Your data matches 448 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000648
(load all 11 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 0
[2,1,3] => [1,3,2] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,3,4,2] => 0
[1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,2,3] => 1
[2,1,3,4] => [1,3,4,2] => 0
[2,1,4,3] => [1,4,2,3] => 1
[2,3,1,4] => [1,4,2,3] => 1
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => 0
[2,4,3,1] => [1,2,4,3] => 0
[3,1,2,4] => [1,2,4,3] => 0
[3,1,4,2] => [1,4,2,3] => 1
[3,2,1,4] => [1,4,2,3] => 1
[3,2,4,1] => [1,2,4,3] => 0
[3,4,1,2] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => 0
[4,2,1,3] => [1,3,2,4] => 0
[4,2,3,1] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,5,2,3,4] => [1,5,2,3,4] => 0
[1,5,3,4,2] => [1,5,2,3,4] => 0
[1,5,4,2,3] => [1,5,2,3,4] => 0
[1,5,4,3,2] => [1,5,2,3,4] => 0
[2,1,5,3,4] => [1,5,2,3,4] => 0
[2,1,5,4,3] => [1,5,2,3,4] => 0
[2,3,1,5,4] => [1,5,2,3,4] => 0
[2,3,4,1,5] => [1,5,2,3,4] => 0
[2,3,4,5,1] => [1,2,3,4,5] => 0
[3,1,5,4,2] => [1,5,2,3,4] => 0
[3,2,1,5,4] => [1,5,2,3,4] => 0
[3,4,1,5,2] => [1,5,2,3,4] => 0
[3,4,2,1,5] => [1,5,2,3,4] => 0
[3,4,5,1,2] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [1,2,3,4,5] => 0
[4,1,5,2,3] => [1,5,2,3,4] => 0
Description
The number of 2-excedences of a permutation. This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+2$.
Matching statistic: St000237
(load all 7 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,2,3] => 0
[2,1,3] => [1,3,2] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3] => [1,4,2,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,4,2,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,3,4,2] => [1,2,3,4] => 0
[2,1,4,3] => [1,4,2,3] => [1,2,4,3] => 1
[2,3,1,4] => [1,4,2,3] => [1,2,4,3] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,2,3,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,3,4] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,3,4] => 0
[3,1,4,2] => [1,4,2,3] => [1,2,4,3] => 1
[3,2,1,4] => [1,4,2,3] => [1,2,4,3] => 1
[3,2,4,1] => [1,2,4,3] => [1,2,3,4] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,2,3,4] => 0
[4,2,1,3] => [1,3,2,4] => [1,2,3,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[1,5,4,2,3] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[1,5,4,3,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,1,5,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,1,5,4,3] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,3,1,5,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,3,4,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,1,5,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,2,1,5,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,4,1,5,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,4,2,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1,5,2,3] => [1,5,2,3,4] => [1,2,5,4,3] => 0
Description
The number of small exceedances. This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St000317
(load all 9 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [2,3,1] => 0
[1,3,2] => [1,3,2] => [2,1,3] => 0
[2,1,3] => [1,3,2] => [2,1,3] => 0
[2,3,1] => [1,2,3] => [2,3,1] => 0
[3,1,2] => [1,2,3] => [2,3,1] => 0
[3,2,1] => [1,2,3] => [2,3,1] => 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 0
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 0
[1,3,4,2] => [1,3,4,2] => [2,1,3,4] => 0
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => 1
[1,4,3,2] => [1,4,2,3] => [2,4,1,3] => 1
[2,1,3,4] => [1,3,4,2] => [2,1,3,4] => 0
[2,1,4,3] => [1,4,2,3] => [2,4,1,3] => 1
[2,3,1,4] => [1,4,2,3] => [2,4,1,3] => 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 0
[2,4,1,3] => [1,3,2,4] => [2,4,3,1] => 0
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => 0
[3,1,2,4] => [1,2,4,3] => [2,3,1,4] => 0
[3,1,4,2] => [1,4,2,3] => [2,4,1,3] => 1
[3,2,1,4] => [1,4,2,3] => [2,4,1,3] => 1
[3,2,4,1] => [1,2,4,3] => [2,3,1,4] => 0
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => 0
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => 0
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => 0
[4,1,3,2] => [1,3,2,4] => [2,4,3,1] => 0
[4,2,1,3] => [1,3,2,4] => [2,4,3,1] => 0
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => 0
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => 0
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,5,2,3,4] => [1,5,2,3,4] => [2,4,5,1,3] => 0
[1,5,3,4,2] => [1,5,2,3,4] => [2,4,5,1,3] => 0
[1,5,4,2,3] => [1,5,2,3,4] => [2,4,5,1,3] => 0
[1,5,4,3,2] => [1,5,2,3,4] => [2,4,5,1,3] => 0
[2,1,5,3,4] => [1,5,2,3,4] => [2,4,5,1,3] => 0
[2,1,5,4,3] => [1,5,2,3,4] => [2,4,5,1,3] => 0
[2,3,1,5,4] => [1,5,2,3,4] => [2,4,5,1,3] => 0
[2,3,4,1,5] => [1,5,2,3,4] => [2,4,5,1,3] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[3,1,5,4,2] => [1,5,2,3,4] => [2,4,5,1,3] => 0
[3,2,1,5,4] => [1,5,2,3,4] => [2,4,5,1,3] => 0
[3,4,1,5,2] => [1,5,2,3,4] => [2,4,5,1,3] => 0
[3,4,2,1,5] => [1,5,2,3,4] => [2,4,5,1,3] => 0
[3,4,5,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[3,4,5,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[4,1,5,2,3] => [1,5,2,3,4] => [2,4,5,1,3] => 0
Description
The cycle descent number of a permutation. Let $(i_1,\ldots,i_k)$ be a cycle of a permutation $\pi$ such that $i_1$ is its smallest element. A **cycle descent** of $(i_1,\ldots,i_k)$ is an $i_a$ for $1 \leq a < k$ such that $i_a > i_{a+1}$. The **cycle descent set** of $\pi$ is then the set of descents in all the cycles of $\pi$, and the **cycle descent number** is its cardinality.
Matching statistic: St000534
(load all 8 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000534: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,2,3] => 0
[2,1,3] => [1,3,2] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3] => [1,4,2,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,4,2,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,3,4,2] => [1,2,3,4] => 0
[2,1,4,3] => [1,4,2,3] => [1,2,4,3] => 1
[2,3,1,4] => [1,4,2,3] => [1,2,4,3] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,2,3,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,3,4] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,3,4] => 0
[3,1,4,2] => [1,4,2,3] => [1,2,4,3] => 1
[3,2,1,4] => [1,4,2,3] => [1,2,4,3] => 1
[3,2,4,1] => [1,2,4,3] => [1,2,3,4] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,2,3,4] => 0
[4,2,1,3] => [1,3,2,4] => [1,2,3,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[1,5,4,2,3] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[1,5,4,3,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,1,5,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,1,5,4,3] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,3,1,5,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,3,4,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,1,5,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,2,1,5,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,4,1,5,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,4,2,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1,5,2,3] => [1,5,2,3,4] => [1,2,5,4,3] => 0
Description
The number of 2-rises of a permutation. A 2-rise of a permutation $\pi$ is an index $i$ such that $\pi(i)+2 = \pi(i+1)$. For 1-rises, or successions, see [[St000441]].
Matching statistic: St000663
(load all 6 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000663: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => 0
[2,1,3] => [1,3,2] => [3,1,2] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 0
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 0
[1,3,4,2] => [1,3,4,2] => [3,4,1,2] => 0
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 1
[2,1,3,4] => [1,3,4,2] => [3,4,1,2] => 0
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 1
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => 0
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 0
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => 0
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 1
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => 0
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[1,5,4,2,3] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[1,5,4,3,2] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[2,1,5,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[2,1,5,4,3] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[2,3,1,5,4] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[2,3,4,1,5] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,1,5,4,2] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[3,2,1,5,4] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[3,4,1,5,2] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[3,4,2,1,5] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1,5,2,3] => [1,5,2,3,4] => [1,2,5,3,4] => 0
Description
The number of right floats of a permutation. Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right float is a large ascent not consecutive to any raft of $\pi$. See Definition 3.10 and Example 3.11 in [1].
Matching statistic: St000884
(load all 3 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,2,3] => 0
[2,1,3] => [1,3,2] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3] => [1,4,2,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,4,2,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,3,4,2] => [1,2,3,4] => 0
[2,1,4,3] => [1,4,2,3] => [1,2,4,3] => 1
[2,3,1,4] => [1,4,2,3] => [1,2,4,3] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,2,3,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,3,4] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,3,4] => 0
[3,1,4,2] => [1,4,2,3] => [1,2,4,3] => 1
[3,2,1,4] => [1,4,2,3] => [1,2,4,3] => 1
[3,2,4,1] => [1,2,4,3] => [1,2,3,4] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,2,3,4] => 0
[4,2,1,3] => [1,3,2,4] => [1,2,3,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[1,5,4,2,3] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[1,5,4,3,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,1,5,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,1,5,4,3] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,3,1,5,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,3,4,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,1,5,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,2,1,5,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,4,1,5,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,4,2,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1,5,2,3] => [1,5,2,3,4] => [1,2,5,4,3] => 0
Description
The number of isolated descents of a permutation. A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Matching statistic: St001115
(load all 2 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => 0
[2,1,3] => [1,3,2] => [3,1,2] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 0
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 0
[1,3,4,2] => [1,3,4,2] => [3,1,4,2] => 0
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 1
[2,1,3,4] => [1,3,4,2] => [3,1,4,2] => 0
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 1
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => 0
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 0
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => 0
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 1
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => 0
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[1,5,4,2,3] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[1,5,4,3,2] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[2,1,5,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[2,1,5,4,3] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[2,3,1,5,4] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[2,3,4,1,5] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,1,5,4,2] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[3,2,1,5,4] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[3,4,1,5,2] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[3,4,2,1,5] => [1,5,2,3,4] => [1,2,5,3,4] => 0
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1,5,2,3] => [1,5,2,3,4] => [1,2,5,3,4] => 0
Description
The number of even descents of a permutation.
Matching statistic: St001403
(load all 5 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001403: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,2,3] => 0
[2,1,3] => [1,3,2] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3] => [1,4,2,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,4,2,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,3,4,2] => [1,2,3,4] => 0
[2,1,4,3] => [1,4,2,3] => [1,2,4,3] => 1
[2,3,1,4] => [1,4,2,3] => [1,2,4,3] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,2,3,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,3,4] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,3,4] => 0
[3,1,4,2] => [1,4,2,3] => [1,2,4,3] => 1
[3,2,1,4] => [1,4,2,3] => [1,2,4,3] => 1
[3,2,4,1] => [1,2,4,3] => [1,2,3,4] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,2,3,4] => 0
[4,2,1,3] => [1,3,2,4] => [1,2,3,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[1,5,4,2,3] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[1,5,4,3,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,1,5,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,1,5,4,3] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,3,1,5,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,3,4,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,1,5,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,2,1,5,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,4,1,5,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,4,2,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1,5,2,3] => [1,5,2,3,4] => [1,2,5,4,3] => 0
Description
The number of vertical separators in a permutation. Let $\sigma$ be a permutation. Then $\sigma_i$ is a vertical separator if $|\sigma_{i-1} - \sigma_{i+1}| = 1$.
Matching statistic: St001465
(load all 5 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,2,3] => 0
[2,1,3] => [1,3,2] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3] => [1,4,2,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,4,2,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,3,4,2] => [1,2,3,4] => 0
[2,1,4,3] => [1,4,2,3] => [1,2,4,3] => 1
[2,3,1,4] => [1,4,2,3] => [1,2,4,3] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,2,3,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,3,4] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,3,4] => 0
[3,1,4,2] => [1,4,2,3] => [1,2,4,3] => 1
[3,2,1,4] => [1,4,2,3] => [1,2,4,3] => 1
[3,2,4,1] => [1,2,4,3] => [1,2,3,4] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,2,3,4] => 0
[4,2,1,3] => [1,3,2,4] => [1,2,3,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[1,5,4,2,3] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[1,5,4,3,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,1,5,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,1,5,4,3] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,3,1,5,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,3,4,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,1,5,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,2,1,5,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,4,1,5,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,4,2,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1,5,2,3] => [1,5,2,3,4] => [1,2,5,4,3] => 0
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.
Matching statistic: St001466
(load all 3 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001466: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,2,3] => 0
[2,1,3] => [1,3,2] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3] => [1,4,2,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,4,2,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,3,4,2] => [1,2,3,4] => 0
[2,1,4,3] => [1,4,2,3] => [1,2,4,3] => 1
[2,3,1,4] => [1,4,2,3] => [1,2,4,3] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,2,3,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,3,4] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,3,4] => 0
[3,1,4,2] => [1,4,2,3] => [1,2,4,3] => 1
[3,2,1,4] => [1,4,2,3] => [1,2,4,3] => 1
[3,2,4,1] => [1,2,4,3] => [1,2,3,4] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,2,3,4] => 0
[4,2,1,3] => [1,3,2,4] => [1,2,3,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[1,5,4,2,3] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[1,5,4,3,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,1,5,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,1,5,4,3] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,3,1,5,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,3,4,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,1,5,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,2,1,5,4] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,4,1,5,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,4,2,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => 0
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1,5,2,3] => [1,5,2,3,4] => [1,2,5,4,3] => 0
Description
The number of transpositions swapping cyclically adjacent numbers in a permutation. Put differently, this is the number of adjacent two-cycles in the chord diagram of a permutation.
The following 438 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St000669The number of permutations obtained by switching ascents or descents of size 2. St000142The number of even parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000210Minimum over maximum difference of elements in cycles. St000214The number of adjacencies of a permutation. St000223The number of nestings in the permutation. St000232The number of crossings of a set partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000355The number of occurrences of the pattern 21-3. St000359The number of occurrences of the pattern 23-1. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000441The number of successions of a permutation. St000546The number of global descents of a permutation. St000661The number of rises of length 3 of a Dyck path. St000665The number of rafts of a permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000731The number of double exceedences of a permutation. St000871The number of very big ascents of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001252Half the sum of the even parts of a partition. St001301The first Betti number of the order complex associated with the poset. St001377The major index minus the number of inversions of a permutation. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001394The genus of a permutation. St001402The number of separators in a permutation. St001479The number of bridges of a graph. St001513The number of nested exceedences of a permutation. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001657The number of twos in an integer partition. St001728The number of invisible descents of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001797The number of overfull subgraphs of a graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St001826The maximal number of leaves on a vertex of a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001864The number of excedances of a signed permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000160The multiplicity of the smallest part of a partition. St000183The side length of the Durfee square of an integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000883The number of longest increasing subsequences of a permutation. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001367The smallest number which does not occur as degree of a vertex in a graph. St001399The distinguishing number of a poset. St001484The number of singletons of an integer partition. St001571The Cartan determinant of the integer partition. St001652The length of a longest interval of consecutive numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000759The smallest missing part in an integer partition. St000650The number of 3-rises of a permutation. St000649The number of 3-excedences of a permutation. St000732The number of double deficiencies of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001141The number of occurrences of hills of size 3 in a Dyck path. St001552The number of inversions between excedances and fixed points of a permutation. St001162The minimum jump of a permutation. St000842The breadth of a permutation. St000248The number of anti-singletons of a set partition. St000379The number of Hamiltonian cycles in a graph. St000389The number of runs of ones of odd length in a binary word. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000502The number of successions of a set partitions. St000516The number of stretching pairs of a permutation. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000658The number of rises of length 2 of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000872The number of very big descents of a permutation. St000921The number of internal inversions of a binary word. St000934The 2-degree of an integer partition. St000989The number of final rises of a permutation. St001114The number of odd descents of a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. St001556The number of inversions of the third entry of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St000354The number of recoils of a permutation. St000487The length of the shortest cycle 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. St000633The size of the automorphism group of a poset. St000694The number of affine bounded permutations that project to a given permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000706The product of the factorials of the multiplicities of an integer partition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000988The orbit size of a permutation under Foata's bijection. St000990The first ascent of a permutation. St000993The multiplicity of the largest part of an integer partition. St001052The length of the exterior of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001964The interval resolution global dimension of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A 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. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001498The normalised height of a Nakayama algebra with magnitude 1. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000264The girth of a graph, which is not a tree. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000936The number of even values of the symmetric group character corresponding to the 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. 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. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. 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. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. 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 $K[x]/(x^n)$. St000181The number of connected components of the Hasse diagram for the poset. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001768The number of reduced words of a signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001890The maximum magnitude of the Möbius function of a poset. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000456The monochromatic index of a connected graph. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. 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. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. 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. St001613The binary logarithm of the size of the center of a lattice. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000907The number of maximal antichains of minimal length in a poset. St001720The minimal length of a chain of small intervals in a lattice. St001621The number of atoms of a lattice. St001875The number of simple modules with projective dimension at most 1. St001060The distinguishing index of a graph. St001396Number of triples of incomparable elements in a finite poset. St001570The minimal number of edges to add to make a graph Hamiltonian. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001857The number of edges in the reduced word graph of a signed permutation. St000657The smallest part of an integer composition. St000904The maximal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000084The number of subtrees. St000328The maximum number of child nodes in a tree. St000039The number of crossings of a permutation. St000077The number of boxed and circled entries. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000268The number of strongly connected orientations of a graph. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000295The length of the border of a binary word. St000323The minimal crossing number of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000351The determinant of the adjacency matrix of a graph. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000403The Szeged index minus the Wiener index of a graph. St000407The number of occurrences of the pattern 2143 in a permutation. St000462The major index minus the number of excedences of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000496The rcs statistic of a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000637The length of the longest cycle in a graph. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000943The number of spots the most unlucky car had to go further in a parking function. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000962The 3-shifted major index of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001305The number of induced cycles on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001736The total number of cycles in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001793The difference between the clique number and the chromatic number of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001816Eigenvalues of the top-to-random operator acting on a simple module. St001847The number of occurrences of the pattern 1432 in a permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000021The number of descents of a permutation. St000056The decomposition (or block) number of a permutation. St000154The sum of the descent bottoms of a permutation. St000253The crossing number of a set partition. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000272The treewidth of a graph. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000544The cop number of a graph. St000570The Edelman-Greene number of a permutation. St000640The rank of the largest boolean interval in a poset. St000654The first descent of a permutation. St000729The minimal arc length of a set partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000785The number of distinct colouring schemes of a graph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000948The chromatic discriminant of a graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001271The competition number of a graph. St001277The degeneracy of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001344The neighbouring number of a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001358The largest degree of a regular subgraph of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001363The Euler characteristic of a graph according to Knill. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001592The maximal number of simple paths between any two different vertices of a graph. St001665The number of pure excedances of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001743The discrepancy of a graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001792The arboricity of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001806The upper middle entry of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001928The number of non-overlapping descents in a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000058The order of a permutation. St000061The number of nodes on the left branch of a binary tree. St000105The number of blocks in the set partition. St000325The width of the tree associated to a permutation. St000469The distinguishing number of a graph. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000717The number of ordinal summands of a poset. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St001029The size of the core of a graph. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001111The weak 2-dynamic chromatic number of a graph. St001119The length of a shortest maximal path in a graph. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001316The domatic number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001716The 1-improper chromatic number of a graph. St001108The 2-dynamic chromatic number of a graph. St001926Sparre Andersen's position of the maximum of a signed permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000102The charge of a semistandard tableau. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000873The aix statistic of a permutation. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000879The number of long braid edges in the graph of braid moves of a permutation. St001490The number of connected components of a skew partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St001408The number of maximal entries in a semistandard tableau. St001410The minimal entry of a semistandard tableau. St001409The maximal entry of a semistandard tableau.