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Matching statistic: St000863
(load all 278 compositions to match this statistic)
(load all 278 compositions to match this statistic)
St000863: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 = 2 - 1
[1,2] => 2 = 3 - 1
[2,1] => 2 = 3 - 1
[1,2,3] => 3 = 4 - 1
[1,3,2] => 2 = 3 - 1
[2,1,3] => 3 = 4 - 1
[2,3,1] => 3 = 4 - 1
[3,1,2] => 2 = 3 - 1
[3,2,1] => 3 = 4 - 1
[1,2,3,4] => 4 = 5 - 1
[1,2,4,3] => 3 = 4 - 1
[1,3,2,4] => 3 = 4 - 1
[1,3,4,2] => 3 = 4 - 1
[1,4,2,3] => 3 = 4 - 1
[1,4,3,2] => 3 = 4 - 1
[2,1,3,4] => 4 = 5 - 1
[2,1,4,3] => 3 = 4 - 1
[2,3,1,4] => 4 = 5 - 1
[2,3,4,1] => 4 = 5 - 1
[2,4,1,3] => 3 = 4 - 1
[2,4,3,1] => 3 = 4 - 1
[3,1,2,4] => 3 = 4 - 1
[3,1,4,2] => 3 = 4 - 1
[3,2,1,4] => 4 = 5 - 1
[3,2,4,1] => 4 = 5 - 1
[3,4,1,2] => 3 = 4 - 1
[3,4,2,1] => 4 = 5 - 1
[4,1,2,3] => 3 = 4 - 1
[4,1,3,2] => 3 = 4 - 1
[4,2,1,3] => 3 = 4 - 1
[4,2,3,1] => 3 = 4 - 1
[4,3,1,2] => 3 = 4 - 1
[4,3,2,1] => 4 = 5 - 1
Description
The length of the first row 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 length of the first row of $P$ and $Q$.
Matching statistic: St000050
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000050: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000050: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> 1 = 2 - 1
[1,2] => [.,[.,.]]
=> 2 = 3 - 1
[2,1] => [[.,.],.]
=> 2 = 3 - 1
[1,2,3] => [.,[.,[.,.]]]
=> 3 = 4 - 1
[1,3,2] => [.,[[.,.],.]]
=> 3 = 4 - 1
[2,1,3] => [[.,.],[.,.]]
=> 2 = 3 - 1
[2,3,1] => [[.,[.,.]],.]
=> 3 = 4 - 1
[3,1,2] => [[.,.],[.,.]]
=> 2 = 3 - 1
[3,2,1] => [[[.,.],.],.]
=> 3 = 4 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 4 = 5 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> 4 = 5 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> 3 = 4 - 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> 4 = 5 - 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> 3 = 4 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> 4 = 5 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> 3 = 4 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> 3 = 4 - 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> 3 = 4 - 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> 4 = 5 - 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> 3 = 4 - 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> 4 = 5 - 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> 3 = 4 - 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> 3 = 4 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> 3 = 4 - 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> 3 = 4 - 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> 4 = 5 - 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> 3 = 4 - 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> 3 = 4 - 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> 3 = 4 - 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> 3 = 4 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> 4 = 5 - 1
Description
The depth or height of a binary tree.
The depth (or height) of a binary tree is the maximal depth (or height) of one of its vertices.
The '''height''' of a vertex is the number of edges on the longest path between that node and a leaf. The '''depth''' of a vertex is the number of edges from the vertex to the root.
See [1] and [2] for this terminology.
The depth (or height) of a tree $T$ can be recursively defined: $\operatorname{depth}(T) = 0$ if $T$ is empty and
$$\operatorname{depth}(T) = 1 + max(\operatorname{depth}(L),\operatorname{depth}(R))$$
if $T$ is nonempty with left and right subtrees $L$ and $R$, respectively.
The upper and lower bounds on the depth of a binary tree $T$ of size $n$ are $log_2(n) \leq \operatorname{depth}(T) \leq n$.
Matching statistic: St000062
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 2 - 1
[1,2] => [1,2] => 2 = 3 - 1
[2,1] => [1,2] => 2 = 3 - 1
[1,2,3] => [1,2,3] => 3 = 4 - 1
[1,3,2] => [1,3,2] => 2 = 3 - 1
[2,1,3] => [1,3,2] => 2 = 3 - 1
[2,3,1] => [1,2,3] => 3 = 4 - 1
[3,1,2] => [1,2,3] => 3 = 4 - 1
[3,2,1] => [1,2,3] => 3 = 4 - 1
[1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[1,3,2,4] => [1,3,2,4] => 3 = 4 - 1
[1,3,4,2] => [1,3,4,2] => 3 = 4 - 1
[1,4,2,3] => [1,4,2,3] => 3 = 4 - 1
[1,4,3,2] => [1,4,2,3] => 3 = 4 - 1
[2,1,3,4] => [1,3,4,2] => 3 = 4 - 1
[2,1,4,3] => [1,4,2,3] => 3 = 4 - 1
[2,3,1,4] => [1,4,2,3] => 3 = 4 - 1
[2,3,4,1] => [1,2,3,4] => 4 = 5 - 1
[2,4,1,3] => [1,3,2,4] => 3 = 4 - 1
[2,4,3,1] => [1,2,4,3] => 3 = 4 - 1
[3,1,2,4] => [1,2,4,3] => 3 = 4 - 1
[3,1,4,2] => [1,4,2,3] => 3 = 4 - 1
[3,2,1,4] => [1,4,2,3] => 3 = 4 - 1
[3,2,4,1] => [1,2,4,3] => 3 = 4 - 1
[3,4,1,2] => [1,2,3,4] => 4 = 5 - 1
[3,4,2,1] => [1,2,3,4] => 4 = 5 - 1
[4,1,2,3] => [1,2,3,4] => 4 = 5 - 1
[4,1,3,2] => [1,3,2,4] => 3 = 4 - 1
[4,2,1,3] => [1,3,2,4] => 3 = 4 - 1
[4,2,3,1] => [1,2,3,4] => 4 = 5 - 1
[4,3,1,2] => [1,2,3,4] => 4 = 5 - 1
[4,3,2,1] => [1,2,3,4] => 4 = 5 - 1
Description
The length of the longest increasing subsequence of the permutation.
Matching statistic: St000308
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 2 - 1
[1,2] => [1,2] => 2 = 3 - 1
[2,1] => [1,2] => 2 = 3 - 1
[1,2,3] => [1,2,3] => 3 = 4 - 1
[1,3,2] => [1,3,2] => 2 = 3 - 1
[2,1,3] => [1,3,2] => 2 = 3 - 1
[2,3,1] => [1,2,3] => 3 = 4 - 1
[3,1,2] => [1,2,3] => 3 = 4 - 1
[3,2,1] => [1,2,3] => 3 = 4 - 1
[1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[1,3,2,4] => [1,3,2,4] => 3 = 4 - 1
[1,3,4,2] => [1,3,4,2] => 3 = 4 - 1
[1,4,2,3] => [1,4,2,3] => 3 = 4 - 1
[1,4,3,2] => [1,4,2,3] => 3 = 4 - 1
[2,1,3,4] => [1,3,4,2] => 3 = 4 - 1
[2,1,4,3] => [1,4,2,3] => 3 = 4 - 1
[2,3,1,4] => [1,4,2,3] => 3 = 4 - 1
[2,3,4,1] => [1,2,3,4] => 4 = 5 - 1
[2,4,1,3] => [1,3,2,4] => 3 = 4 - 1
[2,4,3,1] => [1,2,4,3] => 3 = 4 - 1
[3,1,2,4] => [1,2,4,3] => 3 = 4 - 1
[3,1,4,2] => [1,4,2,3] => 3 = 4 - 1
[3,2,1,4] => [1,4,2,3] => 3 = 4 - 1
[3,2,4,1] => [1,2,4,3] => 3 = 4 - 1
[3,4,1,2] => [1,2,3,4] => 4 = 5 - 1
[3,4,2,1] => [1,2,3,4] => 4 = 5 - 1
[4,1,2,3] => [1,2,3,4] => 4 = 5 - 1
[4,1,3,2] => [1,3,2,4] => 3 = 4 - 1
[4,2,1,3] => [1,3,2,4] => 3 = 4 - 1
[4,2,3,1] => [1,2,3,4] => 4 = 5 - 1
[4,3,1,2] => [1,2,3,4] => 4 = 5 - 1
[4,3,2,1] => [1,2,3,4] => 4 = 5 - 1
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
Matching statistic: St001554
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
St001554: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001554: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> 1 = 2 - 1
[1,2] => [.,[.,.]]
=> 2 = 3 - 1
[2,1] => [[.,.],.]
=> 2 = 3 - 1
[1,2,3] => [.,[.,[.,.]]]
=> 3 = 4 - 1
[1,3,2] => [.,[[.,.],.]]
=> 3 = 4 - 1
[2,1,3] => [[.,.],[.,.]]
=> 2 = 3 - 1
[2,3,1] => [[.,[.,.]],.]
=> 3 = 4 - 1
[3,1,2] => [[.,.],[.,.]]
=> 2 = 3 - 1
[3,2,1] => [[[.,.],.],.]
=> 3 = 4 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 4 = 5 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> 4 = 5 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> 3 = 4 - 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> 4 = 5 - 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> 3 = 4 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> 4 = 5 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> 3 = 4 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> 3 = 4 - 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> 3 = 4 - 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> 4 = 5 - 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> 3 = 4 - 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> 4 = 5 - 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> 3 = 4 - 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> 3 = 4 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> 3 = 4 - 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> 3 = 4 - 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> 4 = 5 - 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> 3 = 4 - 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> 3 = 4 - 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> 3 = 4 - 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> 3 = 4 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> 4 = 5 - 1
Description
The number of distinct nonempty subtrees of a binary tree.
Matching statistic: St000245
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 2 - 2
[1,2] => [1,2] => 1 = 3 - 2
[2,1] => [1,2] => 1 = 3 - 2
[1,2,3] => [1,2,3] => 2 = 4 - 2
[1,3,2] => [1,3,2] => 1 = 3 - 2
[2,1,3] => [1,3,2] => 1 = 3 - 2
[2,3,1] => [1,2,3] => 2 = 4 - 2
[3,1,2] => [1,2,3] => 2 = 4 - 2
[3,2,1] => [1,2,3] => 2 = 4 - 2
[1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,2,4,3] => [1,2,4,3] => 2 = 4 - 2
[1,3,2,4] => [1,3,2,4] => 2 = 4 - 2
[1,3,4,2] => [1,3,4,2] => 2 = 4 - 2
[1,4,2,3] => [1,4,2,3] => 2 = 4 - 2
[1,4,3,2] => [1,4,2,3] => 2 = 4 - 2
[2,1,3,4] => [1,3,4,2] => 2 = 4 - 2
[2,1,4,3] => [1,4,2,3] => 2 = 4 - 2
[2,3,1,4] => [1,4,2,3] => 2 = 4 - 2
[2,3,4,1] => [1,2,3,4] => 3 = 5 - 2
[2,4,1,3] => [1,3,2,4] => 2 = 4 - 2
[2,4,3,1] => [1,2,4,3] => 2 = 4 - 2
[3,1,2,4] => [1,2,4,3] => 2 = 4 - 2
[3,1,4,2] => [1,4,2,3] => 2 = 4 - 2
[3,2,1,4] => [1,4,2,3] => 2 = 4 - 2
[3,2,4,1] => [1,2,4,3] => 2 = 4 - 2
[3,4,1,2] => [1,2,3,4] => 3 = 5 - 2
[3,4,2,1] => [1,2,3,4] => 3 = 5 - 2
[4,1,2,3] => [1,2,3,4] => 3 = 5 - 2
[4,1,3,2] => [1,3,2,4] => 2 = 4 - 2
[4,2,1,3] => [1,3,2,4] => 2 = 4 - 2
[4,2,3,1] => [1,2,3,4] => 3 = 5 - 2
[4,3,1,2] => [1,2,3,4] => 3 = 5 - 2
[4,3,2,1] => [1,2,3,4] => 3 = 5 - 2
Description
The number of ascents of a permutation.
Matching statistic: St000672
(load all 44 compositions to match this statistic)
(load all 44 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 2 - 2
[1,2] => [1,2] => 1 = 3 - 2
[2,1] => [1,2] => 1 = 3 - 2
[1,2,3] => [1,2,3] => 2 = 4 - 2
[1,3,2] => [1,3,2] => 1 = 3 - 2
[2,1,3] => [1,3,2] => 1 = 3 - 2
[2,3,1] => [1,2,3] => 2 = 4 - 2
[3,1,2] => [1,2,3] => 2 = 4 - 2
[3,2,1] => [1,2,3] => 2 = 4 - 2
[1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,2,4,3] => [1,2,4,3] => 2 = 4 - 2
[1,3,2,4] => [1,3,2,4] => 2 = 4 - 2
[1,3,4,2] => [1,3,4,2] => 2 = 4 - 2
[1,4,2,3] => [1,4,2,3] => 2 = 4 - 2
[1,4,3,2] => [1,4,2,3] => 2 = 4 - 2
[2,1,3,4] => [1,3,4,2] => 2 = 4 - 2
[2,1,4,3] => [1,4,2,3] => 2 = 4 - 2
[2,3,1,4] => [1,4,2,3] => 2 = 4 - 2
[2,3,4,1] => [1,2,3,4] => 3 = 5 - 2
[2,4,1,3] => [1,3,2,4] => 2 = 4 - 2
[2,4,3,1] => [1,2,4,3] => 2 = 4 - 2
[3,1,2,4] => [1,2,4,3] => 2 = 4 - 2
[3,1,4,2] => [1,4,2,3] => 2 = 4 - 2
[3,2,1,4] => [1,4,2,3] => 2 = 4 - 2
[3,2,4,1] => [1,2,4,3] => 2 = 4 - 2
[3,4,1,2] => [1,2,3,4] => 3 = 5 - 2
[3,4,2,1] => [1,2,3,4] => 3 = 5 - 2
[4,1,2,3] => [1,2,3,4] => 3 = 5 - 2
[4,1,3,2] => [1,3,2,4] => 2 = 4 - 2
[4,2,1,3] => [1,3,2,4] => 2 = 4 - 2
[4,2,3,1] => [1,2,3,4] => 3 = 5 - 2
[4,3,1,2] => [1,2,3,4] => 3 = 5 - 2
[4,3,2,1] => [1,2,3,4] => 3 = 5 - 2
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St001298
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
St001298: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001298: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 2 - 2
[1,2] => [1,2] => 1 = 3 - 2
[2,1] => [1,2] => 1 = 3 - 2
[1,2,3] => [1,2,3] => 2 = 4 - 2
[1,3,2] => [1,3,2] => 1 = 3 - 2
[2,1,3] => [1,3,2] => 1 = 3 - 2
[2,3,1] => [1,2,3] => 2 = 4 - 2
[3,1,2] => [1,2,3] => 2 = 4 - 2
[3,2,1] => [1,2,3] => 2 = 4 - 2
[1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,2,4,3] => [1,2,4,3] => 2 = 4 - 2
[1,3,2,4] => [1,3,2,4] => 2 = 4 - 2
[1,3,4,2] => [1,3,4,2] => 2 = 4 - 2
[1,4,2,3] => [1,4,2,3] => 2 = 4 - 2
[1,4,3,2] => [1,4,2,3] => 2 = 4 - 2
[2,1,3,4] => [1,3,4,2] => 2 = 4 - 2
[2,1,4,3] => [1,4,2,3] => 2 = 4 - 2
[2,3,1,4] => [1,4,2,3] => 2 = 4 - 2
[2,3,4,1] => [1,2,3,4] => 3 = 5 - 2
[2,4,1,3] => [1,3,2,4] => 2 = 4 - 2
[2,4,3,1] => [1,2,4,3] => 2 = 4 - 2
[3,1,2,4] => [1,2,4,3] => 2 = 4 - 2
[3,1,4,2] => [1,4,2,3] => 2 = 4 - 2
[3,2,1,4] => [1,4,2,3] => 2 = 4 - 2
[3,2,4,1] => [1,2,4,3] => 2 = 4 - 2
[3,4,1,2] => [1,2,3,4] => 3 = 5 - 2
[3,4,2,1] => [1,2,3,4] => 3 = 5 - 2
[4,1,2,3] => [1,2,3,4] => 3 = 5 - 2
[4,1,3,2] => [1,3,2,4] => 2 = 4 - 2
[4,2,1,3] => [1,3,2,4] => 2 = 4 - 2
[4,2,3,1] => [1,2,3,4] => 3 = 5 - 2
[4,3,1,2] => [1,2,3,4] => 3 = 5 - 2
[4,3,2,1] => [1,2,3,4] => 3 = 5 - 2
Description
The number of repeated entries in the Lehmer code of a permutation.
The Lehmer code of a permutation $\pi$ is the sequence $(v_1,\dots,v_n)$, with $v_i=|\{j > i: \pi(j) < \pi(i)\}$. This statistic counts the number of distinct elements in this sequence.
Matching statistic: St000380
(load all 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000380: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000380: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 2
[1,2] => [1,2] => [2]
=> 3
[2,1] => [1,2] => [2]
=> 3
[1,2,3] => [1,2,3] => [3]
=> 4
[1,3,2] => [1,2,3] => [3]
=> 4
[2,1,3] => [1,2,3] => [3]
=> 4
[2,3,1] => [1,2,3] => [3]
=> 4
[3,1,2] => [1,3,2] => [2,1]
=> 3
[3,2,1] => [1,3,2] => [2,1]
=> 3
[1,2,3,4] => [1,2,3,4] => [4]
=> 5
[1,2,4,3] => [1,2,3,4] => [4]
=> 5
[1,3,2,4] => [1,2,3,4] => [4]
=> 5
[1,3,4,2] => [1,2,3,4] => [4]
=> 5
[1,4,2,3] => [1,2,4,3] => [3,1]
=> 4
[1,4,3,2] => [1,2,4,3] => [3,1]
=> 4
[2,1,3,4] => [1,2,3,4] => [4]
=> 5
[2,1,4,3] => [1,2,3,4] => [4]
=> 5
[2,3,1,4] => [1,2,3,4] => [4]
=> 5
[2,3,4,1] => [1,2,3,4] => [4]
=> 5
[2,4,1,3] => [1,2,4,3] => [3,1]
=> 4
[2,4,3,1] => [1,2,4,3] => [3,1]
=> 4
[3,1,2,4] => [1,3,2,4] => [3,1]
=> 4
[3,1,4,2] => [1,3,4,2] => [3,1]
=> 4
[3,2,1,4] => [1,3,2,4] => [3,1]
=> 4
[3,2,4,1] => [1,3,4,2] => [3,1]
=> 4
[3,4,1,2] => [1,3,2,4] => [3,1]
=> 4
[3,4,2,1] => [1,3,2,4] => [3,1]
=> 4
[4,1,2,3] => [1,4,3,2] => [2,1,1]
=> 4
[4,1,3,2] => [1,4,2,3] => [3,1]
=> 4
[4,2,1,3] => [1,4,3,2] => [2,1,1]
=> 4
[4,2,3,1] => [1,4,2,3] => [3,1]
=> 4
[4,3,1,2] => [1,4,2,3] => [3,1]
=> 4
[4,3,2,1] => [1,4,2,3] => [3,1]
=> 4
Description
Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition.
Put differently, this is the smallest number $n$ such that the partition fits into the triangular partition $(n-1,n-2,\dots,1)$.
Matching statistic: St001237
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001237: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001237: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 2
[1,2] => [2] => [1,1,0,0]
=> 3
[2,1] => [1,1] => [1,0,1,0]
=> 3
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 4
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 3
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 4
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 3
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 4
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 5
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 4
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 4
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 4
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 4
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 5
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 4
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 4
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 4
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 5
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 5
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 4
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 5
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 5
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 5
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
Description
The number of simple modules with injective dimension at most one or dominant dimension at least one.
The following 384 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000010The length of the partition. St000031The number of cycles in the cycle decomposition of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000384The maximal part of the shifted composition of an integer partition. St000393The number of strictly increasing runs in a binary word. St000470The number of runs in a permutation. St000507The number of ascents of a standard tableau. St000528The height of a poset. St000553The number of blocks of a graph. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000784The maximum of the length and the largest part of the integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000912The number of maximal antichains in a poset. St000991The number of right-to-left minima of a permutation. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001286The annihilation number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001391The disjunction number of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001566The length of the longest arithmetic progression in a permutation. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001717The largest size of an interval in a poset. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000021The number of descents of a permutation. St000080The rank of the poset. St000155The number of exceedances (also excedences) of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000662The staircase size of the code of a permutation. St000778The metric dimension of a graph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001176The size of a partition minus its first part. St001489The maximum of the number of descents and the number of inverse descents. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001935The number of ascents in a parking function. St001949The rigidity index of a graph. St000094The depth of an ordered tree. St000521The number of distinct subtrees of an ordered tree. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001619The number of non-isomorphic sublattices of a lattice. St001626The number of maximal proper sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001720The minimal length of a chain of small intervals in a lattice. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000054The first entry of the permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St000071The number of maximal chains in a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000105The number of blocks in the set partition. St000108The number of partitions contained in the given partition. St000166The depth minus 1 of an ordered tree. St000167The number of leaves of an ordered tree. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000443The number of long tunnels of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000453The number of distinct Laplacian eigenvalues of a graph. St000482The (zero)-forcing number of a graph. St000527The width of the poset. St000532The total number of rook placements on a Ferrers board. St000542The number of left-to-right-minima of a permutation. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000636The hull number of a graph. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000822The Hadwiger number of the graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001400The total number of Littlewood-Richardson tableaux of given shape. St001430The number of positive entries in a signed permutation. St001439The number of even weak deficiencies and of odd weak exceedences. St001494The Alon-Tarsi number of a graph. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001616The number of neutral elements in a lattice. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001654The monophonic hull number of a graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001733The number of weak left to right maxima of a Dyck path. St001820The size of the image of the pop stack sorting operator. St001863The number of weak excedances of a signed permutation. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000053The number of valleys of the Dyck path. St000074The number of special entries. St000141The maximum drop size of a permutation. St000148The number of odd parts of a partition. St000157The number of descents of a standard tableau. St000160The multiplicity of the smallest part of a partition. St000168The number of internal nodes of an ordered tree. St000228The size of a partition. St000272The treewidth of a graph. St000304The load of a permutation. St000306The bounce count of a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000536The pathwidth of a graph. St000548The number of different non-empty partial sums of an integer partition. St000632The jump number of the poset. St000670The reversal length of a permutation. St000703The number of deficiencies of a permutation. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000867The sum of the hook lengths in the first row of an integer partition. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001127The sum of the squares of the parts of a partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001358The largest degree of a regular subgraph of a graph. St001427The number of descents of a signed permutation. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001644The dimension of a graph. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001812The biclique partition number of a graph. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001946The number of descents in a parking function. St001955The number of natural descents for set-valued two row standard Young tableaux. St001962The proper pathwidth of a graph. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000941The number of characters of the symmetric group whose value on the partition is even. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000668The least common multiple of the parts of the partition. St000702The number of weak deficiencies of a permutation. St000708The product of the parts of an integer partition. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000619The number of cyclic descents of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000886The number of permutations with the same antidiagonal sums. St000922The minimal number such that all substrings of this length are unique. St000937The number of positive values of the symmetric group character corresponding to the partition. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001668The number of points of the poset minus the width of the poset. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000925The number of topologically connected components of a set partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000058The order of a permutation. St000216The absolute length of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000485The length of the longest cycle of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000654The first descent of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000876The number of factors in the Catalan decomposition of a binary word. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000932The number of occurrences of the pattern UDU in a Dyck path. St000933The number of multipartitions of sizes given by an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000982The length of the longest constant subword. St000984The number of boxes below precisely one peak. St000989The number of final rises of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001081The number of minimal length factorizations of a permutation into star transpositions. St001128The exponens consonantiae of a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001220The width of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001313The number of Dyck paths above the lattice path given by a binary word. St001360The number of covering relations in Young's lattice below a partition. St001372The length of a longest cyclic run of ones of a binary word. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001480The number of simple summands of the module J^2/J^3. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000461The rix statistic of a permutation. St000539The number of odd inversions of a permutation. St000682The Grundy value of Welter's game on a binary word. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000837The number of ascents of distance 2 of a permutation. St000877The depth of the binary word interpreted as a path. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001082The number of boxed occurrences of 123 in a permutation. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001388The number of non-attacking neighbors of a permutation. St001557The number of inversions of the second entry of a permutation. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001651The Frankl number of a lattice. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001811The Castelnuovo-Mumford regularity of a permutation. St000928The sum of the coefficients of the character polynomial of an integer partition. St000741The Colin de Verdière graph invariant. St000259The diameter of a connected graph. St000045The number of linear extensions of a binary tree. St000219The number of occurrences of the pattern 231 in a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000455The second largest eigenvalue of a graph if it is integral. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St000522The number of 1-protected nodes of a rooted tree. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001645The pebbling number of a connected graph. St000717The number of ordinal summands of a poset. St000075The orbit size of a standard tableau under promotion. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000264The girth of a graph, which is not a tree. St001875The number of simple modules with projective dimension at most 1. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000422The energy of a graph, if it is integral. St000906The length of the shortest maximal chain in a poset. St000273The domination number of a graph. St000387The matching number of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000808The number of up steps of the associated bargraph. St000916The packing number of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001114The number of odd descents of a permutation. St001339The irredundance number of a graph. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001928The number of non-overlapping descents in a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000735The last entry on the main diagonal of a standard tableau. St000850The number of 1/2-balanced pairs in a poset. St001469The holeyness of a permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001926Sparre Andersen's position of the maximum of a signed permutation. St000649The number of 3-excedences of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001637The number of (upper) dissectors of a poset. St000307The number of rowmotion orbits of a poset. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St000101The cocharge of a semistandard tableau. St001117The game chromatic index of a graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001642The Prague dimension of a graph. St001649The length of a longest trail in a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000736The last entry in the first row of a semistandard tableau. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001569The maximal modular displacement of a permutation. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001742The difference of the maximal and the minimal degree in a graph. St000102The charge of a semistandard tableau. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001556The number of inversions of the third entry of a permutation. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001856The number of edges in the reduced word graph of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001964The interval resolution global dimension of a poset.
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