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Your data matches 154 different statistics following compositions of up to 3 maps.
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Matching statistic: St000092
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ 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
[1,2] => [1] => 1
[2,1] => [1] => 1
[1,2,3] => [1,2] => 1
[1,3,2] => [1,2] => 1
[2,1,3] => [2,1] => 1
[2,3,1] => [2,1] => 1
[3,1,2] => [1,2] => 1
[3,2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => 1
[1,3,4,2] => [1,3,2] => 1
[1,4,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3] => 2
[2,1,4,3] => [2,1,3] => 2
[2,3,1,4] => [2,3,1] => 1
[2,3,4,1] => [2,3,1] => 1
[2,4,1,3] => [2,1,3] => 2
[2,4,3,1] => [2,3,1] => 1
[3,1,2,4] => [3,1,2] => 2
[3,1,4,2] => [3,1,2] => 2
[3,2,1,4] => [3,2,1] => 1
[3,2,4,1] => [3,2,1] => 1
[3,4,1,2] => [3,1,2] => 2
[3,4,2,1] => [3,2,1] => 1
[4,1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => 1
[4,2,1,3] => [2,1,3] => 2
[4,2,3,1] => [2,3,1] => 1
[4,3,1,2] => [3,1,2] => 2
[4,3,2,1] => [3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3] => 1
[1,2,4,5,3] => [1,2,4,3] => 1
[1,2,5,3,4] => [1,2,3,4] => 1
[1,2,5,4,3] => [1,2,4,3] => 1
[1,3,2,4,5] => [1,3,2,4] => 2
[1,3,2,5,4] => [1,3,2,4] => 2
[1,3,4,2,5] => [1,3,4,2] => 1
[1,3,4,5,2] => [1,3,4,2] => 1
[1,3,5,2,4] => [1,3,2,4] => 2
[1,3,5,4,2] => [1,3,4,2] => 1
[1,4,2,3,5] => [1,4,2,3] => 2
[1,4,2,5,3] => [1,4,2,3] => 2
[1,4,3,2,5] => [1,4,3,2] => 1
[1,4,3,5,2] => [1,4,3,2] => 1
[1,4,5,2,3] => [1,4,2,3] => 2
[1,4,5,3,2] => [1,4,3,2] => 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 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ 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
[1,2] => [1] => 1
[2,1] => [1] => 1
[1,2,3] => [1,2] => 1
[1,3,2] => [1,2] => 1
[2,1,3] => [2,1] => 1
[2,3,1] => [2,1] => 1
[3,1,2] => [1,2] => 1
[3,2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => 2
[1,3,4,2] => [1,3,2] => 2
[1,4,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => 2
[2,1,3,4] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => 2
[2,3,4,1] => [2,3,1] => 2
[2,4,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => 2
[3,1,2,4] => [3,1,2] => 1
[3,1,4,2] => [3,1,2] => 1
[3,2,1,4] => [3,2,1] => 1
[3,2,4,1] => [3,2,1] => 1
[3,4,1,2] => [3,1,2] => 1
[3,4,2,1] => [3,2,1] => 1
[4,1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => 2
[4,2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => 2
[4,3,1,2] => [3,1,2] => 1
[4,3,2,1] => [3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3] => 2
[1,2,4,5,3] => [1,2,4,3] => 2
[1,2,5,3,4] => [1,2,3,4] => 1
[1,2,5,4,3] => [1,2,4,3] => 2
[1,3,2,4,5] => [1,3,2,4] => 2
[1,3,2,5,4] => [1,3,2,4] => 2
[1,3,4,2,5] => [1,3,4,2] => 2
[1,3,4,5,2] => [1,3,4,2] => 2
[1,3,5,2,4] => [1,3,2,4] => 2
[1,3,5,4,2] => [1,3,4,2] => 2
[1,4,2,3,5] => [1,4,2,3] => 2
[1,4,2,5,3] => [1,4,2,3] => 2
[1,4,3,2,5] => [1,4,3,2] => 2
[1,4,3,5,2] => [1,4,3,2] => 2
[1,4,5,2,3] => [1,4,2,3] => 2
[1,4,5,3,2] => [1,4,3,2] => 2
Description
The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is [[St000353]].
Matching statistic: St000023
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ 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
[1,2] => [1] => 0 = 1 - 1
[2,1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => 0 = 1 - 1
[2,3,1] => [2,1] => 0 = 1 - 1
[3,1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => 0 = 1 - 1
[2,1,4,3] => [2,1,3] => 0 = 1 - 1
[2,3,1,4] => [2,3,1] => 1 = 2 - 1
[2,3,4,1] => [2,3,1] => 1 = 2 - 1
[2,4,1,3] => [2,1,3] => 0 = 1 - 1
[2,4,3,1] => [2,3,1] => 1 = 2 - 1
[3,1,2,4] => [3,1,2] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => 0 = 1 - 1
[4,2,3,1] => [2,3,1] => 1 = 2 - 1
[4,3,1,2] => [3,1,2] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,2] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => 1 = 2 - 1
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
Matching statistic: St000201
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ 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
[1,2] => [1] => [.,.]
=> 1
[2,1] => [1] => [.,.]
=> 1
[1,2,3] => [1,2] => [.,[.,.]]
=> 1
[1,3,2] => [1,2] => [.,[.,.]]
=> 1
[2,1,3] => [2,1] => [[.,.],.]
=> 1
[2,3,1] => [2,1] => [[.,.],.]
=> 1
[3,1,2] => [1,2] => [.,[.,.]]
=> 1
[3,2,1] => [2,1] => [[.,.],.]
=> 1
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> 1
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> 1
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 1
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> 2
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> 2
[2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
=> 1
[2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
=> 1
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 2
[2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
=> 1
[3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
=> 2
[3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
=> 2
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> 1
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> 1
[3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
=> 2
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> 1
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 1
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 1
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 2
[4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> 1
[4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> 2
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> 1
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 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: St000325
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 1
[2,1] => [1] => [1] => 1
[1,2,3] => [1,2] => [1,2] => 1
[1,3,2] => [1,2] => [1,2] => 1
[2,1,3] => [2,1] => [1,2] => 1
[2,3,1] => [2,1] => [1,2] => 1
[3,1,2] => [1,2] => [1,2] => 1
[3,2,1] => [2,1] => [1,2] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 2
[1,3,4,2] => [1,3,2] => [1,3,2] => 2
[1,4,2,3] => [1,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3,4] => [2,1,3] => [1,3,2] => 2
[2,1,4,3] => [2,1,3] => [1,3,2] => 2
[2,3,1,4] => [2,3,1] => [1,2,3] => 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 2
[2,4,3,1] => [2,3,1] => [1,2,3] => 1
[3,1,2,4] => [3,1,2] => [1,2,3] => 1
[3,1,4,2] => [3,1,2] => [1,2,3] => 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 1
[3,4,1,2] => [3,1,2] => [1,2,3] => 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 2
[4,2,1,3] => [2,1,3] => [1,3,2] => 2
[4,2,3,1] => [2,3,1] => [1,2,3] => 1
[4,3,1,2] => [3,1,2] => [1,2,3] => 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 2
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 2
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 2
Description
The width 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 width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]].
See also [[St000308]] for the height of this tree.
Matching statistic: St000470
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 1
[2,1] => [1] => [1] => 1
[1,2,3] => [1,2] => [1,2] => 1
[1,3,2] => [1,2] => [1,2] => 1
[2,1,3] => [2,1] => [1,2] => 1
[2,3,1] => [2,1] => [1,2] => 1
[3,1,2] => [1,2] => [1,2] => 1
[3,2,1] => [2,1] => [1,2] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 2
[1,3,4,2] => [1,3,2] => [1,3,2] => 2
[1,4,2,3] => [1,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3,4] => [2,1,3] => [1,3,2] => 2
[2,1,4,3] => [2,1,3] => [1,3,2] => 2
[2,3,1,4] => [2,3,1] => [1,2,3] => 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 2
[2,4,3,1] => [2,3,1] => [1,2,3] => 1
[3,1,2,4] => [3,1,2] => [1,2,3] => 1
[3,1,4,2] => [3,1,2] => [1,2,3] => 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 1
[3,4,1,2] => [3,1,2] => [1,2,3] => 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 2
[4,2,1,3] => [2,1,3] => [1,3,2] => 2
[4,2,3,1] => [2,3,1] => [1,2,3] => 1
[4,3,1,2] => [3,1,2] => [1,2,3] => 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 2
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 2
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 2
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000021
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => [1,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,2] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 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
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => [1,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,2] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [1,2,3] => 0 = 1 - 1
[1,3,4,2] => [1,3,2] => [1,2,3] => 0 = 1 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [1,2,3] => 0 = 1 - 1
[2,1,3,4] => [2,1,3] => [1,2,3] => 0 = 1 - 1
[2,1,4,3] => [2,1,3] => [1,2,3] => 0 = 1 - 1
[2,3,1,4] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [1,2,3] => 0 = 1 - 1
[2,4,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[3,1,4,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[3,2,1,4] => [3,2,1] => [1,3,2] => 1 = 2 - 1
[3,2,4,1] => [3,2,1] => [1,3,2] => 1 = 2 - 1
[3,4,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[3,4,2,1] => [3,2,1] => [1,3,2] => 1 = 2 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [1,2,3] => 0 = 1 - 1
[4,2,1,3] => [2,1,3] => [1,2,3] => 0 = 1 - 1
[4,2,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[4,3,2,1] => [3,2,1] => [1,3,2] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,2,3,4] => 0 = 1 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,2,3,4] => 0 = 1 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,2,3,4] => 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,2,4,3] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,2,4,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,2,4,3] => 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,2,4,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,2,4,3] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,2,4,3] => 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 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ 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
[1,2] => [1] => [.,.]
=> 0 = 1 - 1
[2,1] => [1] => [.,.]
=> 0 = 1 - 1
[1,2,3] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[1,3,2] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[2,1,3] => [2,1] => [[.,.],.]
=> 0 = 1 - 1
[2,3,1] => [2,1] => [[.,.],.]
=> 0 = 1 - 1
[3,1,2] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[3,2,1] => [2,1] => [[.,.],.]
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
=> 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
=> 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
=> 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
=> 1 = 2 - 1
[3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
=> 1 = 2 - 1
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> 1 = 2 - 1
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0 = 1 - 1
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0 = 1 - 1
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 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 $2^{n-2}$.
Matching statistic: St000291
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0 => 0 = 1 - 1
[2,1] => [1,2] => 0 => 0 = 1 - 1
[1,2,3] => [1,2,3] => 00 => 0 = 1 - 1
[1,3,2] => [1,2,3] => 00 => 0 = 1 - 1
[2,1,3] => [1,2,3] => 00 => 0 = 1 - 1
[2,3,1] => [1,2,3] => 00 => 0 = 1 - 1
[3,1,2] => [1,3,2] => 01 => 0 = 1 - 1
[3,2,1] => [1,3,2] => 01 => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => 001 => 0 = 1 - 1
[1,4,3,2] => [1,2,4,3] => 001 => 0 = 1 - 1
[2,1,3,4] => [1,2,3,4] => 000 => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => 000 => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => 000 => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => 000 => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => 001 => 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => 001 => 0 = 1 - 1
[3,1,2,4] => [1,3,2,4] => 010 => 1 = 2 - 1
[3,1,4,2] => [1,3,4,2] => 001 => 0 = 1 - 1
[3,2,1,4] => [1,3,2,4] => 010 => 1 = 2 - 1
[3,2,4,1] => [1,3,4,2] => 001 => 0 = 1 - 1
[3,4,1,2] => [1,3,2,4] => 010 => 1 = 2 - 1
[3,4,2,1] => [1,3,2,4] => 010 => 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => 011 => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => 010 => 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => 011 => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => 010 => 1 = 2 - 1
[4,3,1,2] => [1,4,2,3] => 010 => 1 = 2 - 1
[4,3,2,1] => [1,4,2,3] => 010 => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,5,4] => 0001 => 0 = 1 - 1
[1,2,5,4,3] => [1,2,3,5,4] => 0001 => 0 = 1 - 1
[1,3,2,4,5] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,3,5,2,4] => [1,2,3,5,4] => 0001 => 0 = 1 - 1
[1,3,5,4,2] => [1,2,3,5,4] => 0001 => 0 = 1 - 1
[1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,4,2,5,3] => [1,2,4,5,3] => 0001 => 0 = 1 - 1
[1,4,3,2,5] => [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,4,3,5,2] => [1,2,4,5,3] => 0001 => 0 = 1 - 1
[1,4,5,2,3] => [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,4,5,3,2] => [1,2,4,3,5] => 0010 => 1 = 2 - 1
Description
The number of descents of a binary word.
The following 144 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. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000390The number of runs of ones in a binary word. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000527The width of the poset. St000659The number of rises of length at least 2 of a Dyck path. St001280The number of parts of an integer partition that are at least two. St000245The number of ascents of a permutation. St000292The number of ascents of a binary word. St000312The number of leaves in a graph. St000386The number of factors DDU in a Dyck path. St000523The number of 2-protected nodes of a rooted tree. St000632The jump number of the poset. St000636The hull number of a graph. St000834The number of right outer peaks of a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001307The number of induced stars on four vertices in a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001883The mutual visibility number of a graph. St000353The number of inner valleys of a permutation. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St000354The number of recoils of a permutation. St001638The book thickness of a graph. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000259The diameter of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000260The radius of a connected graph. St000252The number of nodes of degree 3 of a binary tree. St001487The number of inner corners of a skew partition. St001470The cyclic holeyness of a permutation. St001964The interval resolution global dimension of a poset. St001960The number of descents of a permutation minus one if its first entry is not one. St000455The second largest eigenvalue of a graph if it is integral. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St000862The number of parts of the shifted shape of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St000624The normalized sum of the minimal distances to a greater element. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001330The hat guessing number of a graph. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001396Number of triples of incomparable elements in a finite poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001845The number of join irreducibles minus the rank of a lattice. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001875The number of simple modules with projective dimension at most 1. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St001866The nesting alignments of a signed permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001490The number of connected components of a skew partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001694The number of maximal dissociation sets in a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001413Half the length of the longest even length palindromic prefix of a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001768The number of reduced words of a signed permutation. St001946The number of descents in a parking function. St000679The pruning number of an ordered tree. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001857The number of edges in the reduced word graph of a signed permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001624The breadth of a lattice. St000322The skewness of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. 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. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St001271The competition number of a graph. St001765The number of connected components of the friends and strangers graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001570The minimal number of edges to add to make a graph Hamiltonian.
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