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Your data matches 44 different statistics following compositions of up to 3 maps.
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Matching statistic: St000092
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ 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
[+,+] => [1,2] => 1
[-,+] => [2,1] => 1
[+,-] => [1,2] => 1
[-,-] => [1,2] => 1
[2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => 1
[-,+,+] => [2,3,1] => 1
[+,-,+] => [1,3,2] => 1
[+,+,-] => [1,2,3] => 1
[-,-,+] => [3,1,2] => 2
[-,+,-] => [2,1,3] => 2
[+,-,-] => [1,2,3] => 1
[-,-,-] => [1,2,3] => 1
[+,3,2] => [1,3,2] => 1
[-,3,2] => [3,1,2] => 2
[2,1,+] => [2,3,1] => 1
[2,1,-] => [2,1,3] => 2
[2,3,1] => [3,1,2] => 2
[3,1,2] => [2,3,1] => 1
[3,+,1] => [2,3,1] => 1
[3,-,1] => [3,1,2] => 2
[+,+,+,+] => [1,2,3,4] => 1
[-,+,+,+] => [2,3,4,1] => 1
[+,-,+,+] => [1,3,4,2] => 1
[+,+,-,+] => [1,2,4,3] => 1
[+,+,+,-] => [1,2,3,4] => 1
[-,-,+,+] => [3,4,1,2] => 2
[-,+,-,+] => [2,4,1,3] => 2
[-,+,+,-] => [2,3,1,4] => 2
[+,-,-,+] => [1,4,2,3] => 2
[+,-,+,-] => [1,3,2,4] => 2
[+,+,-,-] => [1,2,3,4] => 1
[-,-,-,+] => [4,1,2,3] => 2
[-,-,+,-] => [3,1,2,4] => 2
[-,+,-,-] => [2,1,3,4] => 2
[+,-,-,-] => [1,2,3,4] => 1
[-,-,-,-] => [1,2,3,4] => 1
[+,+,4,3] => [1,2,4,3] => 1
[-,+,4,3] => [2,4,1,3] => 2
[+,-,4,3] => [1,4,2,3] => 2
[-,-,4,3] => [4,1,2,3] => 2
[+,3,2,+] => [1,3,4,2] => 1
[-,3,2,+] => [3,4,1,2] => 2
[+,3,2,-] => [1,3,2,4] => 2
[-,3,2,-] => [3,1,2,4] => 2
[+,3,4,2] => [1,4,2,3] => 2
[-,3,4,2] => [4,1,2,3] => 2
[+,4,2,3] => [1,3,4,2] => 1
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: St000099
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => 1
[-] => [1] => 1
[+,+] => [1,2] => 1
[-,+] => [2,1] => 1
[+,-] => [1,2] => 1
[-,-] => [1,2] => 1
[2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => 1
[-,+,+] => [2,3,1] => 2
[+,-,+] => [1,3,2] => 2
[+,+,-] => [1,2,3] => 1
[-,-,+] => [3,1,2] => 1
[-,+,-] => [2,1,3] => 1
[+,-,-] => [1,2,3] => 1
[-,-,-] => [1,2,3] => 1
[+,3,2] => [1,3,2] => 2
[-,3,2] => [3,1,2] => 1
[2,1,+] => [2,3,1] => 2
[2,1,-] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 1
[3,1,2] => [2,3,1] => 2
[3,+,1] => [2,3,1] => 2
[3,-,1] => [3,1,2] => 1
[+,+,+,+] => [1,2,3,4] => 1
[-,+,+,+] => [2,3,4,1] => 2
[+,-,+,+] => [1,3,4,2] => 2
[+,+,-,+] => [1,2,4,3] => 2
[+,+,+,-] => [1,2,3,4] => 1
[-,-,+,+] => [3,4,1,2] => 2
[-,+,-,+] => [2,4,1,3] => 2
[-,+,+,-] => [2,3,1,4] => 2
[+,-,-,+] => [1,4,2,3] => 2
[+,-,+,-] => [1,3,2,4] => 2
[+,+,-,-] => [1,2,3,4] => 1
[-,-,-,+] => [4,1,2,3] => 1
[-,-,+,-] => [3,1,2,4] => 1
[-,+,-,-] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => 1
[-,-,-,-] => [1,2,3,4] => 1
[+,+,4,3] => [1,2,4,3] => 2
[-,+,4,3] => [2,4,1,3] => 2
[+,-,4,3] => [1,4,2,3] => 2
[-,-,4,3] => [4,1,2,3] => 1
[+,3,2,+] => [1,3,4,2] => 2
[-,3,2,+] => [3,4,1,2] => 2
[+,3,2,-] => [1,3,2,4] => 2
[-,3,2,-] => [3,1,2,4] => 1
[+,3,4,2] => [1,4,2,3] => 2
[-,3,4,2] => [4,1,2,3] => 1
[+,4,2,3] => [1,3,4,2] => 2
Description
The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is [[St000353]].
Matching statistic: St000862
(load all 42 compositions to match this statistic)
(load all 42 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000862: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => 1
[-] => [1] => 1
[+,+] => [1,2] => 1
[-,+] => [2,1] => 1
[+,-] => [1,2] => 1
[-,-] => [1,2] => 1
[2,1] => [2,1] => 1
[+,+,+] => [1,2,3] => 1
[-,+,+] => [2,3,1] => 1
[+,-,+] => [1,3,2] => 2
[+,+,-] => [1,2,3] => 1
[-,-,+] => [3,1,2] => 2
[-,+,-] => [2,1,3] => 1
[+,-,-] => [1,2,3] => 1
[-,-,-] => [1,2,3] => 1
[+,3,2] => [1,3,2] => 2
[-,3,2] => [3,1,2] => 2
[2,1,+] => [2,3,1] => 1
[2,1,-] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 2
[3,1,2] => [2,3,1] => 1
[3,+,1] => [2,3,1] => 1
[3,-,1] => [3,1,2] => 2
[+,+,+,+] => [1,2,3,4] => 1
[-,+,+,+] => [2,3,4,1] => 1
[+,-,+,+] => [1,3,4,2] => 2
[+,+,-,+] => [1,2,4,3] => 2
[+,+,+,-] => [1,2,3,4] => 1
[-,-,+,+] => [3,4,1,2] => 2
[-,+,-,+] => [2,4,1,3] => 2
[-,+,+,-] => [2,3,1,4] => 1
[+,-,-,+] => [1,4,2,3] => 2
[+,-,+,-] => [1,3,2,4] => 2
[+,+,-,-] => [1,2,3,4] => 1
[-,-,-,+] => [4,1,2,3] => 2
[-,-,+,-] => [3,1,2,4] => 2
[-,+,-,-] => [2,1,3,4] => 1
[+,-,-,-] => [1,2,3,4] => 1
[-,-,-,-] => [1,2,3,4] => 1
[+,+,4,3] => [1,2,4,3] => 2
[-,+,4,3] => [2,4,1,3] => 2
[+,-,4,3] => [1,4,2,3] => 2
[-,-,4,3] => [4,1,2,3] => 2
[+,3,2,+] => [1,3,4,2] => 2
[-,3,2,+] => [3,4,1,2] => 2
[+,3,2,-] => [1,3,2,4] => 2
[-,3,2,-] => [3,1,2,4] => 2
[+,3,4,2] => [1,4,2,3] => 2
[-,3,4,2] => [4,1,2,3] => 2
[+,4,2,3] => [1,3,4,2] => 2
Description
The number of parts of the shifted shape of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of parts of the shifted shape.
Matching statistic: St000023
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => 0 = 1 - 1
[-] => [1] => 0 = 1 - 1
[+,+] => [1,2] => 0 = 1 - 1
[-,+] => [2,1] => 0 = 1 - 1
[+,-] => [1,2] => 0 = 1 - 1
[-,-] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => 0 = 1 - 1
[+,+,+] => [1,2,3] => 0 = 1 - 1
[-,+,+] => [2,3,1] => 1 = 2 - 1
[+,-,+] => [1,3,2] => 1 = 2 - 1
[+,+,-] => [1,2,3] => 0 = 1 - 1
[-,-,+] => [3,1,2] => 0 = 1 - 1
[-,+,-] => [2,1,3] => 0 = 1 - 1
[+,-,-] => [1,2,3] => 0 = 1 - 1
[-,-,-] => [1,2,3] => 0 = 1 - 1
[+,3,2] => [1,3,2] => 1 = 2 - 1
[-,3,2] => [3,1,2] => 0 = 1 - 1
[2,1,+] => [2,3,1] => 1 = 2 - 1
[2,1,-] => [2,1,3] => 0 = 1 - 1
[2,3,1] => [3,1,2] => 0 = 1 - 1
[3,1,2] => [2,3,1] => 1 = 2 - 1
[3,+,1] => [2,3,1] => 1 = 2 - 1
[3,-,1] => [3,1,2] => 0 = 1 - 1
[+,+,+,+] => [1,2,3,4] => 0 = 1 - 1
[-,+,+,+] => [2,3,4,1] => 1 = 2 - 1
[+,-,+,+] => [1,3,4,2] => 1 = 2 - 1
[+,+,-,+] => [1,2,4,3] => 1 = 2 - 1
[+,+,+,-] => [1,2,3,4] => 0 = 1 - 1
[-,-,+,+] => [3,4,1,2] => 1 = 2 - 1
[-,+,-,+] => [2,4,1,3] => 1 = 2 - 1
[-,+,+,-] => [2,3,1,4] => 1 = 2 - 1
[+,-,-,+] => [1,4,2,3] => 1 = 2 - 1
[+,-,+,-] => [1,3,2,4] => 1 = 2 - 1
[+,+,-,-] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,+] => [4,1,2,3] => 0 = 1 - 1
[-,-,+,-] => [3,1,2,4] => 0 = 1 - 1
[-,+,-,-] => [2,1,3,4] => 0 = 1 - 1
[+,-,-,-] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => 0 = 1 - 1
[+,+,4,3] => [1,2,4,3] => 1 = 2 - 1
[-,+,4,3] => [2,4,1,3] => 1 = 2 - 1
[+,-,4,3] => [1,4,2,3] => 1 = 2 - 1
[-,-,4,3] => [4,1,2,3] => 0 = 1 - 1
[+,3,2,+] => [1,3,4,2] => 1 = 2 - 1
[-,3,2,+] => [3,4,1,2] => 1 = 2 - 1
[+,3,2,-] => [1,3,2,4] => 1 = 2 - 1
[-,3,2,-] => [3,1,2,4] => 0 = 1 - 1
[+,3,4,2] => [1,4,2,3] => 1 = 2 - 1
[-,3,4,2] => [4,1,2,3] => 0 = 1 - 1
[+,4,2,3] => [1,3,4,2] => 1 = 2 - 1
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
Matching statistic: St000201
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ 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
[+,+] => [1,2] => [.,[.,.]]
=> 1
[-,+] => [2,1] => [[.,.],.]
=> 1
[+,-] => [1,2] => [.,[.,.]]
=> 1
[-,-] => [1,2] => [.,[.,.]]
=> 1
[2,1] => [2,1] => [[.,.],.]
=> 1
[+,+,+] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[-,+,+] => [2,3,1] => [[.,[.,.]],.]
=> 1
[+,-,+] => [1,3,2] => [.,[[.,.],.]]
=> 1
[+,+,-] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[-,-,+] => [3,1,2] => [[.,.],[.,.]]
=> 2
[-,+,-] => [2,1,3] => [[.,.],[.,.]]
=> 2
[+,-,-] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[-,-,-] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[+,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 1
[-,3,2] => [3,1,2] => [[.,.],[.,.]]
=> 2
[2,1,+] => [2,3,1] => [[.,[.,.]],.]
=> 1
[2,1,-] => [2,1,3] => [[.,.],[.,.]]
=> 2
[2,3,1] => [3,1,2] => [[.,.],[.,.]]
=> 2
[3,1,2] => [2,3,1] => [[.,[.,.]],.]
=> 1
[3,+,1] => [2,3,1] => [[.,[.,.]],.]
=> 1
[3,-,1] => [3,1,2] => [[.,.],[.,.]]
=> 2
[+,+,+,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[-,+,+,+] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 1
[+,-,+,+] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
[+,+,-,+] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[+,+,+,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[-,-,+,+] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2
[-,+,-,+] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2
[-,+,+,-] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2
[+,-,-,+] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[+,-,+,-] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2
[+,+,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[-,-,-,+] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2
[-,-,+,-] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 2
[-,+,-,-] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 2
[+,-,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[-,-,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[+,+,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[-,+,4,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2
[+,-,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[-,-,4,3] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2
[+,3,2,+] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
[-,3,2,+] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2
[+,3,2,-] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2
[-,3,2,-] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 2
[+,3,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[-,3,4,2] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2
[+,4,2,3] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
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: St000396
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [.,.]
=> 1
[-] => [1] => [.,.]
=> 1
[+,+] => [1,2] => [.,[.,.]]
=> 1
[-,+] => [2,1] => [[.,.],.]
=> 1
[+,-] => [1,2] => [.,[.,.]]
=> 1
[-,-] => [1,2] => [.,[.,.]]
=> 1
[2,1] => [2,1] => [[.,.],.]
=> 1
[+,+,+] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[-,+,+] => [2,3,1] => [[.,[.,.]],.]
=> 1
[+,-,+] => [1,3,2] => [.,[[.,.],.]]
=> 1
[+,+,-] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[-,-,+] => [3,1,2] => [[.,.],[.,.]]
=> 2
[-,+,-] => [2,1,3] => [[.,.],[.,.]]
=> 2
[+,-,-] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[-,-,-] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[+,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 1
[-,3,2] => [3,1,2] => [[.,.],[.,.]]
=> 2
[2,1,+] => [2,3,1] => [[.,[.,.]],.]
=> 1
[2,1,-] => [2,1,3] => [[.,.],[.,.]]
=> 2
[2,3,1] => [3,1,2] => [[.,.],[.,.]]
=> 2
[3,1,2] => [2,3,1] => [[.,[.,.]],.]
=> 1
[3,+,1] => [2,3,1] => [[.,[.,.]],.]
=> 1
[3,-,1] => [3,1,2] => [[.,.],[.,.]]
=> 2
[+,+,+,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[-,+,+,+] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 1
[+,-,+,+] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
[+,+,-,+] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[+,+,+,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[-,-,+,+] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2
[-,+,-,+] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2
[-,+,+,-] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2
[+,-,-,+] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[+,-,+,-] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2
[+,+,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[-,-,-,+] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2
[-,-,+,-] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 2
[-,+,-,-] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 2
[+,-,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[-,-,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[+,+,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[-,+,4,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2
[+,-,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[-,-,4,3] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2
[+,3,2,+] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
[-,3,2,+] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2
[+,3,2,-] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2
[-,3,2,-] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 2
[+,3,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[-,3,4,2] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2
[+,4,2,3] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
Description
The register function (or Horton-Strahler number) of a binary tree.
This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.
Matching statistic: St000758
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000758: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
St000758: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => 1
[-] => [1] => [1] => 1
[+,+] => [1,2] => [2] => 1
[-,+] => [2,1] => [1,1] => 1
[+,-] => [1,2] => [2] => 1
[-,-] => [1,2] => [2] => 1
[2,1] => [2,1] => [1,1] => 1
[+,+,+] => [1,2,3] => [3] => 1
[-,+,+] => [2,3,1] => [2,1] => 1
[+,-,+] => [1,3,2] => [2,1] => 1
[+,+,-] => [1,2,3] => [3] => 1
[-,-,+] => [3,1,2] => [1,2] => 2
[-,+,-] => [2,1,3] => [1,2] => 2
[+,-,-] => [1,2,3] => [3] => 1
[-,-,-] => [1,2,3] => [3] => 1
[+,3,2] => [1,3,2] => [2,1] => 1
[-,3,2] => [3,1,2] => [1,2] => 2
[2,1,+] => [2,3,1] => [2,1] => 1
[2,1,-] => [2,1,3] => [1,2] => 2
[2,3,1] => [3,1,2] => [1,2] => 2
[3,1,2] => [2,3,1] => [2,1] => 1
[3,+,1] => [2,3,1] => [2,1] => 1
[3,-,1] => [3,1,2] => [1,2] => 2
[+,+,+,+] => [1,2,3,4] => [4] => 1
[-,+,+,+] => [2,3,4,1] => [3,1] => 1
[+,-,+,+] => [1,3,4,2] => [3,1] => 1
[+,+,-,+] => [1,2,4,3] => [3,1] => 1
[+,+,+,-] => [1,2,3,4] => [4] => 1
[-,-,+,+] => [3,4,1,2] => [2,2] => 2
[-,+,-,+] => [2,4,1,3] => [2,2] => 2
[-,+,+,-] => [2,3,1,4] => [2,2] => 2
[+,-,-,+] => [1,4,2,3] => [2,2] => 2
[+,-,+,-] => [1,3,2,4] => [2,2] => 2
[+,+,-,-] => [1,2,3,4] => [4] => 1
[-,-,-,+] => [4,1,2,3] => [1,3] => 2
[-,-,+,-] => [3,1,2,4] => [1,3] => 2
[-,+,-,-] => [2,1,3,4] => [1,3] => 2
[+,-,-,-] => [1,2,3,4] => [4] => 1
[-,-,-,-] => [1,2,3,4] => [4] => 1
[+,+,4,3] => [1,2,4,3] => [3,1] => 1
[-,+,4,3] => [2,4,1,3] => [2,2] => 2
[+,-,4,3] => [1,4,2,3] => [2,2] => 2
[-,-,4,3] => [4,1,2,3] => [1,3] => 2
[+,3,2,+] => [1,3,4,2] => [3,1] => 1
[-,3,2,+] => [3,4,1,2] => [2,2] => 2
[+,3,2,-] => [1,3,2,4] => [2,2] => 2
[-,3,2,-] => [3,1,2,4] => [1,3] => 2
[+,3,4,2] => [1,4,2,3] => [2,2] => 2
[-,3,4,2] => [4,1,2,3] => [1,3] => 2
[+,4,2,3] => [1,3,4,2] => [3,1] => 1
Description
The length of the longest staircase fitting into an integer composition.
For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]
Matching statistic: St000021
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [] => 0 = 1 - 1
[-] => [1] => [] => 0 = 1 - 1
[+,+] => [1,2] => [1] => 0 = 1 - 1
[-,+] => [2,1] => [1] => 0 = 1 - 1
[+,-] => [1,2] => [1] => 0 = 1 - 1
[-,-] => [1,2] => [1] => 0 = 1 - 1
[2,1] => [2,1] => [1] => 0 = 1 - 1
[+,+,+] => [1,2,3] => [1,2] => 0 = 1 - 1
[-,+,+] => [2,3,1] => [2,1] => 1 = 2 - 1
[+,-,+] => [1,3,2] => [1,2] => 0 = 1 - 1
[+,+,-] => [1,2,3] => [1,2] => 0 = 1 - 1
[-,-,+] => [3,1,2] => [1,2] => 0 = 1 - 1
[-,+,-] => [2,1,3] => [2,1] => 1 = 2 - 1
[+,-,-] => [1,2,3] => [1,2] => 0 = 1 - 1
[-,-,-] => [1,2,3] => [1,2] => 0 = 1 - 1
[+,3,2] => [1,3,2] => [1,2] => 0 = 1 - 1
[-,3,2] => [3,1,2] => [1,2] => 0 = 1 - 1
[2,1,+] => [2,3,1] => [2,1] => 1 = 2 - 1
[2,1,-] => [2,1,3] => [2,1] => 1 = 2 - 1
[2,3,1] => [3,1,2] => [1,2] => 0 = 1 - 1
[3,1,2] => [2,3,1] => [2,1] => 1 = 2 - 1
[3,+,1] => [2,3,1] => [2,1] => 1 = 2 - 1
[3,-,1] => [3,1,2] => [1,2] => 0 = 1 - 1
[+,+,+,+] => [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[-,+,+,+] => [2,3,4,1] => [2,3,1] => 1 = 2 - 1
[+,-,+,+] => [1,3,4,2] => [1,3,2] => 1 = 2 - 1
[+,+,-,+] => [1,2,4,3] => [1,2,3] => 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[-,-,+,+] => [3,4,1,2] => [3,1,2] => 1 = 2 - 1
[-,+,-,+] => [2,4,1,3] => [2,1,3] => 1 = 2 - 1
[-,+,+,-] => [2,3,1,4] => [2,3,1] => 1 = 2 - 1
[+,-,-,+] => [1,4,2,3] => [1,2,3] => 0 = 1 - 1
[+,-,+,-] => [1,3,2,4] => [1,3,2] => 1 = 2 - 1
[+,+,-,-] => [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[-,-,-,+] => [4,1,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,+,-] => [3,1,2,4] => [3,1,2] => 1 = 2 - 1
[-,+,-,-] => [2,1,3,4] => [2,1,3] => 1 = 2 - 1
[+,-,-,-] => [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[+,+,4,3] => [1,2,4,3] => [1,2,3] => 0 = 1 - 1
[-,+,4,3] => [2,4,1,3] => [2,1,3] => 1 = 2 - 1
[+,-,4,3] => [1,4,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,4,3] => [4,1,2,3] => [1,2,3] => 0 = 1 - 1
[+,3,2,+] => [1,3,4,2] => [1,3,2] => 1 = 2 - 1
[-,3,2,+] => [3,4,1,2] => [3,1,2] => 1 = 2 - 1
[+,3,2,-] => [1,3,2,4] => [1,3,2] => 1 = 2 - 1
[-,3,2,-] => [3,1,2,4] => [3,1,2] => 1 = 2 - 1
[+,3,4,2] => [1,4,2,3] => [1,2,3] => 0 = 1 - 1
[-,3,4,2] => [4,1,2,3] => [1,2,3] => 0 = 1 - 1
[+,4,2,3] => [1,3,4,2] => [1,3,2] => 1 = 2 - 1
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000035
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [] => 0 = 1 - 1
[-] => [1] => [] => 0 = 1 - 1
[+,+] => [1,2] => [1] => 0 = 1 - 1
[-,+] => [2,1] => [1] => 0 = 1 - 1
[+,-] => [1,2] => [1] => 0 = 1 - 1
[-,-] => [1,2] => [1] => 0 = 1 - 1
[2,1] => [2,1] => [1] => 0 = 1 - 1
[+,+,+] => [1,2,3] => [1,2] => 0 = 1 - 1
[-,+,+] => [2,3,1] => [2,1] => 1 = 2 - 1
[+,-,+] => [1,3,2] => [1,2] => 0 = 1 - 1
[+,+,-] => [1,2,3] => [1,2] => 0 = 1 - 1
[-,-,+] => [3,1,2] => [1,2] => 0 = 1 - 1
[-,+,-] => [2,1,3] => [2,1] => 1 = 2 - 1
[+,-,-] => [1,2,3] => [1,2] => 0 = 1 - 1
[-,-,-] => [1,2,3] => [1,2] => 0 = 1 - 1
[+,3,2] => [1,3,2] => [1,2] => 0 = 1 - 1
[-,3,2] => [3,1,2] => [1,2] => 0 = 1 - 1
[2,1,+] => [2,3,1] => [2,1] => 1 = 2 - 1
[2,1,-] => [2,1,3] => [2,1] => 1 = 2 - 1
[2,3,1] => [3,1,2] => [1,2] => 0 = 1 - 1
[3,1,2] => [2,3,1] => [2,1] => 1 = 2 - 1
[3,+,1] => [2,3,1] => [2,1] => 1 = 2 - 1
[3,-,1] => [3,1,2] => [1,2] => 0 = 1 - 1
[+,+,+,+] => [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[-,+,+,+] => [2,3,4,1] => [2,3,1] => 1 = 2 - 1
[+,-,+,+] => [1,3,4,2] => [1,3,2] => 1 = 2 - 1
[+,+,-,+] => [1,2,4,3] => [1,2,3] => 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[-,-,+,+] => [3,4,1,2] => [3,1,2] => 1 = 2 - 1
[-,+,-,+] => [2,4,1,3] => [2,1,3] => 1 = 2 - 1
[-,+,+,-] => [2,3,1,4] => [2,3,1] => 1 = 2 - 1
[+,-,-,+] => [1,4,2,3] => [1,2,3] => 0 = 1 - 1
[+,-,+,-] => [1,3,2,4] => [1,3,2] => 1 = 2 - 1
[+,+,-,-] => [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[-,-,-,+] => [4,1,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,+,-] => [3,1,2,4] => [3,1,2] => 1 = 2 - 1
[-,+,-,-] => [2,1,3,4] => [2,1,3] => 1 = 2 - 1
[+,-,-,-] => [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[+,+,4,3] => [1,2,4,3] => [1,2,3] => 0 = 1 - 1
[-,+,4,3] => [2,4,1,3] => [2,1,3] => 1 = 2 - 1
[+,-,4,3] => [1,4,2,3] => [1,2,3] => 0 = 1 - 1
[-,-,4,3] => [4,1,2,3] => [1,2,3] => 0 = 1 - 1
[+,3,2,+] => [1,3,4,2] => [1,3,2] => 1 = 2 - 1
[-,3,2,+] => [3,4,1,2] => [3,1,2] => 1 = 2 - 1
[+,3,2,-] => [1,3,2,4] => [1,3,2] => 1 = 2 - 1
[-,3,2,-] => [3,1,2,4] => [3,1,2] => 1 = 2 - 1
[+,3,4,2] => [1,4,2,3] => [1,2,3] => 0 = 1 - 1
[-,3,4,2] => [4,1,2,3] => [1,2,3] => 0 = 1 - 1
[+,4,2,3] => [1,3,4,2] => [1,3,2] => 1 = 2 - 1
Description
The number of left outer peaks of a permutation.
A left 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 $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Matching statistic: St000196
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ 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] => [.,.]
=> 0 = 1 - 1
[-] => [1] => [.,.]
=> 0 = 1 - 1
[+,+] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[-,+] => [2,1] => [[.,.],.]
=> 0 = 1 - 1
[+,-] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[-,-] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[2,1] => [2,1] => [[.,.],.]
=> 0 = 1 - 1
[+,+,+] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[-,+,+] => [2,3,1] => [[.,[.,.]],.]
=> 0 = 1 - 1
[+,-,+] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
[+,+,-] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[-,-,+] => [3,1,2] => [[.,.],[.,.]]
=> 1 = 2 - 1
[-,+,-] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[+,-,-] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[-,-,-] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[+,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
[-,3,2] => [3,1,2] => [[.,.],[.,.]]
=> 1 = 2 - 1
[2,1,+] => [2,3,1] => [[.,[.,.]],.]
=> 0 = 1 - 1
[2,1,-] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[2,3,1] => [3,1,2] => [[.,.],[.,.]]
=> 1 = 2 - 1
[3,1,2] => [2,3,1] => [[.,[.,.]],.]
=> 0 = 1 - 1
[3,+,1] => [2,3,1] => [[.,[.,.]],.]
=> 0 = 1 - 1
[3,-,1] => [3,1,2] => [[.,.],[.,.]]
=> 1 = 2 - 1
[+,+,+,+] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[-,+,+,+] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 0 = 1 - 1
[+,-,+,+] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[+,+,-,+] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[-,-,+,+] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 1 = 2 - 1
[-,+,-,+] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 1 = 2 - 1
[-,+,+,-] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> 1 = 2 - 1
[+,-,-,+] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[+,-,+,-] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[+,+,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[-,-,-,+] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[-,-,+,-] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[-,+,-,-] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[+,-,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[+,+,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
[-,+,4,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 1 = 2 - 1
[+,-,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[-,-,4,3] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[+,3,2,+] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[-,3,2,+] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 1 = 2 - 1
[+,3,2,-] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[-,3,2,-] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[+,3,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[-,3,4,2] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[+,4,2,3] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0 = 1 - 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 $2^{n-2}$.
The following 34 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000884The number of isolated descents of a permutation. St000390The number of runs of ones in a binary word. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000920The logarithmic height of a Dyck path. St001111The weak 2-dynamic chromatic number of a graph. St001716The 1-improper chromatic number of a graph. St000091The descent variation of a composition. St000292The number of ascents of a binary word. St000552The number of cut vertices of a graph. St000761The number of ascents in an integer composition. St001689The number of celebrities in a graph. St001712The number of natural descents of a standard Young tableau. St000353The number of inner valleys of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000568The hook number of a binary tree. St000291The number of descents of a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000260The radius of a connected graph. St001487The number of inner corners of a skew partition. St001960The number of descents of a permutation minus one if its first entry is not one. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000456The monochromatic index of a connected graph. St001896The number of right descents of a signed permutations.
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