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Matching statistic: St000099
(load all 37 compositions to match this statistic)
(load all 37 compositions to match this statistic)
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 1
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 2
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 1
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 1
[4,3,2,1] => 1
[1,2,3,4,5] => 1
[1,2,3,5,4] => 2
[1,2,4,3,5] => 2
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 2
[1,3,2,4,5] => 2
[1,3,2,5,4] => 3
[1,3,4,2,5] => 2
[1,3,4,5,2] => 2
[1,3,5,2,4] => 2
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 3
[1,4,3,2,5] => 2
[1,4,3,5,2] => 3
[1,4,5,2,3] => 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 37 compositions to match this statistic)
(load all 37 compositions to match this statistic)
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 0 = 1 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 0 = 1 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 0 = 1 - 1
[3,2,1] => 0 = 1 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => 1 = 2 - 1
[1,3,4,2] => 1 = 2 - 1
[1,4,2,3] => 1 = 2 - 1
[1,4,3,2] => 1 = 2 - 1
[2,1,3,4] => 0 = 1 - 1
[2,1,4,3] => 1 = 2 - 1
[2,3,1,4] => 1 = 2 - 1
[2,3,4,1] => 1 = 2 - 1
[2,4,1,3] => 1 = 2 - 1
[2,4,3,1] => 1 = 2 - 1
[3,1,2,4] => 0 = 1 - 1
[3,1,4,2] => 1 = 2 - 1
[3,2,1,4] => 0 = 1 - 1
[3,2,4,1] => 1 = 2 - 1
[3,4,1,2] => 1 = 2 - 1
[3,4,2,1] => 1 = 2 - 1
[4,1,2,3] => 0 = 1 - 1
[4,1,3,2] => 1 = 2 - 1
[4,2,1,3] => 0 = 1 - 1
[4,2,3,1] => 1 = 2 - 1
[4,3,1,2] => 0 = 1 - 1
[4,3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => 1 = 2 - 1
[1,2,4,3,5] => 1 = 2 - 1
[1,2,4,5,3] => 1 = 2 - 1
[1,2,5,3,4] => 1 = 2 - 1
[1,2,5,4,3] => 1 = 2 - 1
[1,3,2,4,5] => 1 = 2 - 1
[1,3,2,5,4] => 2 = 3 - 1
[1,3,4,2,5] => 1 = 2 - 1
[1,3,4,5,2] => 1 = 2 - 1
[1,3,5,2,4] => 1 = 2 - 1
[1,3,5,4,2] => 1 = 2 - 1
[1,4,2,3,5] => 1 = 2 - 1
[1,4,2,5,3] => 2 = 3 - 1
[1,4,3,2,5] => 1 = 2 - 1
[1,4,3,5,2] => 2 = 3 - 1
[1,4,5,2,3] => 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 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 1
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 1
[1,3,2] => [3,1,2] => 2
[2,1,3] => [2,3,1] => 1
[2,3,1] => [2,1,3] => 2
[3,1,2] => [1,3,2] => 1
[3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [4,1,2,3] => 2
[1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [3,1,2,4] => 2
[1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [4,3,1,2] => 2
[2,1,3,4] => [2,3,4,1] => 1
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [2,3,1,4] => 2
[2,3,4,1] => [2,1,3,4] => 2
[2,4,1,3] => [2,4,1,3] => 2
[2,4,3,1] => [4,2,1,3] => 2
[3,1,2,4] => [1,3,4,2] => 1
[3,1,4,2] => [3,1,4,2] => 2
[3,2,1,4] => [3,4,2,1] => 1
[3,2,4,1] => [3,2,4,1] => 2
[3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => [3,2,1,4] => 2
[4,1,2,3] => [1,2,4,3] => 1
[4,1,3,2] => [4,1,3,2] => 2
[4,2,1,3] => [2,4,3,1] => 1
[4,2,3,1] => [4,2,3,1] => 2
[4,3,1,2] => [1,4,3,2] => 1
[4,3,2,1] => [4,3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [5,1,2,3,4] => 2
[1,2,4,3,5] => [1,4,2,3,5] => 2
[1,2,4,5,3] => [4,1,2,3,5] => 2
[1,2,5,3,4] => [1,5,2,3,4] => 2
[1,2,5,4,3] => [5,4,1,2,3] => 2
[1,3,2,4,5] => [1,3,4,2,5] => 2
[1,3,2,5,4] => [3,1,5,2,4] => 3
[1,3,4,2,5] => [1,3,2,4,5] => 2
[1,3,4,5,2] => [3,1,2,4,5] => 2
[1,3,5,2,4] => [3,5,1,2,4] => 2
[1,3,5,4,2] => [5,3,1,2,4] => 2
[1,4,2,3,5] => [1,2,4,3,5] => 2
[1,4,2,5,3] => [4,1,5,2,3] => 3
[1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,3,5,2] => [4,1,3,2,5] => 3
[1,4,5,2,3] => [4,5,1,2,3] => 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 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00069: Permutations —complement⟶ 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%
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] => [1] => [.,.]
=> 1
[1,2] => [2,1] => [[.,.],.]
=> 1
[2,1] => [1,2] => [.,[.,.]]
=> 1
[1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 1
[1,3,2] => [3,1,2] => [[.,.],[.,.]]
=> 2
[2,1,3] => [2,3,1] => [[.,[.,.]],.]
=> 1
[2,3,1] => [2,1,3] => [[.,.],[.,.]]
=> 2
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> 1
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 1
[1,2,4,3] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> 2
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> 2
[1,3,4,2] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> 2
[1,4,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> 2
[1,4,3,2] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2
[2,1,3,4] => [3,4,2,1] => [[[.,[.,.]],.],.]
=> 1
[2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2
[2,3,1,4] => [3,2,4,1] => [[[.,.],[.,.]],.]
=> 2
[2,3,4,1] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> 2
[2,4,1,3] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> 2
[2,4,3,1] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 2
[3,1,2,4] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> 1
[3,1,4,2] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2
[3,2,1,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 1
[3,2,4,1] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 2
[3,4,2,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 2
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[4,2,1,3] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2
[4,3,1,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 1
[1,2,3,5,4] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> 2
[1,2,4,3,5] => [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> 2
[1,2,4,5,3] => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> 2
[1,2,5,3,4] => [5,4,1,3,2] => [[[.,.],.],[[.,.],.]]
=> 2
[1,2,5,4,3] => [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> 2
[1,3,2,4,5] => [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> 2
[1,3,2,5,4] => [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> 3
[1,3,4,2,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> 2
[1,3,4,5,2] => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> 2
[1,3,5,2,4] => [5,3,1,4,2] => [[[.,.],.],[[.,.],.]]
=> 2
[1,3,5,4,2] => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> 2
[1,4,2,3,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> 2
[1,4,2,5,3] => [5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
=> 3
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> 2
[1,4,3,5,2] => [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> 3
[1,4,5,2,3] => [5,2,1,4,3] => [[[.,.],.],[[.,.],.]]
=> 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
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
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%
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] => [1] => 1 => 1
[1,2] => [2] => 10 => 1
[2,1] => [1,1] => 11 => 1
[1,2,3] => [3] => 100 => 1
[1,3,2] => [2,1] => 101 => 2
[2,1,3] => [1,2] => 110 => 1
[2,3,1] => [2,1] => 101 => 2
[3,1,2] => [1,2] => 110 => 1
[3,2,1] => [1,1,1] => 111 => 1
[1,2,3,4] => [4] => 1000 => 1
[1,2,4,3] => [3,1] => 1001 => 2
[1,3,2,4] => [2,2] => 1010 => 2
[1,3,4,2] => [3,1] => 1001 => 2
[1,4,2,3] => [2,2] => 1010 => 2
[1,4,3,2] => [2,1,1] => 1011 => 2
[2,1,3,4] => [1,3] => 1100 => 1
[2,1,4,3] => [1,2,1] => 1101 => 2
[2,3,1,4] => [2,2] => 1010 => 2
[2,3,4,1] => [3,1] => 1001 => 2
[2,4,1,3] => [2,2] => 1010 => 2
[2,4,3,1] => [2,1,1] => 1011 => 2
[3,1,2,4] => [1,3] => 1100 => 1
[3,1,4,2] => [1,2,1] => 1101 => 2
[3,2,1,4] => [1,1,2] => 1110 => 1
[3,2,4,1] => [1,2,1] => 1101 => 2
[3,4,1,2] => [2,2] => 1010 => 2
[3,4,2,1] => [2,1,1] => 1011 => 2
[4,1,2,3] => [1,3] => 1100 => 1
[4,1,3,2] => [1,2,1] => 1101 => 2
[4,2,1,3] => [1,1,2] => 1110 => 1
[4,2,3,1] => [1,2,1] => 1101 => 2
[4,3,1,2] => [1,1,2] => 1110 => 1
[4,3,2,1] => [1,1,1,1] => 1111 => 1
[1,2,3,4,5] => [5] => 10000 => 1
[1,2,3,5,4] => [4,1] => 10001 => 2
[1,2,4,3,5] => [3,2] => 10010 => 2
[1,2,4,5,3] => [4,1] => 10001 => 2
[1,2,5,3,4] => [3,2] => 10010 => 2
[1,2,5,4,3] => [3,1,1] => 10011 => 2
[1,3,2,4,5] => [2,3] => 10100 => 2
[1,3,2,5,4] => [2,2,1] => 10101 => 3
[1,3,4,2,5] => [3,2] => 10010 => 2
[1,3,4,5,2] => [4,1] => 10001 => 2
[1,3,5,2,4] => [3,2] => 10010 => 2
[1,3,5,4,2] => [3,1,1] => 10011 => 2
[1,4,2,3,5] => [2,3] => 10100 => 2
[1,4,2,5,3] => [2,2,1] => 10101 => 3
[1,4,3,2,5] => [2,1,2] => 10110 => 2
[1,4,3,5,2] => [2,2,1] => 10101 => 3
[1,4,5,2,3] => [3,2] => 10010 => 2
Description
The number of runs of ones in a binary word.
Matching statistic: St000196
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00069: Permutations —complement⟶ 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%
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] => [1] => [.,.]
=> 0 = 1 - 1
[1,2] => [2,1] => [[.,.],.]
=> 0 = 1 - 1
[2,1] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 0 = 1 - 1
[1,3,2] => [3,1,2] => [[.,.],[.,.]]
=> 1 = 2 - 1
[2,1,3] => [2,3,1] => [[.,[.,.]],.]
=> 0 = 1 - 1
[2,3,1] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 0 = 1 - 1
[1,2,4,3] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> 1 = 2 - 1
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> 1 = 2 - 1
[1,3,4,2] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> 1 = 2 - 1
[1,4,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> 1 = 2 - 1
[1,4,3,2] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[2,1,3,4] => [3,4,2,1] => [[[.,[.,.]],.],.]
=> 0 = 1 - 1
[2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 1 = 2 - 1
[2,3,1,4] => [3,2,4,1] => [[[.,.],[.,.]],.]
=> 1 = 2 - 1
[2,3,4,1] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> 1 = 2 - 1
[2,4,3,1] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[3,1,2,4] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> 0 = 1 - 1
[3,1,4,2] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 1 = 2 - 1
[3,2,1,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 0 = 1 - 1
[3,2,4,1] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 1 = 2 - 1
[3,4,2,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[4,2,1,3] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[4,3,1,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 0 = 1 - 1
[1,2,3,5,4] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> 1 = 2 - 1
[1,2,4,3,5] => [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> 1 = 2 - 1
[1,2,4,5,3] => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> 1 = 2 - 1
[1,2,5,3,4] => [5,4,1,3,2] => [[[.,.],.],[[.,.],.]]
=> 1 = 2 - 1
[1,2,5,4,3] => [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> 1 = 2 - 1
[1,3,2,4,5] => [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> 1 = 2 - 1
[1,3,2,5,4] => [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> 2 = 3 - 1
[1,3,4,2,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> 1 = 2 - 1
[1,3,4,5,2] => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> 1 = 2 - 1
[1,3,5,2,4] => [5,3,1,4,2] => [[[.,.],.],[[.,.],.]]
=> 1 = 2 - 1
[1,3,5,4,2] => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> 1 = 2 - 1
[1,4,2,3,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> 1 = 2 - 1
[1,4,2,5,3] => [5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
=> 2 = 3 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> 1 = 2 - 1
[1,4,3,5,2] => [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> 2 = 3 - 1
[1,4,5,2,3] => [5,2,1,4,3] => [[[.,.],.],[[.,.],.]]
=> 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 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
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%
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] => [1] => 1 => 0 = 1 - 1
[1,2] => [2] => 10 => 0 = 1 - 1
[2,1] => [1,1] => 11 => 0 = 1 - 1
[1,2,3] => [3] => 100 => 0 = 1 - 1
[1,3,2] => [2,1] => 101 => 1 = 2 - 1
[2,1,3] => [1,2] => 110 => 0 = 1 - 1
[2,3,1] => [2,1] => 101 => 1 = 2 - 1
[3,1,2] => [1,2] => 110 => 0 = 1 - 1
[3,2,1] => [1,1,1] => 111 => 0 = 1 - 1
[1,2,3,4] => [4] => 1000 => 0 = 1 - 1
[1,2,4,3] => [3,1] => 1001 => 1 = 2 - 1
[1,3,2,4] => [2,2] => 1010 => 1 = 2 - 1
[1,3,4,2] => [3,1] => 1001 => 1 = 2 - 1
[1,4,2,3] => [2,2] => 1010 => 1 = 2 - 1
[1,4,3,2] => [2,1,1] => 1011 => 1 = 2 - 1
[2,1,3,4] => [1,3] => 1100 => 0 = 1 - 1
[2,1,4,3] => [1,2,1] => 1101 => 1 = 2 - 1
[2,3,1,4] => [2,2] => 1010 => 1 = 2 - 1
[2,3,4,1] => [3,1] => 1001 => 1 = 2 - 1
[2,4,1,3] => [2,2] => 1010 => 1 = 2 - 1
[2,4,3,1] => [2,1,1] => 1011 => 1 = 2 - 1
[3,1,2,4] => [1,3] => 1100 => 0 = 1 - 1
[3,1,4,2] => [1,2,1] => 1101 => 1 = 2 - 1
[3,2,1,4] => [1,1,2] => 1110 => 0 = 1 - 1
[3,2,4,1] => [1,2,1] => 1101 => 1 = 2 - 1
[3,4,1,2] => [2,2] => 1010 => 1 = 2 - 1
[3,4,2,1] => [2,1,1] => 1011 => 1 = 2 - 1
[4,1,2,3] => [1,3] => 1100 => 0 = 1 - 1
[4,1,3,2] => [1,2,1] => 1101 => 1 = 2 - 1
[4,2,1,3] => [1,1,2] => 1110 => 0 = 1 - 1
[4,2,3,1] => [1,2,1] => 1101 => 1 = 2 - 1
[4,3,1,2] => [1,1,2] => 1110 => 0 = 1 - 1
[4,3,2,1] => [1,1,1,1] => 1111 => 0 = 1 - 1
[1,2,3,4,5] => [5] => 10000 => 0 = 1 - 1
[1,2,3,5,4] => [4,1] => 10001 => 1 = 2 - 1
[1,2,4,3,5] => [3,2] => 10010 => 1 = 2 - 1
[1,2,4,5,3] => [4,1] => 10001 => 1 = 2 - 1
[1,2,5,3,4] => [3,2] => 10010 => 1 = 2 - 1
[1,2,5,4,3] => [3,1,1] => 10011 => 1 = 2 - 1
[1,3,2,4,5] => [2,3] => 10100 => 1 = 2 - 1
[1,3,2,5,4] => [2,2,1] => 10101 => 2 = 3 - 1
[1,3,4,2,5] => [3,2] => 10010 => 1 = 2 - 1
[1,3,4,5,2] => [4,1] => 10001 => 1 = 2 - 1
[1,3,5,2,4] => [3,2] => 10010 => 1 = 2 - 1
[1,3,5,4,2] => [3,1,1] => 10011 => 1 = 2 - 1
[1,4,2,3,5] => [2,3] => 10100 => 1 = 2 - 1
[1,4,2,5,3] => [2,2,1] => 10101 => 2 = 3 - 1
[1,4,3,2,5] => [2,1,2] => 10110 => 1 = 2 - 1
[1,4,3,5,2] => [2,2,1] => 10101 => 2 = 3 - 1
[1,4,5,2,3] => [3,2] => 10010 => 1 = 2 - 1
Description
The number of ascents of a binary word.
Matching statistic: St000386
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0 = 1 - 1
[1,2] => [2] => [1,1,0,0]
=> 0 = 1 - 1
[2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000010
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [1] => [1]
=> 1
[1,2] => [[1,2]]
=> [2] => [2]
=> 1
[2,1] => [[1],[2]]
=> [2] => [2]
=> 1
[1,2,3] => [[1,2,3]]
=> [3] => [3]
=> 1
[1,3,2] => [[1,2],[3]]
=> [2,1] => [2,1]
=> 2
[2,1,3] => [[1,3],[2]]
=> [3] => [3]
=> 1
[2,3,1] => [[1,2],[3]]
=> [2,1] => [2,1]
=> 2
[3,1,2] => [[1,3],[2]]
=> [3] => [3]
=> 1
[3,2,1] => [[1],[2],[3]]
=> [3] => [3]
=> 1
[1,2,3,4] => [[1,2,3,4]]
=> [4] => [4]
=> 1
[1,2,4,3] => [[1,2,3],[4]]
=> [3,1] => [3,1]
=> 2
[1,3,2,4] => [[1,2,4],[3]]
=> [2,2] => [2,2]
=> 2
[1,3,4,2] => [[1,2,3],[4]]
=> [3,1] => [3,1]
=> 2
[1,4,2,3] => [[1,2,4],[3]]
=> [2,2] => [2,2]
=> 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [2,2] => [2,2]
=> 2
[2,1,3,4] => [[1,3,4],[2]]
=> [4] => [4]
=> 1
[2,1,4,3] => [[1,3],[2,4]]
=> [3,1] => [3,1]
=> 2
[2,3,1,4] => [[1,2,4],[3]]
=> [2,2] => [2,2]
=> 2
[2,3,4,1] => [[1,2,3],[4]]
=> [3,1] => [3,1]
=> 2
[2,4,1,3] => [[1,2],[3,4]]
=> [2,2] => [2,2]
=> 2
[2,4,3,1] => [[1,2],[3],[4]]
=> [2,2] => [2,2]
=> 2
[3,1,2,4] => [[1,3,4],[2]]
=> [4] => [4]
=> 1
[3,1,4,2] => [[1,3],[2,4]]
=> [3,1] => [3,1]
=> 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [4] => [4]
=> 1
[3,2,4,1] => [[1,3],[2],[4]]
=> [3,1] => [3,1]
=> 2
[3,4,1,2] => [[1,2],[3,4]]
=> [2,2] => [2,2]
=> 2
[3,4,2,1] => [[1,2],[3],[4]]
=> [2,2] => [2,2]
=> 2
[4,1,2,3] => [[1,3,4],[2]]
=> [4] => [4]
=> 1
[4,1,3,2] => [[1,3],[2],[4]]
=> [3,1] => [3,1]
=> 2
[4,2,1,3] => [[1,4],[2],[3]]
=> [4] => [4]
=> 1
[4,2,3,1] => [[1,3],[2],[4]]
=> [3,1] => [3,1]
=> 2
[4,3,1,2] => [[1,4],[2],[3]]
=> [4] => [4]
=> 1
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => [4]
=> 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => [5]
=> 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [4,1] => [4,1]
=> 2
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [3,2] => [3,2]
=> 2
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [4,1] => [4,1]
=> 2
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [3,2] => [3,2]
=> 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [3,2] => [3,2]
=> 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [2,3] => [3,2]
=> 2
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [2,2,1] => [2,2,1]
=> 3
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [3,2] => [3,2]
=> 2
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [4,1] => [4,1]
=> 2
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [3,2] => [3,2]
=> 2
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [3,2] => [3,2]
=> 2
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [2,3] => [3,2]
=> 2
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> [2,2,1] => [2,2,1]
=> 3
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [2,3] => [3,2]
=> 2
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [2,2,1] => [2,2,1]
=> 3
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [3,2] => [3,2]
=> 2
Description
The length of the partition.
Matching statistic: St000068
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> ([],1)
=> 1
[1,2] => [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1
[2,1] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
[1,2,3] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[1,3,4,2] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,4,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,4,3,2] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,1,3,4] => [3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,3,1,4] => [3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[2,3,4,1] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,4,1,3] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,4,3,1] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1,2,4] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,4,2] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,2,1,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,2,4,1] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,4,2,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[4,2,1,3] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[4,3,1,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,2,4,3,5] => [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,2,4,5,3] => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,2,5,3,4] => [5,4,1,3,2] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,3,2,5,4] => [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[1,3,4,2,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,3,4,5,2] => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,3,5,2,4] => [5,3,1,4,2] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,4,2,5,3] => [5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[1,4,3,5,2] => [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[1,4,5,2,3] => [5,2,1,4,3] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
Description
The number of minimal elements in a poset.
The following 50 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000071The number of maximal chains in a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000389The number of runs of ones of odd length in a binary word. St000527The width of the poset. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000159The number of distinct parts of the integer partition. St000272The treewidth of a graph. St000291The number of descents of a binary word. St000356The number of occurrences of the pattern 13-2. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000523The number of 2-protected nodes of a rooted tree. St000536The pathwidth of a graph. St000632The jump number of the poset. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001277The degeneracy of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001689The number of celebrities in a graph. St001712The number of natural descents of a standard Young tableau. St001792The arboricity of a graph. St001812The biclique partition number of a graph. St000353The number of inner valleys of a permutation. St000568The hook number of a binary tree. St000806The semiperimeter of the associated bargraph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001330The hat guessing number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St000256The number of parts from which one can substract 2 and still get an integer partition. St001487The number of inner corners of a skew partition. St000354The number of recoils of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001960The number of descents of a permutation minus one if its first entry is not one.
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