] => [[2,2],[1]]
 => 2
[2,1,3,4] => [2,1,3] => [1,2] => [[2,1],[]]
 => 1
[2,1,4,3] => [2,1,3] => [1,2] => [[2,1],[]]
 => 1
[2,3,1,4] => [2,3,1] => [2,1] => [[2,2],[1]]
 => 2
[2,3,4,1] => [2,3,1] => [2,1] => [[2,2],[1]]
 => 2
[2,4,1,3] => [2,1,3] => [1,2] => [[2,1],[]]
 => 1
[2,4,3,1] => [2,3,1] => [2,1] => [[2,2],[1]]
 => 2
[3,1,2,4] => [3,1,2] => [1,2] => [[2,1],[]]
 => 1
[3,1,4,2] => [3,1,2] => [1,2] => [[2,1],[]]
 => 1
[3,2,1,4] => [3,2,1] => [1,1,1] => [[1,1,1],[]]
 => 1
[3,2,4,1] => [3,2,1] => [1,1,1] => [[1,1,1],[]]
 => 1
[3,4,1,2] => [3,1,2] => [1,2] => [[2,1],[]]
 => 1
[3,4,2,1] => [3,2,1] => [1,1,1] => [[1,1,1],[]]
 => 1
[4,1,2,3] => [1,2,3] => [3] => [[3],[]]
 => 1
[4,1,3,2] => [1,3,2] => [2,1] => [[2,2],[1]]
 => 2
[4,2,1,3] => [2,1,3] => [1,2] => [[2,1],[]]
 => 1
[4,2,3,1] => [2,3,1] => [2,1] => [[2,2],[1]]
 => 2
[4,3,1,2] => [3,1,2] => [1,2] => [[2,1],[]]
 => 1
[4,3,2,1] => [3,2,1] => [1,1,1] => [[1,1,1],[]]
 => 1
[1,2,3,4,5] => [1,2,3,4] => [4] => [[4],[]]
 => 1
[1,2,3,5,4] => [1,2,3,4] => [4] => [[4],[]]
 => 1
[1,2,4,3,5] => [1,2,4,3] => [3,1] => [[3,3],[2]]
 => 2
[1,2,4,5,3] => [1,2,4,3] => [3,1] => [[3,3],[2]]
 => 2
[1,2,5,3,4] => [1,2,3,4] => [4] => [[4],[]]
 => 1
[1,2,5,4,3] => [1,2,4,3] => [3,1] => [[3,3],[2]]
 => 2
[1,3,2,4,5] => [1,3,2,4] => [2,2] => [[3,2],[1]]
 => 2
[1,3,2,5,4] => [1,3,2,4] => [2,2] => [[3,2],[1]]
 => 2
[1,3,4,2,5] => [1,3,4,2] => [3,1] => [[3,3],[2]]
 => 2
[1,3,4,5,2] => [1,3,4,2] => [3,1] => [[3,3],[2]]
 => 2
[1,3,5,2,4] => [1,3,2,4] => [2,2] => [[3,2],[1]]
 => 2
[1,3,5,4,2] => [1,3,4,2] => [3,1] => [[3,3],[2]]
 => 2
[1,4,2,3,5] => [1,4,2,3] => [2,2] => [[3,2],[1]]
 => 2
[1,4,2,5,3] => [1,4,2,3] => [2,2] => [[3,2],[1]]
 => 2
[1,4,3,2,5] => [1,4,3,2] => [2,1,1] => [[2,2,2],[1,1]]
 => 2
[1,4,3,5,2] => [1,4,3,2] => [2,1,1] => [[2,2,2],[1,1]]
 => 2
[1,4,5,2,3] => [1,4,2,3] => [2,2] => [[3,2],[1]]
 => 2
[1,4,5,3,2] => [1,4,3,2] => [2,1,1] => [[2,2,2],[1,1]]
 => 2
[1,2,3,4,5,6,7] => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? = 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? = 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5] => [5,1] => [[5,5],[4]]
 => ? = 2
[1,2,3,4,6,7,5] => [1,2,3,4,6,5] => [5,1] => [[5,5],[4]]
 => ? = 2
[1,2,3,4,7,5,6] => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? = 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,5] => [5,1] => [[5,5],[4]]
 => ? = 2
[1,2,3,5,4,6,7] => [1,2,3,5,4,6] => [4,2] => [[5,4],[3]]
 => ? = 2
[1,2,3,5,4,7,6] => [1,2,3,5,4,6] => [4,2] => [[5,4],[3]]
 => ? = 2
[1,2,3,5,6,4,7] => [1,2,3,5,6,4] => [5,1] => [[5,5],[4]]
 => ? = 2
[1,2,3,5,6,7,4] => [1,2,3,5,6,4] => [5,1] => [[5,5],[4]]
 => ? = 2
[1,2,3,5,7,4,6] => [1,2,3,5,4,6] => [4,2] => [[5,4],[3]]
 => ? = 2
[1,2,3,5,7,6,4] => [1,2,3,5,6,4] => [5,1] => [[5,5],[4]]
 => ? = 2
[1,2,3,6,4,5,7] => [1,2,3,6,4,5] => [4,2] => [[5,4],[3]]
 => ? = 2
[1,2,3,6,4,7,5] => [1,2,3,6,4,5] => [4,2] => [[5,4],[3]]
 => ? = 2
[1,2,3,6,5,4,7] => [1,2,3,6,5,4] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 2
[1,2,3,6,5,7,4] => [1,2,3,6,5,4] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 2
[1,2,3,6,7,4,5] => [1,2,3,6,4,5] => [4,2] => [[5,4],[3]]
 => ? = 2
[1,2,3,6,7,5,4] => [1,2,3,6,5,4] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 2
[1,2,3,7,4,5,6] => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? = 1
[1,2,3,7,4,6,5] => [1,2,3,4,6,5] => [5,1] => [[5,5],[4]]
 => ? = 2
[1,2,3,7,5,4,6] => [1,2,3,5,4,6] => [4,2] => [[5,4],[3]]
 => ? = 2
[1,2,3,7,5,6,4] => [1,2,3,5,6,4] => [5,1] => [[5,5],[4]]
 => ? = 2
[1,2,3,7,6,4,5] => [1,2,3,6,4,5] => [4,2] => [[5,4],[3]]
 => ? = 2
[1,2,3,7,6,5,4] => [1,2,3,6,5,4] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 2
[1,2,4,3,5,6,7] => [1,2,4,3,5,6] => [3,3] => [[5,3],[2]]
 => ? = 2
[1,2,4,3,5,7,6] => [1,2,4,3,5,6] => [3,3] => [[5,3],[2]]
 => ? = 2
[1,2,4,3,6,5,7] => [1,2,4,3,6,5] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 3
[1,2,4,3,6,7,5] => [1,2,4,3,6,5] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 3
[1,2,4,3,7,5,6] => [1,2,4,3,5,6] => [3,3] => [[5,3],[2]]
 => ? = 2
[1,2,4,3,7,6,5] => [1,2,4,3,6,5] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 3
[1,2,4,5,3,6,7] => [1,2,4,5,3,6] => [4,2] => [[5,4],[3]]
 => ? = 2
[1,2,4,5,3,7,6] => [1,2,4,5,3,6] => [4,2] => [[5,4],[3]]
 => ? = 2
[1,2,4,5,6,3,7] => [1,2,4,5,6,3] => [5,1] => [[5,5],[4]]
 => ? = 2
[1,2,4,5,6,7,3] => [1,2,4,5,6,3] => [5,1] => [[5,5],[4]]
 => ? = 2
[1,2,4,5,7,3,6] => [1,2,4,5,3,6] => [4,2] => [[5,4],[3]]
 => ? = 2
[1,2,4,5,7,6,3] => [1,2,4,5,6,3] => [5,1] => [[5,5],[4]]
 => ? = 2
[1,2,4,6,3,5,7] => [1,2,4,6,3,5] => [4,2] => [[5,4],[3]]
 => ? = 2
[1,2,4,6,3,7,5] => [1,2,4,6,3,5] => [4,2] => [[5,4],[3]]
 => ? = 2
[1,2,4,6,5,3,7] => [1,2,4,6,5,3] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 2
[1,2,4,6,5,7,3] => [1,2,4,6,5,3] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 2
[1,2,4,6,7,3,5] => [1,2,4,6,3,5] => [4,2] => [[5,4],[3]]
 => ? = 2
[1,2,4,6,7,5,3] => [1,2,4,6,5,3] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 2
[1,2,4,7,3,5,6] => [1,2,4,3,5,6] => [3,3] => [[5,3],[2]]
 => ? = 2
[1,2,4,7,3,6,5] => [1,2,4,3,6,5] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 3
[1,2,4,7,5,3,6] => [1,2,4,5,3,6] => [4,2] => [[5,4],[3]]
 => ? = 2
[1,2,4,7,5,6,3] => [1,2,4,5,6,3] => [5,1] => [[5,5],[4]]
 => ? = 2
[1,2,4,7,6,3,5] => [1,2,4,6,3,5] => [4,2] => [[5,4],[3]]
 => ? = 2
[1,2,4,7,6,5,3] => [1,2,4,6,5,3] => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 2
[1,2,5,3,4,6,7] => [1,2,5,3,4,6] => [3,3] => [[5,3],[2]]
 => ? = 2
[1,2,5,3,4,7,6] => [1,2,5,3,4,6] => [3,3] => [[5,3],[2]]
 => ? = 2

Description

The number of inner corners of a skew partition.

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St000068The number of minimal elements in a poset.