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Your data matches 191 different statistics following compositions of up to 3 maps.
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Matching statistic: St000243
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
St000243: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 = 0 + 1
[2,1] => 1 = 0 + 1
[1,2,3] => 1 = 0 + 1
[1,3,2] => 1 = 0 + 1
[2,1,3] => 1 = 0 + 1
[2,3,1] => 1 = 0 + 1
[3,1,2] => 1 = 0 + 1
[3,2,1] => 1 = 0 + 1
[1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => 1 = 0 + 1
[1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => 1 = 0 + 1
[1,4,2,3] => 2 = 1 + 1
[1,4,3,2] => 1 = 0 + 1
[2,1,3,4] => 1 = 0 + 1
[2,1,4,3] => 1 = 0 + 1
[2,3,1,4] => 2 = 1 + 1
[2,3,4,1] => 1 = 0 + 1
[2,4,1,3] => 2 = 1 + 1
[2,4,3,1] => 1 = 0 + 1
[3,1,2,4] => 1 = 0 + 1
[3,1,4,2] => 2 = 1 + 1
[3,2,1,4] => 1 = 0 + 1
[3,2,4,1] => 2 = 1 + 1
[3,4,1,2] => 1 = 0 + 1
[3,4,2,1] => 1 = 0 + 1
[4,1,2,3] => 1 = 0 + 1
[4,1,3,2] => 2 = 1 + 1
[4,2,1,3] => 1 = 0 + 1
[4,2,3,1] => 2 = 1 + 1
[4,3,1,2] => 1 = 0 + 1
[4,3,2,1] => 1 = 0 + 1
[1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => 1 = 0 + 1
[1,2,4,3,5] => 2 = 1 + 1
[1,2,4,5,3] => 1 = 0 + 1
[1,2,5,3,4] => 2 = 1 + 1
[1,2,5,4,3] => 1 = 0 + 1
[1,3,2,4,5] => 2 = 1 + 1
[1,3,2,5,4] => 2 = 1 + 1
[1,3,4,2,5] => 2 = 1 + 1
[1,3,4,5,2] => 1 = 0 + 1
[1,3,5,2,4] => 2 = 1 + 1
[1,3,5,4,2] => 1 = 0 + 1
[1,4,2,3,5] => 2 = 1 + 1
[1,4,2,5,3] => 2 = 1 + 1
[1,4,3,2,5] => 2 = 1 + 1
[1,4,3,5,2] => 2 = 1 + 1
[1,4,5,2,3] => 2 = 1 + 1
[1,4,5,3,2] => 1 = 0 + 1
Description
The number of cyclic valleys and cyclic peaks of a permutation.
This is given by the number of indices $i$ such that $\pi_{i-1} > \pi_i < \pi_{i+1}$ with indices considered cyclically. Equivalently, this is the number of indices $i$ such that $\pi_{i-1} < \pi_i > \pi_{i+1}$ with indices considered cyclically.
Matching statistic: St000023
(load all 25 compositions to match this statistic)
(load all 25 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
[2,1] => [1] => 0
[1,2,3] => [1,2] => 0
[1,3,2] => [1,2] => 0
[2,1,3] => [2,1] => 0
[2,3,1] => [2,1] => 0
[3,1,2] => [1,2] => 0
[3,2,1] => [2,1] => 0
[1,2,3,4] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => 1
[1,3,4,2] => [1,3,2] => 1
[1,4,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3] => 0
[2,1,4,3] => [2,1,3] => 0
[2,3,1,4] => [2,3,1] => 1
[2,3,4,1] => [2,3,1] => 1
[2,4,1,3] => [2,1,3] => 0
[2,4,3,1] => [2,3,1] => 1
[3,1,2,4] => [3,1,2] => 0
[3,1,4,2] => [3,1,2] => 0
[3,2,1,4] => [3,2,1] => 0
[3,2,4,1] => [3,2,1] => 0
[3,4,1,2] => [3,1,2] => 0
[3,4,2,1] => [3,2,1] => 0
[4,1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => 1
[4,2,1,3] => [2,1,3] => 0
[4,2,3,1] => [2,3,1] => 1
[4,3,1,2] => [3,1,2] => 0
[4,3,2,1] => [3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,4] => 0
[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] => 0
[1,2,5,4,3] => [1,2,4,3] => 1
[1,3,2,4,5] => [1,3,2,4] => 1
[1,3,2,5,4] => [1,3,2,4] => 1
[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] => 1
[1,3,5,4,2] => [1,3,4,2] => 1
[1,4,2,3,5] => [1,4,2,3] => 1
[1,4,2,5,3] => [1,4,2,3] => 1
[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] => 1
[1,4,5,3,2] => [1,4,3,2] => 1
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
Matching statistic: St000353
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000353: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => 0
[2,1,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,3,2] => 0
[3,2,1] => [1,3,2] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => 0
[1,4,3,2] => [1,2,4,3] => 0
[2,1,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => 0
[2,4,3,1] => [1,2,4,3] => 0
[3,1,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => 0
[3,2,1,4] => [1,3,2,4] => 1
[3,2,4,1] => [1,3,4,2] => 0
[3,4,1,2] => [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => 0
[4,1,3,2] => [1,4,2,3] => 1
[4,2,1,3] => [1,4,3,2] => 0
[4,2,3,1] => [1,4,2,3] => 1
[4,3,1,2] => [1,4,2,3] => 1
[4,3,2,1] => [1,4,2,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => 0
[1,2,5,4,3] => [1,2,3,5,4] => 0
[1,3,2,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => 0
[1,3,5,4,2] => [1,2,3,5,4] => 0
[1,4,2,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => 0
[1,4,3,2,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => 0
[1,4,5,2,3] => [1,2,4,3,5] => 1
[1,4,5,3,2] => [1,2,4,3,5] => 1
Description
The number of inner valleys of a permutation.
The number of valleys including the boundary is [[St000099]].
Matching statistic: St000092
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ 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] => [1,2] => 1 = 0 + 1
[2,1] => [1,2] => 1 = 0 + 1
[1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,2,3] => 1 = 0 + 1
[2,1,3] => [1,2,3] => 1 = 0 + 1
[2,3,1] => [1,2,3] => 1 = 0 + 1
[3,1,2] => [1,3,2] => 1 = 0 + 1
[3,2,1] => [1,3,2] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => 1 = 0 + 1
[1,4,3,2] => [1,2,4,3] => 1 = 0 + 1
[2,1,3,4] => [1,2,3,4] => 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => 1 = 0 + 1
[2,4,3,1] => [1,2,4,3] => 1 = 0 + 1
[3,1,2,4] => [1,3,2,4] => 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => 1 = 0 + 1
[3,2,1,4] => [1,3,2,4] => 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => 1 = 0 + 1
[3,4,1,2] => [1,3,2,4] => 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => 1 = 0 + 1
[4,1,3,2] => [1,4,2,3] => 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => 1 = 0 + 1
[4,2,3,1] => [1,4,2,3] => 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => 1 = 0 + 1
[1,2,5,4,3] => [1,2,3,5,4] => 1 = 0 + 1
[1,3,2,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => 1 = 0 + 1
[1,3,5,4,2] => [1,2,3,5,4] => 1 = 0 + 1
[1,4,2,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => 1 = 0 + 1
[1,4,3,2,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => 1 = 0 + 1
[1,4,5,2,3] => [1,2,4,3,5] => 2 = 1 + 1
[1,4,5,3,2] => [1,2,4,3,5] => 2 = 1 + 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 25 compositions to match this statistic)
(load all 25 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 = 0 + 1
[2,1] => [1] => 1 = 0 + 1
[1,2,3] => [1,2] => 1 = 0 + 1
[1,3,2] => [1,2] => 1 = 0 + 1
[2,1,3] => [2,1] => 1 = 0 + 1
[2,3,1] => [2,1] => 1 = 0 + 1
[3,1,2] => [1,2] => 1 = 0 + 1
[3,2,1] => [2,1] => 1 = 0 + 1
[1,2,3,4] => [1,2,3] => 1 = 0 + 1
[1,2,4,3] => [1,2,3] => 1 = 0 + 1
[1,3,2,4] => [1,3,2] => 2 = 1 + 1
[1,3,4,2] => [1,3,2] => 2 = 1 + 1
[1,4,2,3] => [1,2,3] => 1 = 0 + 1
[1,4,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3,4] => [2,1,3] => 1 = 0 + 1
[2,1,4,3] => [2,1,3] => 1 = 0 + 1
[2,3,1,4] => [2,3,1] => 2 = 1 + 1
[2,3,4,1] => [2,3,1] => 2 = 1 + 1
[2,4,1,3] => [2,1,3] => 1 = 0 + 1
[2,4,3,1] => [2,3,1] => 2 = 1 + 1
[3,1,2,4] => [3,1,2] => 1 = 0 + 1
[3,1,4,2] => [3,1,2] => 1 = 0 + 1
[3,2,1,4] => [3,2,1] => 1 = 0 + 1
[3,2,4,1] => [3,2,1] => 1 = 0 + 1
[3,4,1,2] => [3,1,2] => 1 = 0 + 1
[3,4,2,1] => [3,2,1] => 1 = 0 + 1
[4,1,2,3] => [1,2,3] => 1 = 0 + 1
[4,1,3,2] => [1,3,2] => 2 = 1 + 1
[4,2,1,3] => [2,1,3] => 1 = 0 + 1
[4,2,3,1] => [2,3,1] => 2 = 1 + 1
[4,3,1,2] => [3,1,2] => 1 = 0 + 1
[4,3,2,1] => [3,2,1] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,3] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,5,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2] => 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,2] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,4,2] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,3] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,3,2] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,3] => 2 = 1 + 1
[1,4,5,3,2] => [1,4,3,2] => 2 = 1 + 1
Description
The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is [[St000353]].
Matching statistic: St000834
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 1 = 0 + 1
[2,1] => [1,2] => 1 = 0 + 1
[1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,2,3] => 1 = 0 + 1
[2,1,3] => [1,2,3] => 1 = 0 + 1
[2,3,1] => [1,2,3] => 1 = 0 + 1
[3,1,2] => [1,3,2] => 1 = 0 + 1
[3,2,1] => [1,3,2] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => 1 = 0 + 1
[1,4,3,2] => [1,2,4,3] => 1 = 0 + 1
[2,1,3,4] => [1,2,3,4] => 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => 1 = 0 + 1
[2,4,3,1] => [1,2,4,3] => 1 = 0 + 1
[3,1,2,4] => [1,3,2,4] => 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => 1 = 0 + 1
[3,2,1,4] => [1,3,2,4] => 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => 1 = 0 + 1
[3,4,1,2] => [1,3,2,4] => 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => 1 = 0 + 1
[4,1,3,2] => [1,4,2,3] => 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => 1 = 0 + 1
[4,2,3,1] => [1,4,2,3] => 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => 1 = 0 + 1
[1,2,5,4,3] => [1,2,3,5,4] => 1 = 0 + 1
[1,3,2,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => 1 = 0 + 1
[1,3,5,4,2] => [1,2,3,5,4] => 1 = 0 + 1
[1,4,2,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => 1 = 0 + 1
[1,4,3,2,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => 1 = 0 + 1
[1,4,5,2,3] => [1,2,4,3,5] => 2 = 1 + 1
[1,4,5,3,2] => [1,2,4,3,5] => 2 = 1 + 1
Description
The number of right outer peaks of a permutation.
A right 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 $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000021
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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
[2,1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => 0
[1,3,2] => [1,2] => [1,2] => 0
[2,1,3] => [2,1] => [1,2] => 0
[2,3,1] => [2,1] => [1,2] => 0
[3,1,2] => [1,2] => [1,2] => 0
[3,2,1] => [2,1] => [1,2] => 0
[1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => [1,3,2] => 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3] => [1,3,2] => 1
[2,1,4,3] => [2,1,3] => [1,3,2] => 1
[2,3,1,4] => [2,3,1] => [1,2,3] => 0
[2,3,4,1] => [2,3,1] => [1,2,3] => 0
[2,4,1,3] => [2,1,3] => [1,3,2] => 1
[2,4,3,1] => [2,3,1] => [1,2,3] => 0
[3,1,2,4] => [3,1,2] => [1,2,3] => 0
[3,1,4,2] => [3,1,2] => [1,2,3] => 0
[3,2,1,4] => [3,2,1] => [1,2,3] => 0
[3,2,4,1] => [3,2,1] => [1,2,3] => 0
[3,4,1,2] => [3,1,2] => [1,2,3] => 0
[3,4,2,1] => [3,2,1] => [1,2,3] => 0
[4,1,2,3] => [1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => [1,3,2] => 1
[4,2,1,3] => [2,1,3] => [1,3,2] => 1
[4,2,3,1] => [2,3,1] => [1,2,3] => 0
[4,3,1,2] => [3,1,2] => [1,2,3] => 0
[4,3,2,1] => [3,2,1] => [1,2,3] => 0
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 1
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 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 24 compositions to match this statistic)
(load all 24 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
[2,1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => 0
[1,3,2] => [1,2] => [1,2] => 0
[2,1,3] => [2,1] => [1,2] => 0
[2,3,1] => [2,1] => [1,2] => 0
[3,1,2] => [1,2] => [1,2] => 0
[3,2,1] => [2,1] => [1,2] => 0
[1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => [1,2,3] => 0
[1,3,4,2] => [1,3,2] => [1,2,3] => 0
[1,4,2,3] => [1,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => [1,2,3] => 0
[2,1,3,4] => [2,1,3] => [1,2,3] => 0
[2,1,4,3] => [2,1,3] => [1,2,3] => 0
[2,3,1,4] => [2,3,1] => [1,2,3] => 0
[2,3,4,1] => [2,3,1] => [1,2,3] => 0
[2,4,1,3] => [2,1,3] => [1,2,3] => 0
[2,4,3,1] => [2,3,1] => [1,2,3] => 0
[3,1,2,4] => [3,1,2] => [1,3,2] => 1
[3,1,4,2] => [3,1,2] => [1,3,2] => 1
[3,2,1,4] => [3,2,1] => [1,3,2] => 1
[3,2,4,1] => [3,2,1] => [1,3,2] => 1
[3,4,1,2] => [3,1,2] => [1,3,2] => 1
[3,4,2,1] => [3,2,1] => [1,3,2] => 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => [1,2,3] => 0
[4,2,1,3] => [2,1,3] => [1,2,3] => 0
[4,2,3,1] => [2,3,1] => [1,2,3] => 0
[4,3,1,2] => [3,1,2] => [1,3,2] => 1
[4,3,2,1] => [3,2,1] => [1,3,2] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3,5] => [1,2,4,3] => [1,2,3,4] => 0
[1,2,4,5,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,5,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4,5] => [1,3,2,4] => [1,2,3,4] => 0
[1,3,2,5,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2,5] => [1,3,4,2] => [1,2,3,4] => 0
[1,3,4,5,2] => [1,3,4,2] => [1,2,3,4] => 0
[1,3,5,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,3,5,4,2] => [1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3,5] => [1,4,2,3] => [1,2,4,3] => 1
[1,4,2,5,3] => [1,4,2,3] => [1,2,4,3] => 1
[1,4,3,2,5] => [1,4,3,2] => [1,2,4,3] => 1
[1,4,3,5,2] => [1,4,3,2] => [1,2,4,3] => 1
[1,4,5,2,3] => [1,4,2,3] => [1,2,4,3] => 1
[1,4,5,3,2] => [1,4,3,2] => [1,2,4,3] => 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 5 compositions to match this statistic)
(load all 5 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
[2,1] => [1] => [.,.]
=> 0
[1,2,3] => [1,2] => [.,[.,.]]
=> 0
[1,3,2] => [1,2] => [.,[.,.]]
=> 0
[2,1,3] => [2,1] => [[.,.],.]
=> 0
[2,3,1] => [2,1] => [[.,.],.]
=> 0
[3,1,2] => [1,2] => [.,[.,.]]
=> 0
[3,2,1] => [2,1] => [[.,.],.]
=> 0
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> 0
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> 0
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 0
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> 1
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> 1
[2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
=> 0
[2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
=> 0
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 1
[2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
=> 0
[3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
=> 1
[3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
=> 1
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> 0
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> 0
[3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
=> 1
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> 0
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 0
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 1
[4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> 0
[4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> 1
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> 0
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0
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: St000252
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000252: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000252: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> 0
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> 0
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> 0
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0
[4,1,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0
[4,2,3,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 0
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 0
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 0
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 0
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 0
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 0
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
Description
The number of nodes of degree 3 of a binary tree.
Equivalently, the number of internal triangles in the associated triangulation of an $(n+2)$-gon.
The following 181 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000291The number of descents of a binary word. 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. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001928The number of non-overlapping descents in a permutation. St000201The number of leaf nodes in a binary tree. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000245The number of ascents of a permutation. St000292The number of ascents of a binary word. St000354The number of recoils of a permutation. 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. St000647The number of big descents of a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001298The number of repeated entries in the Lehmer code of a permutation. St001307The number of induced stars on four vertices in a graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001665The number of pure excedances of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001777The number of weak descents in an integer composition. St001964The interval resolution global dimension of a poset. 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. St000808The number of up steps of the associated bargraph. St001280The number of parts of an integer partition that are at least two. St001487The number of inner corners of a skew partition. St000312The number of leaves in a graph. St000636The hull number of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001883The mutual visibility number of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St000628The balance of a binary word. St000711The number of big exceedences of a permutation. St001638The book thickness of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St001323The independence gap of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000455The second largest eigenvalue of a graph if it is integral. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. 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 $K[x]/(x^n)$. 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. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. 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. St000934The 2-degree of an integer partition. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St001570The minimal number of edges to add to make a graph Hamiltonian. 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. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. 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. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001845The number of join irreducibles minus the rank of a lattice. 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. St000259The diameter 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. 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. St001330The hat guessing number of a graph. St001866The nesting alignments of a signed permutation. 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. 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". 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. St001490The number of connected components of a skew partition. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001694The number of maximal dissociation sets in a graph. 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 $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001857The number of edges in the reduced word graph of a signed permutation. 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. St000093The cardinality of a maximal independent set of vertices of a graph. St000679The pruning number of an ordered tree. 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. 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. St000322The skewness 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. 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. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001765The number of connected components of the friends and strangers graph. St001271The competition number of a graph.
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